FREE TRADE NETWORKS WITH TRANSFERS*

The Japanese Economic ReviewVol 56 No 2 June 2005

ndash 144 ndashcopy Japanese Economic Association 2005

FREE TRADE NETWORKS WITH TRANSFERS

By TAIJI FURUSAWAdagger and HIDEO KONISHIDagger

daggerHitotsubashi University DaggerBoston College

We investigate the network of bilateral free trade agreements (FTAs) in the context of anetwork formation game with transfers In a previous paper we showed that withoutinternational transfers countries with different industrialization levels cannot sign an FTAso that the global free trade network in which every pair of countries signs an FTA is notin general pairwise-stable In this paper we show that even if the world consists of fairlyasymmetric countries the global free trade network is pairwise-stable when transfersbetween FTA signatories are allowed Moreover it is a unique pairwise-stable networkunless industrial commodities are highly substitutableJEL Classification Numbers C70 F15

1 Introduction

Preferential trade agreements (PTAs) such as customs unions and free trade agreements(FTAs) have become increasingly popular among World Trade Organization (WTO)member countries The WTO reports that over 170 PTAs are currently in force It is easyto imagine that the web of PTAs worldwide is fairly complex

The network formation game developed by Jackson and Wolinsky (1996) provides asuitable framework for the analysis of such a complex network of PTAs1 Goyal and Joshi(2001) and Furusawa and Konishi (2002) analyse FTA formation in the framework of thenetwork formation game The latter for example shows that if all countries are symmetric theglobal free trade network in which every pair of countries forms an FTA is pairwise-stable(Jackson and Wolinsky 1996) Moreover the global free trade network is the unique pairwise-stable network unless industrial commodities which are subject to import tariffs whentraded are highly substitutable Although they also analyse countriesrsquo incentives for FTAs inasymmetric countries most of the strong results are obtained apply to symmetric countries

Indeed the asymmetry of countries is a major obstacle to FTA formation Consider forexample an FTA between a highly industrialized small country and a less industrializedlarge country The FTA increases trade flows from the former to the latter disproportionatelydramatically increasing the trade surplus of the small highly industrialized country anddecreasing that of the large less industrialized country The direct impact of the FTA

This paper was in part prepared for Konishirsquos invited lecture at the Annual Meeting of the JapaneseEconomic Association at Meiji University Tokyo 12ndash13 October 2003 We both gratefully acknowledgefinancial support from the Japan Society for Promotion of Science Research Furusawa also gratefullyacknowledges financial supports from the Seimeikai Foundation and the Ministry of Education CultureSports Science and Technology of Japan (the 21st Century Centre of Excellence Project on the NormativeEvaluation and Social Choice of Contemporary Economic Systems) We are grateful to Richard Baldwinand other participants of Hitotsubashi Conference on International Trade and FDI for helpful commentsWe also greatly benefited from discussions with Tomoya Mori who is working with us on a dynamic exten-sion of this model We acknowledge helpful comments from an anonymous referee and thank SabinaPogorelec and Yoichi Sugita for research assistance

1 The network formation game is especially suitable for an analysis of FTAs since it can deal with a hub-and-spoke system of FTA links The hub-and-spoke system is a prominent feature of FTAs since unlikecustoms unions they allow member countries to impose different tariff rates on non-member countries

T Furusawa H Konishi Free Trade Networks with Transfers


on the industrial good trade surplus is favourable for the former and unfavourable forthe latter Unless opening the market benefits domestic consumers sufficiently the lessindustrialized country will object to the FTA

This rather intuitive discussion is formally supported with the aid of the welfare decom-position proposed by Furusawa and Konishi (2004) who show that if each consumerrsquospreferences are characterized by a quasi-linear utility function social welfare can be decomposedinto consumersrsquo gross utility and the trade surplus of the non-numeraire good FollowingFurusawa and Konishi (2002) the present paper adopts the quasi-linear utility functionwith a continuum of non-numeraire good commodities which is developed by Ottaviano et al(2002) Now the impact of an FTA on industrial good trade surplus can be decomposedinto a direct surplus effect and a third-country effect The former is the effect on the tradesurplus of the partner country of the FTA in question and the latter is the effect on the tradesurplus of all other countries The direct surplus effect can be positive or negative dependingon the countryrsquos relative industrialization level compared with its partner country as discussedabove On the other hand the third-country effect is always positive since the countryrsquosexports to third countries are not affected by the FTA while its imports from them decreasebecause such commodities become relatively more expensive in the domestic market

Even if a pair of countries is so asymmetric that one of them has no incentive to signan FTA the FTA is likely to raise the joint welfare since the third-country effect ispositive and the direct surplus effects are cancelled out between the two countries (Eachcountryrsquos exports to the other are the other countryrsquos imports and vice versa) Thereforeif international transfers are possible they may sign an FTA and the country that benefitsfrom the FTA will transfer part of the benefit to the other country which would incur aloss otherwise International transfers are not uncommon in reality A well known exampleis the monetary transfers that take place among EU members The European Commission(2003) reports that in 2002 the ldquooperationalrdquo budgetary balance of Germany for examplewas minus$50678 million while that of Spain was $88708 million indicating that fairlylarge amounts of transfers are made among EU members International transfers may notnecessarily be made in the form of monetary transfers recent FTAs have often includedagreements on economic issues other than tariff reduction such as competition policylabour and environmental regulations In many cases one signatory will demand thatother signatories harmonize their standards with its own Such unilateral changes ofstandards can be considered effectively as an international transfer

We extend Furusawa and Konishirsquos (2002) model to allow international transfersbetween FTA signatories in order to examine FTA formation between asymmetric coun-tries We find that their results are significantly strengthened if we allow internationaltransfers We obtained the same results for the global free trade network mentioned aboveeven where the world consists of fairly asymmetric countries Mutual tariff elimination byany pair of countries benefits these countries as a whole unless industrial commodities arehighly substitutable so that they will sign an FTA if a transfer between them is possibleHatta and Fukushima (1979) and Fukushima and Kim (1989) show that proportionalreduction of all tariff rates of all countries improves world welfare2 Mutual tariff

2 Strictly speaking world welfare improves if all the countries in the world (a) reduce all their ad valoremtariff rates proportionally (Hatta and Fukushima 1979) or (b) reduce all their specific tariffs rates and sub-sidies proportionally (Fukushima and Kim 1989) However Kowalczyk (1989) demonstrates that worldwelfare may decline if all countries proportionally reduce all of their ad valorem tariff rates and subsidies

The Japanese Economic Review


reduction by an FTA is significantly different from their exercises in that countries eliminatetheir tariffs only for their FTA partners A discriminatory tariff profile caused by FTAsgenerally creates distortions in the domestic consumption of industrial commodities Ifindustrial commodities are not highly substitutable these distortions are relatively small

In this paper we propose two concepts that are suitable for network formation gameswith transfers The first is a simple extension of Jackson and Wolinskyrsquos (1996) pairwisestability Roughly speaking a pair of network and transfer systems is pairwise-stable inthis context if there is a bilateral transfer system under which no country has an incentiveto cut a link and no pair of unlinked countries have incentives to form a link with an appro-priate transfer between themselves The second concept is what we call the ldquocontractualpairwise stabilityrdquo whereby the effect of a link between a pair of countries spills over toother players In our FTA network formation game an FTA between a pair of countriesadversely affects all other countries by reducing their exports to these two countries3 Thena country that has been an FTA partner of one of the new FTA signatories may be againstthis FTA possibly threatening to terminate the existing link We consider the situation inwhich countries need to compensate their FTA partners for possible resulting losses whenthey cut or form an FTA link Roughly speaking a pair of network and transfer systemsis contractually pairwise-stable if it is pairwise-stable in this environment

Two recent independent studies have also analysed transfers in network formationgames although unlike ours their approaches are based on non-cooperative gamesBloch and Jackson (2004) set up a general non-cooperative network formation game inwhich each player announces the amount of transfer (which can be negative) to each ofother players A pair of players form a link if and only if their proposed transfers to eachother are compatible4 They first analyse the Nash equilibrium of this game and thenintroduce its refinement which they call the pairwise equilibrium The set of outcomes inthis equilibrium concept is a refinement of our cooperative game solution concept iepairwise stability with transfers Matsubayashi and Yamakawa (2003) also analyse a non-cooperative link-cost sharing game in the context of Jackson and Wolinskyrsquos (1996) con-nections model They investigate whether there is an efficient Nash equilibrium in this game

The rest of the paper is organized as follows The next section sets out a multi-countrymodel of the quasi-linear economy with a continuum of industrial commodities Section3 discusses incentives to sign a bilateral FTA and defines the pairwise stability in thepresence of transfers Section 4 derives main results about the global free trade networkand Section 5 shows that the same results obtain even with the contractual pairwisestability Section 6 concludes

2 The model

The basic model structure is the same as that of Furusawa and Konishi (2002) to whichwe refer for detailed derivation of the formulae presented below

3 Our FTA network formation game is what Goyal and Joshi (2003) call a ldquolocal spillover gamerdquo Theydefine this as the game in which the marginal returns for the two players from the link depend only onthe numbers of links that they each have

4 Bloch and Jackson (2004) also consider indirect transfer games which allow player i to make transfersto players j and k contingent on the formation of a link between j and k



21 Overview

Let N = 1 2 n be the set of n countries (n ge 2) each of which is populatedby a continuum of identical consumers who consume a numeraire good and a continuumof horizontally differentiated commodities that are indexed by ω isin [0 1] A differentiatedcommodity can be considered as a variety of an industrial good Each industrial commodityω is produced by one firm which is also indexed by the same ω We assume that there isno entry of firms to this industry Each firm is owned equally by all domestic consumerswho receive equal shares of all firmsrsquo profits The numeraire good is produced competi-tively on the other hand Each consumer is endowed with l units of labour which is usedfor the production of the industrial and numeraire goods Each unit of labour producesone unit of the numeraire good so that the wage rate equals 1 We also assume that indus-trial commodities are produced with a linear technology and we normalize the unit labourrequirement to be equal to 0 for each industrial commodity without loss of generalityAlternatively we can interpret the model such that each consumer is endowed with l unitsof the numeraire good which can be transformed by a linear technology into industrialcommodities

In country i isin N there are microi consumers and s i firms each producing a differentindustrial commodity Thus country i produces si different industrial commodities whichare consumed in every country in the world We assume that the markets are segmentedso that firms can perfectly price-discriminate among different countries We normalizethe size of total population so that as well as The ratio θ i equiv s imicroi

measures country irsquos industrialization level the higher the ratio the higher the industrial-ization level Country i imposes a specific tariff at a rate of on the imports of industrialcommodities produced in country k The GATT Most Favoured Nation obligation will notallow any country to differentiate the tariff on like products based on originating countriesexcept for products imported from partner countries of preferential trade agreementsTherefore we suppose that for country i the tariff rates are the same at t i for all commoditiesimported from countries that have no FTA with country i We assume that there is nocommodity tax so that = 0 and that the countries do not impose tariffs on the numerairegood which may be traded internationally to balance the trade Tariff revenue is redistributedequally to domestic consumers

We examine the network of bilateral FTAs in the context of a network formation gameswith transfers5 An FTA between countries i and j is described by a link which is anunordered pair of two countries An FTA graph is an undirected graph (N Γ ) consistingof the set of countries N and an FTA network Γ that is a collection of links We sayΓ comp = S sub N | S | = 2 is a complete network and the set of all networks is denoted byG = Γ | Γ sube Γ comp Let Ci(Γ) = k isin N | = 0 under Γ represent the set of countriesproducing commodities on which country i does not impose tariffs It follows from = 0that Ci(Γ) is the set of country irsquos FTA partners and country i itself ie Ci(Γ) = i cupk isin N (i k) isin Γ For simplicity we omit the argument and write just Ci as long asthe network Γ is fixed

5 An FTA that involves more than two countries can be considered as a collection of bilateral FTAsbetween member countries

sum ==kn k

1 1micro sum ==kn ks1 1

tki

tii

tki

tii



22 Consumer demands and equilibrium

The utility function of a representative consumer in every country is given by the followingquasi-linear utility function

(1)

where q [0 1] rarr R+ is an integrable consumption function and q0 denotes theconsumption level of the numeraire good6 The second-last term represents the substi-tutability among differentiated commodities which may become clearer if we notice that

Letting y and π [0 1] rarr R+ denote the consumerrsquos

income and the consumer price function respectively the budget constraint can be written as

(2)

Letting pi(ω) and P i denote the producer price for commodity ω sold in country i and theaverage consumer price in country i respectively a representative consumerrsquos demands incountry i for a commodity ω produced in country k can be written as

(3)

Firm ω in country k chooses in order to maximize its profits The first-order condition for this maximization gives us

(4)

for any i Notice that p i(ω) does not vary with ω Prices charged by firms depend only on theimporting countryrsquos tariff policies For simplicity we henceforth suppress the argument ω

It follows from (4) that country irsquos average consumer price P i is given by

(5)

where Substituting (5) into (4) yields the equilibrium producer price thateach firm in country k charges for the market of country i as a function of country irsquostariff vector

6 This utility function is a continuous-goods version of those of Shubik (1984) and Yi (1996 2000) whoanalyse the case where there are a finitely number of differentiated commodities Our continuum-of-commodity setup is based on Ottaviano et al (2002)

U q q q d q d q d q( ) ( ) ( ) ( ) 00

1

0

1

2

0

12

02 2

= minus minus

+α ω ω β ω ω δ ω ω

0

12

0

1

0

1

q d q q d d( ) ( ) ( )ω ω ω ω ω ω

= prime prime

y q d q ( ) ( ) = +0

1

0 π ω ω ω

q p ti iki i( ) ( ( ) )

( )( )ω α

β βω δ

β β δα= minus + minus

+minus

1P

( )piinω =1 π ω micro ω ω( ) ( ) ( )= sum =i

n i i ip q1

p tiki i( )

ω αβ

β δδ

β δ=

+minus +

+

1

2P

Pi i

=

++

++

αββ δ

β δβ δ2 2

t

t ikn k

kis t equiv sum =1

ti init t ( )= 1

p tki i

ki i( )

( )t =

+minus +

+αββ δ

δβ δ2

1

2 2 2t



Then it follows from (3) that a representative consumerrsquos demand in country i for a com-modity produced in country k is

(6)

Notice that holds for any tariff vector ti

23 Social welfare

Under the world tariff vector t = (t1 tn) each firm in country i earns profits

(7)

Country irsquos per capita tariff revenue is

(8)

A representative consumerrsquos income in country i is the sum of labour income redistributedtariff revenue and the profit shares of the firms in country i

(9)

Then it follows from (2) that

where if ω is produced in country kSubstituting this equilibrium demand into (1) and letting qi(ti) represent country irsquos equi-

librium consumption function under the tariff ti ie we obtain arepresentative consumerrsquos utility in country i as a function of the world tariff vectorCountry irsquos social welfare is the sum of the utilities across all consumers in country i7

Lemma 1 (Furusawa and Konishi 2002) Country irsquos social welfare can be written as follows

(10)

7 In the presence of transfers between countries it is more convenient to work with (aggregate) socialwelfare than with per capita social welfare as in Furusawa and Konishi (2002) The analysis itself is notaffected by this modification

q tki i

ki i( )

( )t =

+minus +

+α

β δ βδ

β β δ2

1

2 2 2t

p qki i

ki i( ) ( )t t= β

π micro micro βik

ik k

ik k

k

n

kik k

k

n

p q q( ) ( ) ( ) ( ) t = == =

sum sumt t t1

2

1

R t s qi iki k

ki i

k

n

( ) ( )t t==

sum1

y l Rsi i

ii

i ( )

( )= + +t

tπmicro

q l Rs

s p t q

l s t qs

p q

i i ii

i

ik

ki i

ki

ki

ki

k

n

kki

ki i

k

n i

ik

ik k

ik k

k

n

01

1 1

( ) ( ) ( )

[ ( ) ] ( )

( ) ( ) ( )

t tt

t t

t t t

= + + minus +

= + + minus

=

= =

sum

sum sum

πmicro

micromicro [ ( ) ] ( )

( ) ( ) ( ) ( )

s p t q

l s p qs

p q

kki i

ki

ki i

k

n

kki i

ki i

k i

i

ik

ik k

ik k

k i

t t

t t t t

+

= minus +

=

ne ne

sum

sum sum

1

micromicro

q qiki i( ) ( )ω = t

q qi iki i

k N( ) ( ( ))t t= isin

W U q q V X Mi i i i i i i i i i i( ) ( ( ) ( )) ( ) ( ) ( )t t t t t tequiv = + minusminusmicro 0



where

V i(ti) = microiU (qi(ti) l) (11)

(12)

(13)

with t minusi = (t1 timinus1 ti+1 t n) The functions V i(ti) M i(t i) and Xi(tminusi) representconsumersrsquo gross utility (industrial) import payments and (industrial) export profitsrespectively8 Country irsquos social welfare consists of consumersrsquo gross utility V i(ti) and theindustrial trade surplus X i(tminusi) minus M i(ti)

Notice from (11)ndash(13) that an increase in a tariff rate affects V i(ti) X i(tminusi) and M i(ti)only through the changes in the consumption of industrial commodities We see from (6)that the consumption of an industrial commodity depends on the tariff rate imposed onthat commodity and the average tariff rate ie Thus we can write forexample An increase in affects not only directlybut also indirectly for all k = 1 2 n These changes in consumption affect V i(ti) andM i(ti) in turn

Country irsquos optimal tariff profile maximizes V i(ti) minus M i(ti) since X i(tminusi) does not dependon ti We consider here the situation in which country i has signed FTAs rather than CUswith all other countries in Ci Therefore country i chooses its external tariffs without anycoordination with other countries in Ci As the following lemma shows a countryrsquosoptimal tariff rates depend only on its own characteristics owing to the separability of aconsumerrsquos utility function

Lemma 2 (Furusawa and Konishi 2002) Country irsquos optimal tariff rates are the same forall other countries This common optimal tariff rate is a function of si and

which is increasing in si and decreasing in

3 Free trade agreements

31 Incentives to sign an FTA without transfers

We examine incentives for country i to sign an FTA with country j If countries i and jsign an FTA they eliminate all tariffs imposed on commodities imported from each other

8 See also Furusawa and Konishi (2004) for this welfare decomposition Notice that the consumption ofthe numeraire good is fixed at l so that V i(t i) effectively measures the gross utility derived from theconsumption of industrial commodities alone

M s p q s qi i i kki i

ki i

k i

i kki i

k i

( ) ( ) ( ) ( ) t t t t= =ne ne

sum summicro βmicro 2

X s p q s qi i i kik k

ik k

k i

i kik k

k i

( ) ( ) ( ) ( ) tminus

ne ne= =sum summicro β microt t t 2

q tki i

ki

ki i( ) ( )t equiv Q t

V t ti i i i i ini

ni i( ) ( ( ) ( ))t equiv V Q Q1 1 t t t j

i q ji

qki

s sCk C

kii

( )equiv sum isin

t s ss

s si C

i

C ii

i( )

( )

( ) ( )[ ( ) ( )] =

++ minus minus + minus minus

gt4

3 2 1 4 2 1 20

2

α β β δβ δ δ β δ δ

sCi



while keeping all other tariffs at their original levels Letting t and tprime denote the world tariffvectors before and after the FTA respectively tprime differs from t only in that Wheninternational transfers are not possible country i has an incentive to sign an FTA withcountry j if and only if W i(tprime) ge W i(t) which can be written as

∆V i(ti) + [∆ X i(tminusi) minus ∆M i(ti)] ge 0 (14)

where ∆ represents a change in the respective function values caused by an FTA betweencountries i and j such that ∆V i(ti) equiv V i(tiprime) minus V i(ti) for example As we will see shortly tariffreduction is likely to increase consumersrsquo gross utility unless the industrial commoditiesare highly substitutable On the other hand since the FTA increases the countryrsquos exportprofits and is also likely to increase the import payments the FTA has an ambiguousimpact on country irsquos industrial trade surplus

Let us first investigate the sign of ∆V i(ti) The next lemma shows that an FTA increasesconsumersrsquo gross utility either if the substitutability among the industrial commodities islow or if the original tariff rate is small

Lemma 3 (Furusawa and Konishi 2002) A bilateral FTA with country j increases consumersrsquogross utility in country i ie ∆V i(t i) gt 0 if either of the following conditions is satisfied

In particular when country i levies a tariff rate t i that is not greater than the optimal tariffrate t(si ) it is sufficient that δ le 10β for ∆V i(t i) to be positive

These conditions reflect the second-best effect that partial removal of tax distortionsmay even reduce efficiency (Dixit 1975 Hatta 1977) In this case the removal of countryirsquos tariffs against country j enhances country irsquos consumersrsquo gross utility if tariff distortionsare not so widespread that holds (condition (i) in Lemma 3) Even whencondition (i) is violated the second-best effect is negligible if tax distortions are so smallthat condition (ii) is satisfied

We next turn to the effect of an FTA between countries i and j on the industrial tradesurplus Let and be country irsquos import payments to country k and country irsquos exportprofits from country k respectively

Then we can rewrite country irsquos industrial trade surplus as

An FTA between i and j involves changes only in t i and t j so it does not affect forany k ne i j Consequently a change in country irsquos industrial trade surplus can be written as

t tji

ijprime = prime = 0

( ) ( ) ( ) i s sC ji4 1 2 02β β δ δ+ minus minus minus ge

( ) ( ) ( )

ii ts s

iC ji

leminus minus minus +

8

1 2 4

2

2

α βδ β β δ

sCi

s sC ji ( ) + ge1 2 1 2

Mki X k

i

M s qki i i k

ki i( ) ( ) t t= βmicro 2

X s q Mki k k i

ik k

ik k( ) ( ) ( ( ))t t t= =βmicro 2

X M X Mi i i iki k

ki i

k i

( ) ( ) [ ( ) ( )]t t t tminus

neminus = minussum

X ki k( )t



The third-country effect represented by the last term is positive as long as country irsquostariffs against countries other than j remain the same since the reduction of makescommodities imported from country j relatively less expensive so that country irsquos importsfrom third countries decrease ie The third-country effect plays an importantrole in the later analysis of FTAs with transfers It works positively for the FTA betweencountries i and j at the sacrifice of all other countries

Having shown that the third-country effect is positive let us now investigate thedirect surplus effect which can be rewritten as follows from the definitions of and

where θ i = simicroi as defined above The higher are θ i and the lower θ j the larger will be anincrease in country irsquos industrial trade surplus That is other things being equal the rela-tively more industrialized country will be more enthusiastic than the less industrializedcountry about signing a bilateral FTA The more industrialized country derives a largerbenefit from the opening of the partnerrsquos relatively large market In addition opening itsown market to the partnerrsquos firms does not significantly increase import payments sincethe resulting penetration by the partnerrsquos firms is relatively small Another importantfactor affecting incentives to sign an FTA is the difference in the original tariff rates It iseasy to see that if t i lt t j for example then Country irsquos exports tocountry j increase more than its imports from country j and hence the FTA between i andj tends to be more beneficial to country i

The larger the difference in the two countriesrsquo characteristics the greater is the directsurplus effect Therefore it is likely in such cases that the direct surplus effect for one ofthe countries takes a large negative value outweighing the gains from the third-countryeffect and the possibly positive gross consumer utility effect Then these countries willnot sign an FTA (see discussions in Furusawa and Konishi 2002) However if transfersbetween them are allowed the countries may overcome this problem In the next subsectionwe define our solution concept that is used to derive stable FTA networks in the presenceof international transfers

32 Stable free trade networks with transfers

If all tariffs are uniquely determined exogenously or endogenously for each free tradenetwork Γ isin G (eg when all countries set their individual external tariffs at optimal levelsgiven the prevailing network Γ) then country irsquos social welfare without transfers isuniquely assigned to each Γ We write this social welfare function as ui(Γ ) We postponedetermination of the function ui until ldquotariff schemesrdquo are discussed in the next sectionThe set of countries N and their payoff functions define a network formation game

The following concept was first studied by Jackson and Wolinsky (1996) A pairwise-stable network is a network Γ such that (i) for any i isin N and for any (i j) isin Γ u i(Γ)

∆ ∆ ∆[ ( ) ( )] [ ( ) ( )] ( )

X M X M Mi i i iji j

ji i

ki i

k i j

t t t t tminus

neminus = minus minus sum

direct surplus effectthird country effect

1 2444 34441 24 34

t ji

∆Mki i( ) t lt 0

Mij j( )t

M ji i( )t

∆ ∆[ ( ) ( )] [ ( ) ( ) ]X M q qji j

ji i i j i

ij j j

ji it t t tminus = minus

direct surplus effect1 2444 3444

micro micro β θ θ2 2

∆ ∆q qji i

ij j( ) ( )t tlt



9 Note that there is no order in element (i j ) in Γ while transfer Tij has an order such that it is a transferfrom i to j Thus for each element (i j ) isin Γ both Tij and Tji are included in the transfer system

ge u i(Γ(i j)) ie no country has an incentive to cut a link with another and (ii) for any(i j) notin Γ with i ne j if ui(Γ) lt u i(Γ cup (i j )) then u j(Γ) gt u j(Γ cup (i j )) ie for anypair of unlinked countries at least one of them has no incentive to form a link withthe other

In this paper we examine FTA networks where international transfers between FTAsignatories are allowed This modification necessarily changes the definition of pairwisestability Before we define pairwise stability in this context let us first define transfersTwo countries that sign an FTA can make a transfer between them in order to compensatefor any possible damage caused by the FTA We require that transfers be made only withinpairs of linked countries so that transfers need to satisfy pairwise budget balancingcondition

Definition 1 A transfer from i to j is the amount of the numeraire good Tij isin R givenfrom i to j such that Tij = minusTji A transfer system of the network Γ is T = (Tij)(i j )isinΓ such thatTij = minusTji for any (i j) isin Γ A network with transfers (Γ T ) is a pair of a network and anassociated transfer system9

Country irsquos payoff under (Γ T ) is given by v i(Γ T ) = ui(Γ ) + sumjisinNTji For a given (Γ T )with (i j) isin Γ we let (Γ (i j) Tminus(ij ji )) be a network of cutting (i j) by cancelling transfers

and between i and j without affecting any other links and transfers For a given(Γ T ) with (i j) notin Γ and i ne j we let with be anetwork of adding (i j) with transfers and between i and j without affecting anyother links and transfers

Definition 2 A network with transfers (Γ T ) is pairwise-stable if the following twoconditions are satisfied simultaneously(i) For any i isin N and for any (i j) isin Γ ie no

country has an incentive to cut a link and thereby eliminate the transfer from (or to)that partner

(ii) For any (i j) notin Γ with i ne j and for any ( ) with = minus if v i(Γ T) ltv i(Γ cup (i j) ( ) cup T) then v j(Γ T) gt v j(Γ cup (i j) ( ) cup T) ie forany pair of unlinked countries at least one of them has no incentive to form the linkwith any feasible transfer

In concluding this section we provide a useful characterization of pairwise-stable net-works with transfers The proof is relegated to the Appendix

Lemma 4 A network with transfers (Γ T ) is pairwise-stable if and only if the follow-ing two conditions are satisfied simultaneously(i) For any (i j) isin Γ

(ii) For any (i j) notin Γ with i ne j u i(Γ) + u j(Γ) ge u i(Γ cup (i j )) + u j(Γ cup (i j))

Remark For condition (i) in Lemma 4 to hold it is necessary that u i(Γ) + u j(Γ) geu i(Γ(i j)) + u j(Γ(i j)) for any (i j) isin Γ Together with condition (ii) this observation

primeTij primeTji

( ( ) ( ) )Γ cup prime prime cupi j T T Tij ji prime = minus primeT Tij ji primeTij primeTji

v T v i j Ti iij ji( ) ( ( ) )( )Γ Γge minus

primeTij primeTji primeTij primeTji


u u i j T u i j ui iij

j j( ) ( ( )) ( ( )) ( )Γ Γ Γ Γminus ge ge minus



implies that if transfers are allowed then whether or not a pair of countries forms an FTAhinges on its impact on the joint social welfare of the two countries10

4 Pairwise-stable free trade networks

As we have shown in the last section the third-country effect is positive as long as tariffreforms are restricted to the mutual elimination of tariffs against each other Moreoverwe know that consumersrsquo gross utility increases if either one of the conditions in Lemma3 is satisfied Therefore what is left to be determined is the direct surplus effect

Obviously country irsquos imports from country j are country jrsquos exports to country i andhence we have Consequently

which implies that the sum of these two direct surplus effects is always zero If transfersare allowed it is sufficient for countries i and j to sign an FTA that ∆V i and ∆V j arepositive (provided that all other tariffs than and remain the same)

To state our results formally we define a tariff system that is consistent with FTA formationA tariff scheme is a list of functions N rarr R such that forany (i j) notin Γ and for any (i j) isin Γ Notice that shows country irsquos tariff profileunder the FTA network Γ and that a tariff scheme specifies every countryrsquos tariff profileunder every possible FTA network

With a fixed tariff scheme we can formally define country irsquos (aggregate)social welfare under Γ ui(Γ) for each i isin N and Γ isin G

(15)

To see the incentives for unlinked countries to form a new FTA we define for (i j) notin Γ ∆V i equiv

and ∆u i equiv ui(Γ cup (i j)) minus u i(Γ) and similarly for country j Then noticing that country irsquosexports to country j are country jrsquos imports from country i and vice versa we have thefollowing lemma (The proof is relegated to the Appendix)

Lemma 5 For (i j ) notin Γ we have

(i)

(ii)

10 It may appear that our problem can be reformulated as a coalition formation game with partition functionsHowever coalition formation games will not allow a country to be involved in multiple coalitions whichis a serious shortcoming in the analysis of FTA formation

M Xji i

ij i( ) ( )t t=

∆ ∆[ ( ) ( )] [ ( ) ( )]X M X Mji j

ji i

i

ij i

ij j

j

t t t tminus = minus minusrsquos direct surplus effect rsquos direct surplus effect

1 2444 3444 1 2444 3444

t ji ti

j

( ) τΓ Γi

i Nisin isinG τΓi τΓ Γi

jij t( ) ( )=

τΓi j( ) = 0 τΓ

i

( ) τΓ Γi

i Nisin isinG

u V X Mi i i i kk N i

i i( ) ( ) (( ) ) ( ) Γ Γ Γ Γ= + minusisinτ τ τ

V V X X X M M Mii j

i i i i ii j

kk N i

i kk N i

i ii j

i i i( ) ( ) (( ) ) (( ) ) ( ) ( )( ) ( ) ( )τ τ τ τ τ τΓ Γ Γ Γ Γ Γ∆ ∆cup cup isin isin cupminus equiv minus equiv minus

u u V X M

V X M T

i j i iki k

k i jki i

k i j

j jkj k

k i jkj j

k i j

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

Γ Γ Γ Γ Γ

Γ Γ Γ

+ = + minus

+ + minus

ne ne

ne ne

sum sum

sum sum

τ τ τ

τ τ

∆ ∆ ∆ ∆ ∆ ∆u u V V M Mi j i jk i j k

ik i j k

j + = + minus sum minus sumne ne



This lemma formally confirms the observation that whether a pair of countries benefitsfrom forming an FTA depends only on the impacts on consumersrsquo gross utilities andthose on the imports from third countries

Proposition 1 For any arbitrary tariff scheme there is a transfer system Tassociated with Γ comp such that (Γ comp T ) is a pairwise-stable network with transfers

Proof Pick any pair i j isin N and let ∆u i = u i(Γcomp) minus u i(Γcomp (i j)) and similarly forj We appeal to Lemma 5 to show that ∆ui + ∆u j ge 0 and hence neither country has anincentive to cut the link First we show that the third-country effect is non-negative ie

If n = 2 then there is no third-country effect So let us nowsuppose that n ge 3 For any we know that all tariffs but and are zerowhen Γ = Γcomp(i j) Thus an FTA between i and j in Γcomp(i j) will not change the thirdcountriesrsquo tariff rates which implies that Next we showthat ∆V i + ∆V j ge 0 Since Ci includes country i itself we have which immediately implies that Then condition (i) of Lemma 3is satisfied so that ∆V i gt 0 By the same argument we also have ∆V j gt 0 Hence weconclude that ∆ui + ∆u j gt 0 ie ui(Γcomp) minus u i(Γcomp(i j)) ge u j(Γcomp(i j)) minus u j(Γ comp)Then there exists a transfer Tij that satisfies condition (i) of Lemma 4

Condition (ii) of Lemma 4 is vacuously satisfied since there is no unlinked pair underΓ comp

Now we seek a condition under which every pair of countries has its incentives forforming an FTA regardless of the existing FTA network It is obvious that in such a casethe complete network (global free trade) becomes a unique stable network

Here we consider a specific tariff scheme which we call the constant tariff schemeThe constant tariff scheme is a tariff system by which any country specifiesthe same external tariff rate for any FTA network ie for any i isin N there is a constant t i

such that for any j isin N with (i j) notin Γ We now have the following proposition

Proposition 2 Suppose that δ le 10β ie the industrial commodities are not highlysubstitutable Suppose also that the tariff system is the constant tariff scheme such that each countryrsquos external tariff rate is no greater than its optimal tariff when thereis no link ie for any Γ isin G (Recall that t is decreasing in the secondargument) Then under any FTA network with transfers (Γ T ) any unlinked pair of coun-tries i and j has the incentive to sign an FTA with a transfer between them As a resultglobal free trade (the complete network Γ comp) is a unique pairwise-stable network withtransfers

Proof For an arbitrary Γ isin GΓcomp pick any (i j) notin Γ and let ∆ui = u i(Γ cup (i j)) minusu i(Γ) and similarly for j It follows from Lemma 3 that both ∆V i and ∆V j are positive Wealso know that under the constant tariff scheme the third-country effects are non-negativeie Then we find from Lemma 5 that ∆u i + ∆u j gt 0or equivalently u i(Γ cup (i j)) + u j(Γ cup (i j)) gt ui(Γ ) + u j(Γ ) It follows from Lemma 4that (Γ T) is not pairwise-stable for any T Indeed there exists a transfer such that

so that countries i and j have an incentiveto sign an FTA with such a transfer

( ) τΓ Γi

i Nisin isinG

sum + sum lene nek i j ki

k i j kjM M ∆ ∆ 0

( ) τΓ Γi

i Nisin isinG t ji ti

j

sum le sum lene nek i j ki

k i j kjM M ∆ ∆0 0and s sC i j ji

comp( ( )) Γ + = 12 1s sC i j ji

comp( ( )) Γ + ge

( ) τ Γ Γci

i Nisin isinG

τΓci ij t( ) =

( ) τ Γ Γci

i Nisin isinG

τΓci i ik t s s( ) ( )le

primeTij


k i j kjM M ∆ ∆0 0and

primeTij

u i j u T u u i ji iij

j j( ( )) ( ) ( ) ( ( ))Γ Γ Γ Γcup minus ge prime ge minus cupprimeTij



The constant tariff scheme is a realistic tariff system although each countryrsquos tariffsetting is necessarily suboptimal What if countries optimally adjust their external tariffswhen they sign a new FTA Do we still have a result similar to Proposition 2

To answer to this question we consider a tariff system in which when countries i andj sign an FTA they select their individual external tariff rates t i and t j so as to maximizetheir joint social welfare whereas all other countries keep their status quo tariffs11 In thepresence of transfers it is reasonable for countries that sign an new FTA to select theirindividual tariffs so as to maximize their joint welfare since the transfer betweenthem determines each countryrsquos share of the maximized ldquopierdquo Now it follows fromLemma 5 that

where ^ i is country irsquos tariff profile such that if j notin Ci and = 0 if j isin Ci and similarlyfor j It can be readily verified that ˇ i = t(si + s j ) and ˇ j = t(s i + s j )

If countries select their tariff rates in this manner any pair of unlinked countries has anincentive to sign an FTA given that δ le 10β Proposition 2 implies that under any FTAnetwork unlinked countries i and j have an incentive to sign an FTA if they do not adjusttheir tariffs Now under this tariff system countries have more incentives to sign an FTAsince they adjust their tariffs so as to maximize the joint social welfare Therefore thesame result as Proposition 2 obtains even under this version of the optimal tariff system

The significance of Proposition 2 and the following argument is that they apply regardlessof countriesrsquo characteristics and existing FTA networks As long as industrial commoditiesare not highly substitutable any pair of unlinked countries always has an incentive to signan FTA if they can make a transfer between themselves No matter how different thesecountriesrsquo industrialization levels are they have the incentive to sign an FTA In this senseFTAs are likely to be ldquobuilding blocksrdquo rather than ldquostumbling blocksrdquo towards global freetrade as long as transfers are allowed12

We also find that FTA formation and multilateral trade negotiations such as GATTWTO trade negotiation rounds are complementary The proof of Proposition 2 suggeststhat any pair of unlinked countries signs an FTA if their consumersrsquo gross utilities increaseas a consequence However Lemma 3 indicates that this is indeed the case if their currenttariffs are low Thus multilateral trade negotiation may lower every involved countryrsquostariff rates sufficiently for the continuous process of bilateral FTA formation to be kick-started Moreover Lemma 3 also indicates that bilateral FTA formation accelerates sincecondition (i) is more likely to be satisfied as more FTAs are formed

11 Country irsquos external tariff rate depends not only on Γ but also on the ldquopathrdquo to reach Γ since it criticallydepends on the identity of the last country with which country i signed an FTA Therefore this tariffsystem is not a tariff scheme that we have formally defined

12 Although global free trade is not Pareto optimal in our oligopolistic model it can be shown that a pro-portional reduction of all the tariff rates improves world welfare as Fukushima and Kim (1989) show ina perfect competition model so the movement towards global free trade is preferable as a whole Sincecountry irsquos import payments to country j is nothing but country jrsquos export profits derived from country ithe world welfare equals sumiisinNV i(t i) It can be readily shown that for any i isin N V i(t i) increases as ti

proportionally decreases to 0

î i iki i

k i j

V M ( ) ( )

isin minus

ne

sumargmax t t

t ji i = ˇ t j

i

sCi sCi



5 Contractually stable free trade networks

We found in the last section that unless industrial commodities are highly substitutablethe global FTA network is the unique pairwise-stable FTA As Proposition 2 shows anypair of countries has an incentive to sign an FTA regardless of the current FTA networkin such a situation Then it appears that if all countries are myopic countries continue tosign bilateral trade agreements until global free trade is attained in an extended dynamicmodel In the presence of international transfers however this may not be the case unlessthe transfer system continues to be adjusted as new FTA links are formed in order tomaintain countriesrsquo incentives to keep existing FTAs

To see this argument let us consider the situation in which every pair of countries hasan incentive to sign an FTA as in Proposition 2 A new FTA link between i and j benefitsthese countries but adversely affects third countries and thereby changes their incentivesto form or cut FTA links Consider country k which has an FTA with country i inthe FTA network Γ so we have uk(Γ) minus u k(Γ (i k)) ge Tki Country k has an incentive tokeep FTA with country i even after the (i j ) link is formed if u k(Γ cup (i j)) minus uk((Γ cup (ij)) (i k)) ge Tki Even though countries i and k jointly benefit from the (i k) link this con-dition may not hold unless Tki is appropriately changed If country k loses its incentive tokeep the (i k) link as a result of the (i j) link country i may hesitate to form a link withcountry j Countries may not continue to form FTA links until global free trade is attainedin the presence of international transfers

This discussion suggests that if countries compensate their FTA partners when theyform new FTAs they are likely to keep existing FTAs If every pair of countries stillbenefits from a new FTA after full compensations for their partnersrsquo losses we can expectthe global FTA network to be finally attained in an extended dynamic game

To examine this possibility we introduce a new concept that captures the idea thatwhen a pair of countries signs an FTA both must obtain ldquopermissionrdquo from their respectiveFTA partners possibly compensating for their resulting loss In order to describe suchcompensations we first define a new network with transfers obtained by cutting or forminga link from a network with transfers

Definition 3 (i) A network with transfers (Γprime T prime) is agreeable at (ΓΓΓΓ T ) with irsquos cutting of the link (i j )

isinisinisinisin ΓΓΓΓ if T prime is the transfer system associated with Γprime = Γ (i j) such that (a) (thus) if k isin Ci(Γ)i j and for any other (l k) isin Γ and (b) v k(Γ T ) le

vk(Γ (i j) T prime ) for any k isin Ci(Γ) i j(ii) A network with transfers (Γprime T prime) is agreeable at (ΓΓΓΓ T ) with a new link (i j) notinnotinnotinnotin ΓΓΓΓ (for

i ne j) if T prime is the transfer system associated with Γprime = Γ cup (i j) such that (a) (thus ) if k isin Ci(Γ)i (thus ) if k isin Cj(Γ) j and for any other (l k) isin Γ and (b) vk(Γ T ) le vk(Γ cup (i j) T prime ) for any k cup (Ci(Γ) cupCj(Γ)) i j

This definition requires that when countries change the existing network with transferseither by unilaterally cutting a link or forming a link with another country they can neitherreduce their transfers to the original partners nor make their original partners worse off

Definition 4 A network with transfers (Γ T ) is contractually pairwise-stable if thefollowing two conditions are simultaneously satisfied

prime geT Tik ik prime leT Tki ki prime =T Tlk lk

prime geT Tik ik prime leT Tki ki prime geT Tjk jk prime leT Tkj kj prime =T Tlk lk



(i) For any i isin N any (i j) isin Γ and any network with transfers (Γprime T prime ) agreeable at(Γ T ) with irsquos cutting of the link (i j) we have v i(Γ T ) ge v i(Γprime T prime)

(ii) For any (i j) notin Γ with i ne j and for any network with transfers (Γprime T prime ) agreeable at(Γ T) with a new link (i j) if v i(Γ T) lt v i(Γprime T prime ) then v j(Γ T) gt v j(Γprime T prime)

In the present environment where the contractually pairwise stability is an appropriatesolution concept countries need to compensate their partners for possible losses causedby their decisions to form FTA links On the other hand even though their partners arebetter off by such decisions they cannot change transfers to extract those gains

Now we have a counterpart of Lemma 4 The proof of the following lemma is similarto that of Lemma 4 and is relegated to the Appendix

Lemma 6 A network with transfers (Γ T ) is contractually pairwise-stable if and onlyif the following two conditions are simultaneously satisfied (i) For any (i j) isin Γ

(ii) For any (i j) notin Γ with i ne j

An FTA link (i j) affects third countriesrsquo social welfare only through their exports tocountries i and j Since these exports (weakly) decline as a result of the (i j) link wehave (i) uk(Γ) minus u k(Γ (i j)) le 0 for any k isin N i j and for any Γ that includes (i j) and(ii) uk(Γ) minus u k(Γ cup (i j)) ge 0 for any k isin N i j and for any Γ that does not include(i j) Thus we immediately obtain the following lemma

Lemma 7 In the FTA network formation game with transfers (Γ T) is contractu-ally pairwise-stable if and only if the following two conditions are simultaneouslysatisfied (i) For any (i j) isin Γ (ii) For any (i j) notin Γ and i ne j

Since cutting an FTA link benefits all other partner countries a country need notcompensate them when it eliminates an existing link Therefore condition (i) of Lemma7 is the same as that of Lemma 4 However a pair of countries must compensate theirrespective partner countries when they form a new FTA link Their decision as to whetheror not to sign an FTA depends not only on the impact on their own welfare but also ontheir partnersrsquo welfare as condition (ii) of Lemma 7 indicates

Given that u k(Γ cup (i j)) minus u k(Γ) le 0 for any k isin(Ci(Γ) cup Cj(Γ)) i j condition(ii) of Lemma 7 is satisfied more easily than condition (ii) of Lemma 4 Since condition(i) is the same in both of these lemmas we know that if (Γ T ) is pairwise-stable it is

u u i j u u i j T

u i j u u i j

i i

k C i j

k kij

j j

k C i j

k

i

j

( ) ( ( )) max ( ) ( ( ))

( ( )) ( ) min ( (

( )

( )

Γ Γ Γ Γ

Γ Γ Γ

Γ

Γ

minus + minus ge

ge minus +

isin

isin

sum

sum

0

)))) ( ) minus uk Γ 0

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ

cup minus le minus cup= isin cupsum sum 0


j j( ) ( ( )) ( ( )) ( )Γ Γ Γ Γminus ge ge minusΣ Γ Γ

Γ Γk C C

k k

i j

u i j uisin cup

cup minus le( ) ( )

[ ( ( )) ( )] 0



also contractually pairwise-stable in the FTA network formation game with transfersConsequently we obtain the counterpart of Proposition 1

Proposition 3 For any arbitrary tariff scheme there is a transfer system Tassociated with Γ comp such that (Γ comp T ) is a contractually pairwise-stable network withtransfers

Somewhat surprisingly the counterpart of Proposition 2 also obtains since FTA signa-tories need only transfer part of gains from the third-country effect to compensate partnercountries

Proposition 4 Suppose that δ le 10β and that the tariff system is the constant tariff scheme such that each countryrsquos external tariff rate is not greater than its optimal tariff

when there is no link Then under any FTA network with transfers (Γ T ) for any unlinkedpair (i j) there is an FTA network with transfers that is agreeable at (Γ T ) with anew link (i j) and is preferred by both i and j to (Γ T ) Therefore global free trade (thecomplete network Γ comp) is a unique contractually pairwise-stable network with transfers

Proof We show that for any Γ isin G Γcomp condition (ii) of Lemma 7 is violated ifδ le 10β Let and similarly for other changes Since the FTAbetween i and j only changes tariff schedules of these two countries we have

where we have used Lemma 3 and the fact that the (partial) third-country effect is positive

6 Concluding remarks

We have analysed bilateral FTA networks of the world by a network formation game withtransfers When transfers are allowed between FTA signatories a pair of countries formsan FTA if and only if their joint welfare improves as a consequence so that countries aremore likely to sign an FTA Indeed we find that the complete FTA network which effec-tively attains the global free trade is pairwise-stable Surprisingly this result obtains evenif countries are far from symmetric sharply contrasting with Furusawa and Konishi(2002) who derive a similar result in the case of symmetric countries when transfers arenot possible Moreover if industrial commodities are not highly substitutable the globalfree trade network is the unique pairwise-stable network In that case every pair of coun-tries has an incentive to sign an FTA

Since the direct surplus effects are cancelled out between the FTA signatories and their(industrial) trade surplus with third countries improve as a result of the substitution effect

( ) τ Γ Γi

i Nisin isinG

( ) τ Γ Γci

i Nisin isinG

∆ Γ ΓV V Vi ii j

i i i ( ) ( )( )= minuscupτ τ

k C C

k k

i jki

kj

k i j k C C i jik

jk

i

i j

i j

u i j u

V V M M X X

V

isin cup

ne isin cup

sum

sum sum

cup minus

= + minus + + +

= +

( ) ( )

( ( ) ( ))

[ ( ( )) ( )]

( ) ( )

Γ Γ

Γ Γ

Γ Γ

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆∆ ∆ ∆Γ Γ

V M Mj

k C Cki

kj

i j

( )

( ) ( )

minus +

gtnotin cup

sum0



of industrial commodities a bilateral FTA enhances the joint welfare of the signatories ifconsumersrsquo gross utilities from industrial commodities jointly increase This situation islikely to arise if the FTA does not increase distortion in the consumption of industrialcommodities (the second-best effect) The distortion would be small if industrial com-modities were not highly substitutable The resulting distortion can also be negligible ifcountriesrsquo external tariffs are low thanks to the past multilateral trade negotiation underthe auspices of the GATTWTO As Freund (2000) demonstrates multilateral tradenegotiations and regional trade liberalizations based on the GATT Article xxiv are com-plementary in this sense even though they appear contradictory when viewed from theperspective of the GATT principle of non-discrimination

When a pair of countries signs an FTA all other countries including their FTA partnersare worse off Therefore a country may well be against its partner-countryrsquos new FTApossibly threatening to terminate the existing FTA between them When transfers arepossible countries may compensate partner-countries for such possible losses when theyform new FTAs We have proposed contractually pairwise stability as a suitable solutionto this problem We have shown somewhat surprisingly that the results about the pairwise-stable global free trade network obtain even if we use the contractually pairwise stabilityas the solution concept

In concluding we want to point out that we can derive qualitatively the same result inan extended model that includes more than one industrial good Of course includingadditional goods adds more dimensions of similarity between countries and hence enablesus to analyse more problems of interest For the brevity of the current analysis howeverwe leave this extension for future research

Appendix

A1 Proof of Lemma 4

First we show that the definition of pairwise-stable networks with transfers impliesthe conditions in this lemma We start with condition (i) Let (i j) isin Γ Then we have

and from Definition2 Rewriting the former inequality we have

Similarly the latter inequality with regard to country j is equivalent to

Then condition (i) follows from Next we show that condition (ii) holds for any pairwise-stable network with transfers

Suppose to the contrary that ui(Γ) + u j(Γ) lt u i(Γ cup (i j)) + u j(Γ cup (i j)) Let ε equiv u i(Γcup (i j)) + u j(Γ cup (i j)) minus ui(Γ) minus uj(Γ) gt 0 and let Then we have from that Now using

we have

v T v i j Ti iij ji( ) ( ( ) )( )Γ Γge minus v T v i j Tj j

ij ji( ) ( ( ) )( )Γ Γge minus

u T u i j T

T u i j u

i

k C iki

i

k C i jki

jii i

i i

( ) ( ( ))

( ( )) ( )

( ) ( )

Γ Γ

Γ Γ

Γ Γ+ ge +

ge minus

isin isinsum sum

T u i j uijj j ( ( )) ( )ge minusΓ Γ

T Tij ji = minus

prime = minus cupT u u i jjii i ( ) ( ( ))Γ Γ + ε 2

prime + prime =T Tij ji 0 prime = minus cup +T u u i jijj j ( ) ( ( )) Γ Γ ε 2

Tji = 0



(A1)

Similarly we have

(A2)

Equalities (A1) and (A2) imply that countries i and j both gain from forming a new linkwith the transfer system which contradicts the supposition that (Γ T) ispairwise-stable

Now we show that conditions (i) and (ii) of this lemma imply that (Γ T) is pairwise-stable We start by demonstrating that condition (i) implies condition (i) of Definition 2Let (i j) isin Γ Then country irsquos social welfare with transfers satisfies the following

where the last inequality follows from condition (i) of the lemmaLet us turn to condition (ii) Let (i j) notin Γ with i ne j and select an arbitrary pair of transfers

with such that v i(Γ T) lt vi(Γ cup (i j ) cup T) This inequalityis equivalent to Then it follows from

and that Consequently we obtain fromcondition (ii) that Then we imme-diately obtain which shows that condition (ii)is satisfied

A2 Proof of Lemma 5

Since country irsquos exports to country j are country jrsquos imports from country i and viceversa we have Statement (i) follows immediately Moreover thesecountriesrsquo exports to the third countries do not change as a result of the FTA between i and jie for any k isin N i j Consequently we obtain from (i) that

v i j T T T u i j T T

u T

v T

iij ji

iki

k N jji

iki

k N

i

( ( ) ( ) ) ( ( ))

( )

( )

Γ Γ

Γ

Γ

cup prime prime cup = cup + + prime

= + +

= +

isin

isin

sum

sum ε

ε2

2

v i j T T T v Tjij ji

j( ( ) ( ) ) ( ) Γ Γcup prime prime cup = + ε 2

( ) prime prime cupT T Tij ji

v T u T T

u u i j u i j T T

u u i j T

i iki

k N jji

i i iki

k N jji

i i

( ) ( )

( ) ( ( )) ( ( ))

( ) ( ( ))

Γ Γ

Γ Γ Γ

Γ Γ

= + +

= minus + + +

= minus +

isin

isin

sum

sum

jijii

ij ji

iij ji

v i j T

v i j T

( ( ) )

( ( ) )( )

( )

+ge

minus

minus

ΓΓ

( )prime primeT Tij ji prime = minus primeT Tij ji ( )prime primeT Tij ji

u T u i j T Tik N ki

ik N j ki ji( ) ( ( )) Γ Γ+ sum lt cup + sum + primeisin isin

Tji = 0 prime = minus primeT Tij ji prime lt cup minusT u i j uiji i ( ( )) ( )Γ Γ

u u i j u i j u Tj j i iij( ) ( ( )) ( ( )) ( ) Γ Γ Γ Γminus cup ge cup minus gt prime

v T v i j T T Tj jij ji( ) ( ( ) ( ) )Γ Γgt cup prime prime cup

X M X Mji

ij

ij

ji = = and

∆ ∆X Xki

kj = = 0

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆

∆ ∆ ∆

u u V V X M X M

X M X M

V V M

i j i jji

ji

ki

k i jki

ij

ij

kj

k i jkj

i jki

k i j

( ) ( )

( ) ( )

+ = + + minus + minus

+ minus + minus

= + minus

ne

ne

ne

sumsum

sum minusminusnesum

∆Mkj

k i j



A3 Proof of Lemma 6

First we show that Definition 4 (i) implies Lemma 6 (i) for country i We rewrite thecondition vk(Γ T ) le v k(Γ(i j) T prime) for k isin Ci(Γ)i j as

where we have used for l ne i Then it follows from that

(A3)

We fix at the value that satisfies (A3) with equality Then we can rewrite v i(Γ T ) gev i(Γ(i j) T prime ) as

which gives us the first part of Lemma 6 (i) The second part regarding country j obtainssimilarly

Next we show that Lemma 6 (i) implies Definition 4 (i) Tracing the calculation in theabove backward we obtain

It then follows from (A3) that

A similar result obtains for country jNow we demonstrate that Definition 4 (ii) implies Lemma 6 (ii) by showing the contra-

positive Suppose to the contrary that

For any k isin (Ci(Γ) cup Cj(Γ))i j we have from vk(Γ T) le vk(Γ cup (i j) T prime ) that

Then it follows from that

(A4)

u T u i j T

u T u i j T

T T u u i j

klk

l N

klk

l N

kik

kik

ik ikk k

( ) ( ( ))

( ) ( ( ))

( ) ( ( ))

Γ Γ

Γ ΓΓ Γ

+ le + prime

+ le + primeprime ge + minus

isin isinsum sum

prime =T Tlk lk prime geT Tik ik

prime ge + minusT T u u i jik ikk k max ( ) ( ( )) Γ Γ 0

primeTik

u T T u i j T u u i jiij

k C i jik

i

k C i jik

k k

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ

minus minus ge minus + minusisin isin

sum sum 0

u T u i j T u u i ji

k C iik

i

k C i jik

k i

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ

minus ge minus + minusisin isin

sum sum 0

u T u i j T

v T v i j T

i

k C iik

i

k C i jik

i i

i i

( ) ( ( ))

( ) ( ( ) )

( ) ( )

Γ Γ

Γ Γ

Γ Γminus ge minus prime

ge prime

isin isinsum sum

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ

cup minus gt minus cup= isin cupsum sum 0

prime + prime ge + + minus cupT T T T u u i jik jk ik jkk k ( ) ( ( ))Γ Γ

prime ge prime geT T T Tik ik jk jk and

prime + prime ge + + minus cupT T T T u u i jik jk ik jkk k max ( ) ( ( )) Γ Γ 0



Let be the value that satisfies (A4) with equality Then we have

which gives us

Let ε be the left-hand side of the above inequality and let be such that

Then we have vi(Γ cup (i j) T prime) = vi(Γ T) + ε2 Moreover we have from the definitionof ε that v j(Γ cup (i j) T prime) = v j(Γ T) + ε 2 which violates Definition 4 (ii)

Finally we show that Lemma 6 (ii) implies Definition 4 (ii) Again we show the contra-positive of the statement Suppose there exists a transfer system T prime such that v i(Γ cup (i j) T prime)gt v i(Γ T) and v j(Γ cup (i j) T prime) ge v j(Γ T) satisfying all other requirements Then weobtain

v i(Γ cup (i j) T prime) + v j(Γ cup (i j) T prime ) gt v i(Γ T) + v j(Γ T)

which is reduced to

Then it follows from (A4) that

Final version accepted 24 November 2004

REFERENCES

Bloch F and M O Jackson (2004) The Formation of Networks with Transfers among Players Working PaperCalifornia Institute of Technology

Dixit A K (1975) ldquoWelfare Effect and Tax and Price Changesrdquo Journal of Public Economics Vol 4pp 103ndash123

European Commission (2003) ldquoAllocation of 2002 EU Operating Expenditure by Member State at httpeuropaeuintcommbudgetagenda2000reports_enhtm

Freund C L (2000) ldquoMultilateralism and the Endogenous Formation of Preferential Trade AgreementsrdquoJournal of International Economics Vol 52 pp 359ndash376

prime + primeT Tik jk

[ ( ( )) ( )] ( ) ( ( ) ( ))

u i j u T T T Th h

h i j C C i jik jk ik jk

i j

Γ ΓΓ Γ

cup minus gt prime + prime minus minus= cupsum sum

u i j T u T

u i j T u T

iik

k C

iik

k C

jjk

k C

jjk

i i

j

( ( )) ( )

( ( )) ( )

( ) ( )

( )

Γ Γ

Γ Γ

Γ Γ

Γ

cup minus prime

minus minus

+ cup minus prime

minus minus

isin isin

isin

sum sum

sumkk Cjisinsum

gt( )

Γ

0

primeTij

prime = cup minus prime

minus minus

minusisin isinsum sumT u i j T u Tij

iik

k C

iik

k Ci i

( ( )) ( ) ( ) ( )

Γ ΓΓ Γ

ε2

[ ( ( )) ( )] ( ) ( ) ( )

u i j u T T T Th h

h i j k C Cik jk ik jk

i j

Γ ΓΓ Γ

cup minus gt prime + prime minus minus= isin cupsum sum

[ ( ( )) ( )] max ( ) ( ( )) ( ) ( )

u i j u u u i jh h

h i j k C C

k k

i j

Γ Γ Γ ΓΓ Γ




Fukushima T and N Kim (1989) ldquoWelfare Improving Tariff Changes A Case of Many Goods and CountriesrdquoJournal of International Economics Vol 26 pp 383ndash388

Furusawa T and H Konishi (2002) Free Trade Networks Working Paper No 548 Boston College Massmdashmdash and mdashmdash (2004) ldquoA Welfare Decomposition in Quasi-Linear Economiesrdquo Economics Letters Vol 85

pp 29ndash34Goyal S and S Joshi (2001) ldquoBilateralism and Free Traderdquo unpublished papermdashmdash and mdashmdash (2003) ldquoUnequal Connectionsrdquo unpublished paperHatta T (1977) ldquoA Theory of Piecemeal Policy Recommendationsrdquo Review of Economic Studies Vol 44

pp 1ndash21mdashmdash and T Fukushima (1979) ldquoThe Welfare Effect of Tariff Rate Reductions in a Many-Country Worldrdquo

Journal of International Economics Vol 9 pp 503ndash511Jackson M O and A Wolinsky (1996) ldquoA Strategic Model of Social and Economic Networksrdquo Journal of

Economic Theory Vol 71 pp 44ndash74Kowalczyk C (1989) ldquoTrade Negotiations and World Welfarerdquo American Economic Review Vol 79 pp 552ndash

559Matsubayashi N and S Yamakawa (2003) A Network Formation Game with an Endogenous Cost Allocation

Rule Working Paper Tokyo University of ScienceOttaviano G I P T Tabuchi and J-F Thisse (2002) ldquoAgglomeration and Trade Revisitedrdquo International

Economic Review Vol 43 pp 409ndash436Shubik M (1984) A Game-Theoretic Approach to Political Economy Cambridge Mass MIT PressYi S-S (1996) ldquoEndogenous Formation of Customs Unions under Imperfect Competition Open Regionalism

is Goodrdquo Journal of International Economics Vol 41 pp 153ndash177mdashmdash (2000) ldquoFree-Trade Areas and Welfare An Equilibrium Analysisrdquo Review of International Economics

Vol 8 pp 336ndash347



on the industrial good trade surplus is favourable for the former and unfavourable forthe latter Unless opening the market benefits domestic consumers sufficiently the lessindustrialized country will object to the FTA

This rather intuitive discussion is formally supported with the aid of the welfare decom-position proposed by Furusawa and Konishi (2004) who show that if each consumerrsquospreferences are characterized by a quasi-linear utility function social welfare can be decomposedinto consumersrsquo gross utility and the trade surplus of the non-numeraire good FollowingFurusawa and Konishi (2002) the present paper adopts the quasi-linear utility functionwith a continuum of non-numeraire good commodities which is developed by Ottaviano et al(2002) Now the impact of an FTA on industrial good trade surplus can be decomposedinto a direct surplus effect and a third-country effect The former is the effect on the tradesurplus of the partner country of the FTA in question and the latter is the effect on the tradesurplus of all other countries The direct surplus effect can be positive or negative dependingon the countryrsquos relative industrialization level compared with its partner country as discussedabove On the other hand the third-country effect is always positive since the countryrsquosexports to third countries are not affected by the FTA while its imports from them decreasebecause such commodities become relatively more expensive in the domestic market

Even if a pair of countries is so asymmetric that one of them has no incentive to signan FTA the FTA is likely to raise the joint welfare since the third-country effect ispositive and the direct surplus effects are cancelled out between the two countries (Eachcountryrsquos exports to the other are the other countryrsquos imports and vice versa) Thereforeif international transfers are possible they may sign an FTA and the country that benefitsfrom the FTA will transfer part of the benefit to the other country which would incur aloss otherwise International transfers are not uncommon in reality A well known exampleis the monetary transfers that take place among EU members The European Commission(2003) reports that in 2002 the ldquooperationalrdquo budgetary balance of Germany for examplewas minus$50678 million while that of Spain was $88708 million indicating that fairlylarge amounts of transfers are made among EU members International transfers may notnecessarily be made in the form of monetary transfers recent FTAs have often includedagreements on economic issues other than tariff reduction such as competition policylabour and environmental regulations In many cases one signatory will demand thatother signatories harmonize their standards with its own Such unilateral changes ofstandards can be considered effectively as an international transfer

We extend Furusawa and Konishirsquos (2002) model to allow international transfersbetween FTA signatories in order to examine FTA formation between asymmetric coun-tries We find that their results are significantly strengthened if we allow internationaltransfers We obtained the same results for the global free trade network mentioned aboveeven where the world consists of fairly asymmetric countries Mutual tariff elimination byany pair of countries benefits these countries as a whole unless industrial commodities arehighly substitutable so that they will sign an FTA if a transfer between them is possibleHatta and Fukushima (1979) and Fukushima and Kim (1989) show that proportionalreduction of all tariff rates of all countries improves world welfare2 Mutual tariff

2 Strictly speaking world welfare improves if all the countries in the world (a) reduce all their ad valoremtariff rates proportionally (Hatta and Fukushima 1979) or (b) reduce all their specific tariffs rates and sub-sidies proportionally (Fukushima and Kim 1989) However Kowalczyk (1989) demonstrates that worldwelfare may decline if all countries proportionally reduce all of their ad valorem tariff rates and subsidies







2 The model






21 Overview






sum ==kn k


tki

tii

tki

tii





(1)




(2)


(3)


(4)



(5)



U q q q d q d q d q( ) ( ) ( ) ( ) 00

1

0

1

2

0

12

02 2

= minus minus


0

12

0

1

0

1

q d q q d d( ) ( ) ( )ω ω ω ω ω ω

= prime prime

y q d q ( ) ( ) = +0

1

0 π ω ω ω

q p ti iki i( ) ( ( ) )

( )( )ω α

β βω δ


+minus

1P


n i i ip q1

p tiki i( )

ω αβ

β δδ

β δ=

+minus +

+

1

2P

Pi i

=

++

++

αββ δ

β δβ δ2 2

t

t ikn k

kis t equiv sum =1

ti init t ( )= 1

p tki i

ki i( )

( )t =

+minus +

+αββ δ

δβ δ2

1

2 2 2t




(6)


23 Social welfare


(7)


(8)


(9)





(10)


q tki i

ki i( )

( )t =

+minus +

+α

β δ βδ

β β δ2

1

2 2 2t

p qki i

ki i( ) ( )t t= β

π micro micro βik

ik k

ik k

k

n

kik k

k

n

p q q( ) ( ) ( ) ( ) t = == =

sum sumt t t1

2

1

R t s qi iki k

ki i

k

n

( ) ( )t t==

sum1

y l Rsi i

ii

i ( )

( )= + +t

tπmicro

q l Rs

s p t q

l s t qs

p q

i i ii

i

ik

ki i

ki

ki

ki

k

n

kki

ki i

k

n i

ik

ik k

ik k

k

n

01

1 1

( ) ( ) ( )

[ ( ) ] ( )

( ) ( ) ( )

t tt

t t

t t t

= + + minus +

= + + minus

=

= =

sum

sum sum

πmicro


( ) ( ) ( ) ( )

s p t q

l s p qs

p q

kki i

ki

ki i

k

n

kki i

ki i

k i

i

ik

ik k

ik k

k i

t t

t t t t

+

= minus +

=

ne ne

sum

sum sum

1

micromicro


q qi iki i

k N( ) ( ( ))t t= isin




where


(12)

(13)











ki i

k i

i kki i

k i

( ) ( ) ( ) ( ) t t t t= =ne ne



ik k

k i

i kik k

k i

( ) ( ) ( ) ( ) tminus


q tki i

ki


V t ti i i i i ini


i q ji

qki

s sCk C

kii

( )equiv sum isin

t s ss

s si C

i

C ii

i( )

( )

( ) ( )[ ( ) ( )] =


gt4

3 2 1 4 2 1 20

2


sCi













t tji

ijprime = prime = 0


( ) ( ) ( )

ii ts s

iC ji


8

1 2 4

2

2

α βδ β β δ

sCi

s sC ji ( ) + ge1 2 1 2

Mki X k

i

M s qki i i k


X s q Mki k k i

ik k

ik k( ) ( ) ( ( ))t t t= =βmicro 2

X M X Mi i i iki k

ki i

k i

( ) ( ) [ ( ) ( )]t t t tminus

neminus = minussum

X ki k( )t










∆ ∆ ∆[ ( ) ( )] [ ( ) ( )] ( )


ji i

ki i

k i j

t t t t tminus



1 2444 34441 24 34

t ji

∆Mki i( ) t lt 0

Mij j( )t

M ji i( )t

∆ ∆[ ( ) ( )] [ ( ) ( ) ]X M q qji j

ji i i j i

ij j j




∆ ∆q qji i

ij j( ) ( )t tlt
















primeTij primeTji
















(15)




(i)

(ii)


M Xji i

ij i( ) ( )t t=

∆ ∆[ ( ) ( )] [ ( ) ( )]X M X Mji j

ji i

i

ij i

ij j

j


1 2444 3444 1 2444 3444

t ji ti

j

( ) τΓ Γi


jij t( ) ( )=

τΓi j( ) = 0 τΓ

i

( ) τΓ Γi

i Nisin isinG



V V X X X M M Mii j

i i i i ii j

kk N i

i kk N i

i ii j


u u V X M

V X M T

i j i iki k

k i jki i

k i j

j jkj k

k i jkj j

k i j

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

Γ Γ Γ Γ Γ

Γ Γ Γ

+ = + minus

+ + minus

ne ne

ne ne

sum sum

sum sum

τ τ τ

τ τ

∆ ∆ ∆ ∆ ∆ ∆u u V V M Mi j i jk i j k

ik i j k















( ) τΓ Γi

i Nisin isinG



( ) τΓ Γi


j



comp( ( )) Γ + = 12 1s sC i j ji

comp( ( )) Γ + ge

( ) τ Γ Γci

i Nisin isinG

τΓci ij t( ) =

( ) τ Γ Γci

i Nisin isinG


primeTij



primeTij














î i iki i

k i j

V M ( ) ( )

isin minus

ne

sumargmax t t

t ji i = ˇ t j

i

sCi sCi




























u u i j u u i j T

u i j u u i j

i i

k C i j

k kij

j j

k C i j

k

i

j

( ) ( ( )) max ( ) ( ( ))

( ( )) ( ) min ( (

( )

( )

Γ Γ Γ Γ

Γ Γ Γ

Γ

Γ

minus + minus ge

ge minus +

isin

isin

sum

sum

0

)))) ( ) minus uk Γ 0

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ




Γ Γk C C

k k

i j

u i j uisin cup

cup minus le( ) ( )

[ ( ( )) ( )] 0













( ) τ Γ Γi

i Nisin isinG

( ) τ Γ Γci

i Nisin isinG



k C C

k k

i jki

kj

k i j k C C i jik

jk

i

i j

i j

u i j u

V V M M X X

V

isin cup

ne isin cup

sum

sum sum

cup minus

= + minus + + +

= +

( ) ( )

( ( ) ( ))

[ ( ( )) ( )]

( ) ( )

Γ Γ

Γ Γ

Γ Γ

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆∆ ∆ ∆Γ Γ

V M Mj

k C Cki

kj

i j

( )

( ) ( )

minus +

gtnotin cup

sum0






Appendix

A1 Proof of Lemma 4






we have



u T u i j T

T u i j u

i

k C iki

i

k C i jki

jii i

i i

( ) ( ( ))

( ( )) ( )

( ) ( )

Γ Γ

Γ Γ

Γ Γ+ ge +

ge minus

isin isinsum sum


T Tij ji = minus



Tji = 0



(A1)

Similarly we have

(A2)






A2 Proof of Lemma 5



u T

v T

iij ji

iki

k N jji

iki

k N

i

( ( ) ( ) ) ( ( ))

( )

( )

Γ Γ

Γ

Γ


= + +

= +

isin

isin

sum

sum ε

ε2

2




v T u T T

u u i j u i j T T

u u i j T

i iki

k N jji

i i iki

k N jji

i i

( ) ( )

( ) ( ( )) ( ( ))

( ) ( ( ))

Γ Γ

Γ Γ Γ

Γ Γ

= + +

= minus + + +

= minus +

isin

isin

sum

sum

jijii

ij ji

iij ji

v i j T

v i j T

( ( ) )

( ( ) )( )

( )

+ge

minus

minus

ΓΓ







X M X Mji

ij

ij

ji = = and

∆ ∆X Xki

kj = = 0

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆

∆ ∆ ∆

u u V V X M X M

X M X M

V V M

i j i jji

ji

ki

k i jki

ij

ij

kj

k i jkj

i jki

k i j

( ) ( )

( ) ( )


+ minus + minus

= + minus

ne

ne

ne

sumsum

sum minusminusnesum

∆Mkj

k i j



A3 Proof of Lemma 6



(A3)









(A4)

u T u i j T

u T u i j T

T T u u i j

klk

l N

klk

l N

kik

kik

ik ikk k

( ) ( ( ))

( ) ( ( ))

( ) ( ( ))

Γ Γ

Γ ΓΓ Γ

+ le + prime


isin isinsum sum



primeTik


k C i jik

i

k C i jik

k k

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0


k C iik

i

k C i jik

k i

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0

u T u i j T

v T v i j T

i

k C iik

i

k C i jik

i i

i i

( ) ( ( ))

( ) ( ( ) )

( ) ( )

Γ Γ

Γ Γ


ge prime

isin isinsum sum

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ








which gives us





which is reduced to



REFERENCES






[ ( ( )) ( )] ( ) ( ( ) ( ))

u i j u T T T Th h


i j

Γ ΓΓ Γ


u i j T u T

u i j T u T

iik

k C

iik

k C

jjk

k C

jjk

i i

j

( ( )) ( )

( ( )) ( )

( ) ( )

( )

Γ Γ

Γ Γ

Γ Γ

Γ

cup minus prime

minus minus

+ cup minus prime

minus minus

isin isin

isin

sum sum

sumkk Cjisinsum

gt( )

Γ

0

primeTij


minus minus


iik

k C

iik

k Ci i

( ( )) ( ) ( ) ( )

Γ ΓΓ Γ

ε2

[ ( ( )) ( )] ( ) ( ) ( )

u i j u T T T Th h


i j

Γ ΓΓ Γ


[ ( ( )) ( )] max ( ) ( ( )) ( ) ( )

u i j u u u i jh h

h i j k C C

k k

i j

Γ Γ Γ ΓΓ Γ





















2 The model






21 Overview






sum ==kn k


tki

tii

tki

tii





(1)




(2)


(3)


(4)



(5)



U q q q d q d q d q( ) ( ) ( ) ( ) 00

1

0

1

2

0

12

02 2

= minus minus


0

12

0

1

0

1

q d q q d d( ) ( ) ( )ω ω ω ω ω ω

= prime prime

y q d q ( ) ( ) = +0

1

0 π ω ω ω

q p ti iki i( ) ( ( ) )

( )( )ω α

β βω δ


+minus

1P


n i i ip q1

p tiki i( )

ω αβ

β δδ

β δ=

+minus +

+

1

2P

Pi i

=

++

++

αββ δ

β δβ δ2 2

t

t ikn k

kis t equiv sum =1

ti init t ( )= 1

p tki i

ki i( )

( )t =

+minus +

+αββ δ

δβ δ2

1

2 2 2t




(6)


23 Social welfare


(7)


(8)


(9)





(10)


q tki i

ki i( )

( )t =

+minus +

+α

β δ βδ

β β δ2

1

2 2 2t

p qki i

ki i( ) ( )t t= β

π micro micro βik

ik k

ik k

k

n

kik k

k

n

p q q( ) ( ) ( ) ( ) t = == =

sum sumt t t1

2

1

R t s qi iki k

ki i

k

n

( ) ( )t t==

sum1

y l Rsi i

ii

i ( )

( )= + +t

tπmicro

q l Rs

s p t q

l s t qs

p q

i i ii

i

ik

ki i

ki

ki

ki

k

n

kki

ki i

k

n i

ik

ik k

ik k

k

n

01

1 1

( ) ( ) ( )

[ ( ) ] ( )

( ) ( ) ( )

t tt

t t

t t t

= + + minus +

= + + minus

=

= =

sum

sum sum

πmicro


( ) ( ) ( ) ( )

s p t q

l s p qs

p q

kki i

ki

ki i

k

n

kki i

ki i

k i

i

ik

ik k

ik k

k i

t t

t t t t

+

= minus +

=

ne ne

sum

sum sum

1

micromicro


q qi iki i

k N( ) ( ( ))t t= isin




where


(12)

(13)











ki i

k i

i kki i

k i

( ) ( ) ( ) ( ) t t t t= =ne ne



ik k

k i

i kik k

k i

( ) ( ) ( ) ( ) tminus


q tki i

ki


V t ti i i i i ini


i q ji

qki

s sCk C

kii

( )equiv sum isin

t s ss

s si C

i

C ii

i( )

( )

( ) ( )[ ( ) ( )] =


gt4

3 2 1 4 2 1 20

2


sCi













t tji

ijprime = prime = 0


( ) ( ) ( )

ii ts s

iC ji


8

1 2 4

2

2

α βδ β β δ

sCi

s sC ji ( ) + ge1 2 1 2

Mki X k

i

M s qki i i k


X s q Mki k k i

ik k

ik k( ) ( ) ( ( ))t t t= =βmicro 2

X M X Mi i i iki k

ki i

k i

( ) ( ) [ ( ) ( )]t t t tminus

neminus = minussum

X ki k( )t










∆ ∆ ∆[ ( ) ( )] [ ( ) ( )] ( )


ji i

ki i

k i j

t t t t tminus



1 2444 34441 24 34

t ji

∆Mki i( ) t lt 0

Mij j( )t

M ji i( )t

∆ ∆[ ( ) ( )] [ ( ) ( ) ]X M q qji j

ji i i j i

ij j j




∆ ∆q qji i

ij j( ) ( )t tlt
















primeTij primeTji
















(15)




(i)

(ii)


M Xji i

ij i( ) ( )t t=

∆ ∆[ ( ) ( )] [ ( ) ( )]X M X Mji j

ji i

i

ij i

ij j

j


1 2444 3444 1 2444 3444

t ji ti

j

( ) τΓ Γi


jij t( ) ( )=

τΓi j( ) = 0 τΓ

i

( ) τΓ Γi

i Nisin isinG



V V X X X M M Mii j

i i i i ii j

kk N i

i kk N i

i ii j


u u V X M

V X M T

i j i iki k

k i jki i

k i j

j jkj k

k i jkj j

k i j

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

Γ Γ Γ Γ Γ

Γ Γ Γ

+ = + minus

+ + minus

ne ne

ne ne

sum sum

sum sum

τ τ τ

τ τ

∆ ∆ ∆ ∆ ∆ ∆u u V V M Mi j i jk i j k

ik i j k















( ) τΓ Γi

i Nisin isinG



( ) τΓ Γi


j



comp( ( )) Γ + = 12 1s sC i j ji

comp( ( )) Γ + ge

( ) τ Γ Γci

i Nisin isinG

τΓci ij t( ) =

( ) τ Γ Γci

i Nisin isinG


primeTij



primeTij














î i iki i

k i j

V M ( ) ( )

isin minus

ne

sumargmax t t

t ji i = ˇ t j

i

sCi sCi




























u u i j u u i j T

u i j u u i j

i i

k C i j

k kij

j j

k C i j

k

i

j

( ) ( ( )) max ( ) ( ( ))

( ( )) ( ) min ( (

( )

( )

Γ Γ Γ Γ

Γ Γ Γ

Γ

Γ

minus + minus ge

ge minus +

isin

isin

sum

sum

0

)))) ( ) minus uk Γ 0

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ




Γ Γk C C

k k

i j

u i j uisin cup

cup minus le( ) ( )

[ ( ( )) ( )] 0













( ) τ Γ Γi

i Nisin isinG

( ) τ Γ Γci

i Nisin isinG



k C C

k k

i jki

kj

k i j k C C i jik

jk

i

i j

i j

u i j u

V V M M X X

V

isin cup

ne isin cup

sum

sum sum

cup minus

= + minus + + +

= +

( ) ( )

( ( ) ( ))

[ ( ( )) ( )]

( ) ( )

Γ Γ

Γ Γ

Γ Γ

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆∆ ∆ ∆Γ Γ

V M Mj

k C Cki

kj

i j

( )

( ) ( )

minus +

gtnotin cup

sum0






Appendix

A1 Proof of Lemma 4






we have



u T u i j T

T u i j u

i

k C iki

i

k C i jki

jii i

i i

( ) ( ( ))

( ( )) ( )

( ) ( )

Γ Γ

Γ Γ

Γ Γ+ ge +

ge minus

isin isinsum sum


T Tij ji = minus



Tji = 0



(A1)

Similarly we have

(A2)






A2 Proof of Lemma 5



u T

v T

iij ji

iki

k N jji

iki

k N

i

( ( ) ( ) ) ( ( ))

( )

( )

Γ Γ

Γ

Γ


= + +

= +

isin

isin

sum

sum ε

ε2

2




v T u T T

u u i j u i j T T

u u i j T

i iki

k N jji

i i iki

k N jji

i i

( ) ( )

( ) ( ( )) ( ( ))

( ) ( ( ))

Γ Γ

Γ Γ Γ

Γ Γ

= + +

= minus + + +

= minus +

isin

isin

sum

sum

jijii

ij ji

iij ji

v i j T

v i j T

( ( ) )

( ( ) )( )

( )

+ge

minus

minus

ΓΓ







X M X Mji

ij

ij

ji = = and

∆ ∆X Xki

kj = = 0

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆

∆ ∆ ∆

u u V V X M X M

X M X M

V V M

i j i jji

ji

ki

k i jki

ij

ij

kj

k i jkj

i jki

k i j

( ) ( )

( ) ( )


+ minus + minus

= + minus

ne

ne

ne

sumsum

sum minusminusnesum

∆Mkj

k i j



A3 Proof of Lemma 6



(A3)









(A4)

u T u i j T

u T u i j T

T T u u i j

klk

l N

klk

l N

kik

kik

ik ikk k

( ) ( ( ))

( ) ( ( ))

( ) ( ( ))

Γ Γ

Γ ΓΓ Γ

+ le + prime


isin isinsum sum



primeTik


k C i jik

i

k C i jik

k k

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0


k C iik

i

k C i jik

k i

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0

u T u i j T

v T v i j T

i

k C iik

i

k C i jik

i i

i i

( ) ( ( ))

( ) ( ( ) )

( ) ( )

Γ Γ

Γ Γ


ge prime

isin isinsum sum

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ








which gives us





which is reduced to



REFERENCES






[ ( ( )) ( )] ( ) ( ( ) ( ))

u i j u T T T Th h


i j

Γ ΓΓ Γ


u i j T u T

u i j T u T

iik

k C

iik

k C

jjk

k C

jjk

i i

j

( ( )) ( )

( ( )) ( )

( ) ( )

( )

Γ Γ

Γ Γ

Γ Γ

Γ

cup minus prime

minus minus

+ cup minus prime

minus minus

isin isin

isin

sum sum

sumkk Cjisinsum

gt( )

Γ

0

primeTij


minus minus


iik

k C

iik

k Ci i

( ( )) ( ) ( ) ( )

Γ ΓΓ Γ

ε2

[ ( ( )) ( )] ( ) ( ) ( )

u i j u T T T Th h


i j

Γ ΓΓ Γ


[ ( ( )) ( )] max ( ) ( ( )) ( ) ( )

u i j u u u i jh h

h i j k C C

k k

i j

Γ Γ Γ ΓΓ Γ

















21 Overview






sum ==kn k


tki

tii

tki

tii





(1)




(2)


(3)


(4)



(5)



U q q q d q d q d q( ) ( ) ( ) ( ) 00

1

0

1

2

0

12

02 2

= minus minus


0

12

0

1

0

1

q d q q d d( ) ( ) ( )ω ω ω ω ω ω

= prime prime

y q d q ( ) ( ) = +0

1

0 π ω ω ω

q p ti iki i( ) ( ( ) )

( )( )ω α

β βω δ


+minus

1P


n i i ip q1

p tiki i( )

ω αβ

β δδ

β δ=

+minus +

+

1

2P

Pi i

=

++

++

αββ δ

β δβ δ2 2

t

t ikn k

kis t equiv sum =1

ti init t ( )= 1

p tki i

ki i( )

( )t =

+minus +

+αββ δ

δβ δ2

1

2 2 2t




(6)


23 Social welfare


(7)


(8)


(9)





(10)


q tki i

ki i( )

( )t =

+minus +

+α

β δ βδ

β β δ2

1

2 2 2t

p qki i

ki i( ) ( )t t= β

π micro micro βik

ik k

ik k

k

n

kik k

k

n

p q q( ) ( ) ( ) ( ) t = == =

sum sumt t t1

2

1

R t s qi iki k

ki i

k

n

( ) ( )t t==

sum1

y l Rsi i

ii

i ( )

( )= + +t

tπmicro

q l Rs

s p t q

l s t qs

p q

i i ii

i

ik

ki i

ki

ki

ki

k

n

kki

ki i

k

n i

ik

ik k

ik k

k

n

01

1 1

( ) ( ) ( )

[ ( ) ] ( )

( ) ( ) ( )

t tt

t t

t t t

= + + minus +

= + + minus

=

= =

sum

sum sum

πmicro


( ) ( ) ( ) ( )

s p t q

l s p qs

p q

kki i

ki

ki i

k

n

kki i

ki i

k i

i

ik

ik k

ik k

k i

t t

t t t t

+

= minus +

=

ne ne

sum

sum sum

1

micromicro


q qi iki i

k N( ) ( ( ))t t= isin




where


(12)

(13)











ki i

k i

i kki i

k i

( ) ( ) ( ) ( ) t t t t= =ne ne



ik k

k i

i kik k

k i

( ) ( ) ( ) ( ) tminus


q tki i

ki


V t ti i i i i ini


i q ji

qki

s sCk C

kii

( )equiv sum isin

t s ss

s si C

i

C ii

i( )

( )

( ) ( )[ ( ) ( )] =


gt4

3 2 1 4 2 1 20

2


sCi













t tji

ijprime = prime = 0


( ) ( ) ( )

ii ts s

iC ji


8

1 2 4

2

2

α βδ β β δ

sCi

s sC ji ( ) + ge1 2 1 2

Mki X k

i

M s qki i i k


X s q Mki k k i

ik k

ik k( ) ( ) ( ( ))t t t= =βmicro 2

X M X Mi i i iki k

ki i

k i

( ) ( ) [ ( ) ( )]t t t tminus

neminus = minussum

X ki k( )t










∆ ∆ ∆[ ( ) ( )] [ ( ) ( )] ( )


ji i

ki i

k i j

t t t t tminus



1 2444 34441 24 34

t ji

∆Mki i( ) t lt 0

Mij j( )t

M ji i( )t

∆ ∆[ ( ) ( )] [ ( ) ( ) ]X M q qji j

ji i i j i

ij j j




∆ ∆q qji i

ij j( ) ( )t tlt
















primeTij primeTji
















(15)




(i)

(ii)


M Xji i

ij i( ) ( )t t=

∆ ∆[ ( ) ( )] [ ( ) ( )]X M X Mji j

ji i

i

ij i

ij j

j


1 2444 3444 1 2444 3444

t ji ti

j

( ) τΓ Γi


jij t( ) ( )=

τΓi j( ) = 0 τΓ

i

( ) τΓ Γi

i Nisin isinG



V V X X X M M Mii j

i i i i ii j

kk N i

i kk N i

i ii j


u u V X M

V X M T

i j i iki k

k i jki i

k i j

j jkj k

k i jkj j

k i j

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

Γ Γ Γ Γ Γ

Γ Γ Γ

+ = + minus

+ + minus

ne ne

ne ne

sum sum

sum sum

τ τ τ

τ τ

∆ ∆ ∆ ∆ ∆ ∆u u V V M Mi j i jk i j k

ik i j k















( ) τΓ Γi

i Nisin isinG



( ) τΓ Γi


j



comp( ( )) Γ + = 12 1s sC i j ji

comp( ( )) Γ + ge

( ) τ Γ Γci

i Nisin isinG

τΓci ij t( ) =

( ) τ Γ Γci

i Nisin isinG


primeTij



primeTij














î i iki i

k i j

V M ( ) ( )

isin minus

ne

sumargmax t t

t ji i = ˇ t j

i

sCi sCi




























u u i j u u i j T

u i j u u i j

i i

k C i j

k kij

j j

k C i j

k

i

j

( ) ( ( )) max ( ) ( ( ))

( ( )) ( ) min ( (

( )

( )

Γ Γ Γ Γ

Γ Γ Γ

Γ

Γ

minus + minus ge

ge minus +

isin

isin

sum

sum

0

)))) ( ) minus uk Γ 0

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ




Γ Γk C C

k k

i j

u i j uisin cup

cup minus le( ) ( )

[ ( ( )) ( )] 0













( ) τ Γ Γi

i Nisin isinG

( ) τ Γ Γci

i Nisin isinG



k C C

k k

i jki

kj

k i j k C C i jik

jk

i

i j

i j

u i j u

V V M M X X

V

isin cup

ne isin cup

sum

sum sum

cup minus

= + minus + + +

= +

( ) ( )

( ( ) ( ))

[ ( ( )) ( )]

( ) ( )

Γ Γ

Γ Γ

Γ Γ

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆∆ ∆ ∆Γ Γ

V M Mj

k C Cki

kj

i j

( )

( ) ( )

minus +

gtnotin cup

sum0






Appendix

A1 Proof of Lemma 4






we have



u T u i j T

T u i j u

i

k C iki

i

k C i jki

jii i

i i

( ) ( ( ))

( ( )) ( )

( ) ( )

Γ Γ

Γ Γ

Γ Γ+ ge +

ge minus

isin isinsum sum


T Tij ji = minus



Tji = 0



(A1)

Similarly we have

(A2)






A2 Proof of Lemma 5



u T

v T

iij ji

iki

k N jji

iki

k N

i

( ( ) ( ) ) ( ( ))

( )

( )

Γ Γ

Γ

Γ


= + +

= +

isin

isin

sum

sum ε

ε2

2




v T u T T

u u i j u i j T T

u u i j T

i iki

k N jji

i i iki

k N jji

i i

( ) ( )

( ) ( ( )) ( ( ))

( ) ( ( ))

Γ Γ

Γ Γ Γ

Γ Γ

= + +

= minus + + +

= minus +

isin

isin

sum

sum

jijii

ij ji

iij ji

v i j T

v i j T

( ( ) )

( ( ) )( )

( )

+ge

minus

minus

ΓΓ







X M X Mji

ij

ij

ji = = and

∆ ∆X Xki

kj = = 0

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆

∆ ∆ ∆

u u V V X M X M

X M X M

V V M

i j i jji

ji

ki

k i jki

ij

ij

kj

k i jkj

i jki

k i j

( ) ( )

( ) ( )


+ minus + minus

= + minus

ne

ne

ne

sumsum

sum minusminusnesum

∆Mkj

k i j



A3 Proof of Lemma 6



(A3)









(A4)

u T u i j T

u T u i j T

T T u u i j

klk

l N

klk

l N

kik

kik

ik ikk k

( ) ( ( ))

( ) ( ( ))

( ) ( ( ))

Γ Γ

Γ ΓΓ Γ

+ le + prime


isin isinsum sum



primeTik


k C i jik

i

k C i jik

k k

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0


k C iik

i

k C i jik

k i

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0

u T u i j T

v T v i j T

i

k C iik

i

k C i jik

i i

i i

( ) ( ( ))

( ) ( ( ) )

( ) ( )

Γ Γ

Γ Γ


ge prime

isin isinsum sum

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ








which gives us





which is reduced to



REFERENCES






[ ( ( )) ( )] ( ) ( ( ) ( ))

u i j u T T T Th h


i j

Γ ΓΓ Γ


u i j T u T

u i j T u T

iik

k C

iik

k C

jjk

k C

jjk

i i

j

( ( )) ( )

( ( )) ( )

( ) ( )

( )

Γ Γ

Γ Γ

Γ Γ

Γ

cup minus prime

minus minus

+ cup minus prime

minus minus

isin isin

isin

sum sum

sumkk Cjisinsum

gt( )

Γ

0

primeTij


minus minus


iik

k C

iik

k Ci i

( ( )) ( ) ( ) ( )

Γ ΓΓ Γ

ε2

[ ( ( )) ( )] ( ) ( ) ( )

u i j u T T T Th h


i j

Γ ΓΓ Γ


[ ( ( )) ( )] max ( ) ( ( )) ( ) ( )

u i j u u u i jh h

h i j k C C

k k

i j

Γ Γ Γ ΓΓ Γ



















(1)




(2)


(3)


(4)



(5)



U q q q d q d q d q( ) ( ) ( ) ( ) 00

1

0

1

2

0

12

02 2

= minus minus


0

12

0

1

0

1

q d q q d d( ) ( ) ( )ω ω ω ω ω ω

= prime prime

y q d q ( ) ( ) = +0

1

0 π ω ω ω

q p ti iki i( ) ( ( ) )

( )( )ω α

β βω δ


+minus

1P


n i i ip q1

p tiki i( )

ω αβ

β δδ

β δ=

+minus +

+

1

2P

Pi i

=

++

++

αββ δ

β δβ δ2 2

t

t ikn k

kis t equiv sum =1

ti init t ( )= 1

p tki i

ki i( )

( )t =

+minus +

+αββ δ

δβ δ2

1

2 2 2t




(6)


23 Social welfare


(7)


(8)


(9)





(10)


q tki i

ki i( )

( )t =

+minus +

+α

β δ βδ

β β δ2

1

2 2 2t

p qki i

ki i( ) ( )t t= β

π micro micro βik

ik k

ik k

k

n

kik k

k

n

p q q( ) ( ) ( ) ( ) t = == =

sum sumt t t1

2

1

R t s qi iki k

ki i

k

n

( ) ( )t t==

sum1

y l Rsi i

ii

i ( )

( )= + +t

tπmicro

q l Rs

s p t q

l s t qs

p q

i i ii

i

ik

ki i

ki

ki

ki

k

n

kki

ki i

k

n i

ik

ik k

ik k

k

n

01

1 1

( ) ( ) ( )

[ ( ) ] ( )

( ) ( ) ( )

t tt

t t

t t t

= + + minus +

= + + minus

=

= =

sum

sum sum

πmicro


( ) ( ) ( ) ( )

s p t q

l s p qs

p q

kki i

ki

ki i

k

n

kki i

ki i

k i

i

ik

ik k

ik k

k i

t t

t t t t

+

= minus +

=

ne ne

sum

sum sum

1

micromicro


q qi iki i

k N( ) ( ( ))t t= isin




where


(12)

(13)











ki i

k i

i kki i

k i

( ) ( ) ( ) ( ) t t t t= =ne ne



ik k

k i

i kik k

k i

( ) ( ) ( ) ( ) tminus


q tki i

ki


V t ti i i i i ini


i q ji

qki

s sCk C

kii

( )equiv sum isin

t s ss

s si C

i

C ii

i( )

( )

( ) ( )[ ( ) ( )] =


gt4

3 2 1 4 2 1 20

2


sCi













t tji

ijprime = prime = 0


( ) ( ) ( )

ii ts s

iC ji


8

1 2 4

2

2

α βδ β β δ

sCi

s sC ji ( ) + ge1 2 1 2

Mki X k

i

M s qki i i k


X s q Mki k k i

ik k

ik k( ) ( ) ( ( ))t t t= =βmicro 2

X M X Mi i i iki k

ki i

k i

( ) ( ) [ ( ) ( )]t t t tminus

neminus = minussum

X ki k( )t










∆ ∆ ∆[ ( ) ( )] [ ( ) ( )] ( )


ji i

ki i

k i j

t t t t tminus



1 2444 34441 24 34

t ji

∆Mki i( ) t lt 0

Mij j( )t

M ji i( )t

∆ ∆[ ( ) ( )] [ ( ) ( ) ]X M q qji j

ji i i j i

ij j j




∆ ∆q qji i

ij j( ) ( )t tlt
















primeTij primeTji
















(15)




(i)

(ii)


M Xji i

ij i( ) ( )t t=

∆ ∆[ ( ) ( )] [ ( ) ( )]X M X Mji j

ji i

i

ij i

ij j

j


1 2444 3444 1 2444 3444

t ji ti

j

( ) τΓ Γi


jij t( ) ( )=

τΓi j( ) = 0 τΓ

i

( ) τΓ Γi

i Nisin isinG



V V X X X M M Mii j

i i i i ii j

kk N i

i kk N i

i ii j


u u V X M

V X M T

i j i iki k

k i jki i

k i j

j jkj k

k i jkj j

k i j

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

Γ Γ Γ Γ Γ

Γ Γ Γ

+ = + minus

+ + minus

ne ne

ne ne

sum sum

sum sum

τ τ τ

τ τ

∆ ∆ ∆ ∆ ∆ ∆u u V V M Mi j i jk i j k

ik i j k















( ) τΓ Γi

i Nisin isinG



( ) τΓ Γi


j



comp( ( )) Γ + = 12 1s sC i j ji

comp( ( )) Γ + ge

( ) τ Γ Γci

i Nisin isinG

τΓci ij t( ) =

( ) τ Γ Γci

i Nisin isinG


primeTij



primeTij














î i iki i

k i j

V M ( ) ( )

isin minus

ne

sumargmax t t

t ji i = ˇ t j

i

sCi sCi




























u u i j u u i j T

u i j u u i j

i i

k C i j

k kij

j j

k C i j

k

i

j

( ) ( ( )) max ( ) ( ( ))

( ( )) ( ) min ( (

( )

( )

Γ Γ Γ Γ

Γ Γ Γ

Γ

Γ

minus + minus ge

ge minus +

isin

isin

sum

sum

0

)))) ( ) minus uk Γ 0

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ




Γ Γk C C

k k

i j

u i j uisin cup

cup minus le( ) ( )

[ ( ( )) ( )] 0













( ) τ Γ Γi

i Nisin isinG

( ) τ Γ Γci

i Nisin isinG



k C C

k k

i jki

kj

k i j k C C i jik

jk

i

i j

i j

u i j u

V V M M X X

V

isin cup

ne isin cup

sum

sum sum

cup minus

= + minus + + +

= +

( ) ( )

( ( ) ( ))

[ ( ( )) ( )]

( ) ( )

Γ Γ

Γ Γ

Γ Γ

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆∆ ∆ ∆Γ Γ

V M Mj

k C Cki

kj

i j

( )

( ) ( )

minus +

gtnotin cup

sum0






Appendix

A1 Proof of Lemma 4






we have



u T u i j T

T u i j u

i

k C iki

i

k C i jki

jii i

i i

( ) ( ( ))

( ( )) ( )

( ) ( )

Γ Γ

Γ Γ

Γ Γ+ ge +

ge minus

isin isinsum sum


T Tij ji = minus



Tji = 0



(A1)

Similarly we have

(A2)






A2 Proof of Lemma 5



u T

v T

iij ji

iki

k N jji

iki

k N

i

( ( ) ( ) ) ( ( ))

( )

( )

Γ Γ

Γ

Γ


= + +

= +

isin

isin

sum

sum ε

ε2

2




v T u T T

u u i j u i j T T

u u i j T

i iki

k N jji

i i iki

k N jji

i i

( ) ( )

( ) ( ( )) ( ( ))

( ) ( ( ))

Γ Γ

Γ Γ Γ

Γ Γ

= + +

= minus + + +

= minus +

isin

isin

sum

sum

jijii

ij ji

iij ji

v i j T

v i j T

( ( ) )

( ( ) )( )

( )

+ge

minus

minus

ΓΓ







X M X Mji

ij

ij

ji = = and

∆ ∆X Xki

kj = = 0

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆

∆ ∆ ∆

u u V V X M X M

X M X M

V V M

i j i jji

ji

ki

k i jki

ij

ij

kj

k i jkj

i jki

k i j

( ) ( )

( ) ( )


+ minus + minus

= + minus

ne

ne

ne

sumsum

sum minusminusnesum

∆Mkj

k i j



A3 Proof of Lemma 6



(A3)









(A4)

u T u i j T

u T u i j T

T T u u i j

klk

l N

klk

l N

kik

kik

ik ikk k

( ) ( ( ))

( ) ( ( ))

( ) ( ( ))

Γ Γ

Γ ΓΓ Γ

+ le + prime


isin isinsum sum



primeTik


k C i jik

i

k C i jik

k k

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0


k C iik

i

k C i jik

k i

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0

u T u i j T

v T v i j T

i

k C iik

i

k C i jik

i i

i i

( ) ( ( ))

( ) ( ( ) )

( ) ( )

Γ Γ

Γ Γ


ge prime

isin isinsum sum

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ








which gives us





which is reduced to



REFERENCES






[ ( ( )) ( )] ( ) ( ( ) ( ))

u i j u T T T Th h


i j

Γ ΓΓ Γ


u i j T u T

u i j T u T

iik

k C

iik

k C

jjk

k C

jjk

i i

j

( ( )) ( )

( ( )) ( )

( ) ( )

( )

Γ Γ

Γ Γ

Γ Γ

Γ

cup minus prime

minus minus

+ cup minus prime

minus minus

isin isin

isin

sum sum

sumkk Cjisinsum

gt( )

Γ

0

primeTij


minus minus


iik

k C

iik

k Ci i

( ( )) ( ) ( ) ( )

Γ ΓΓ Γ

ε2

[ ( ( )) ( )] ( ) ( ) ( )

u i j u T T T Th h


i j

Γ ΓΓ Γ


[ ( ( )) ( )] max ( ) ( ( )) ( ) ( )

u i j u u u i jh h

h i j k C C

k k

i j

Γ Γ Γ ΓΓ Γ


















(6)


23 Social welfare


(7)


(8)


(9)





(10)


q tki i

ki i( )

( )t =

+minus +

+α

β δ βδ

β β δ2

1

2 2 2t

p qki i

ki i( ) ( )t t= β

π micro micro βik

ik k

ik k

k

n

kik k

k

n

p q q( ) ( ) ( ) ( ) t = == =

sum sumt t t1

2

1

R t s qi iki k

ki i

k

n

( ) ( )t t==

sum1

y l Rsi i

ii

i ( )

( )= + +t

tπmicro

q l Rs

s p t q

l s t qs

p q

i i ii

i

ik

ki i

ki

ki

ki

k

n

kki

ki i

k

n i

ik

ik k

ik k

k

n

01

1 1

( ) ( ) ( )

[ ( ) ] ( )

( ) ( ) ( )

t tt

t t

t t t

= + + minus +

= + + minus

=

= =

sum

sum sum

πmicro


( ) ( ) ( ) ( )

s p t q

l s p qs

p q

kki i

ki

ki i

k

n

kki i

ki i

k i

i

ik

ik k

ik k

k i

t t

t t t t

+

= minus +

=

ne ne

sum

sum sum

1

micromicro


q qi iki i

k N( ) ( ( ))t t= isin




where


(12)

(13)











ki i

k i

i kki i

k i

( ) ( ) ( ) ( ) t t t t= =ne ne



ik k

k i

i kik k

k i

( ) ( ) ( ) ( ) tminus


q tki i

ki


V t ti i i i i ini


i q ji

qki

s sCk C

kii

( )equiv sum isin

t s ss

s si C

i

C ii

i( )

( )

( ) ( )[ ( ) ( )] =


gt4

3 2 1 4 2 1 20

2


sCi













t tji

ijprime = prime = 0


( ) ( ) ( )

ii ts s

iC ji


8

1 2 4

2

2

α βδ β β δ

sCi

s sC ji ( ) + ge1 2 1 2

Mki X k

i

M s qki i i k


X s q Mki k k i

ik k

ik k( ) ( ) ( ( ))t t t= =βmicro 2

X M X Mi i i iki k

ki i

k i

( ) ( ) [ ( ) ( )]t t t tminus

neminus = minussum

X ki k( )t










∆ ∆ ∆[ ( ) ( )] [ ( ) ( )] ( )


ji i

ki i

k i j

t t t t tminus



1 2444 34441 24 34

t ji

∆Mki i( ) t lt 0

Mij j( )t

M ji i( )t

∆ ∆[ ( ) ( )] [ ( ) ( ) ]X M q qji j

ji i i j i

ij j j




∆ ∆q qji i

ij j( ) ( )t tlt
















primeTij primeTji
















(15)




(i)

(ii)


M Xji i

ij i( ) ( )t t=

∆ ∆[ ( ) ( )] [ ( ) ( )]X M X Mji j

ji i

i

ij i

ij j

j


1 2444 3444 1 2444 3444

t ji ti

j

( ) τΓ Γi


jij t( ) ( )=

τΓi j( ) = 0 τΓ

i

( ) τΓ Γi

i Nisin isinG



V V X X X M M Mii j

i i i i ii j

kk N i

i kk N i

i ii j


u u V X M

V X M T

i j i iki k

k i jki i

k i j

j jkj k

k i jkj j

k i j

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

Γ Γ Γ Γ Γ

Γ Γ Γ

+ = + minus

+ + minus

ne ne

ne ne

sum sum

sum sum

τ τ τ

τ τ

∆ ∆ ∆ ∆ ∆ ∆u u V V M Mi j i jk i j k

ik i j k















( ) τΓ Γi

i Nisin isinG



( ) τΓ Γi


j



comp( ( )) Γ + = 12 1s sC i j ji

comp( ( )) Γ + ge

( ) τ Γ Γci

i Nisin isinG

τΓci ij t( ) =

( ) τ Γ Γci

i Nisin isinG


primeTij



primeTij














î i iki i

k i j

V M ( ) ( )

isin minus

ne

sumargmax t t

t ji i = ˇ t j

i

sCi sCi




























u u i j u u i j T

u i j u u i j

i i

k C i j

k kij

j j

k C i j

k

i

j

( ) ( ( )) max ( ) ( ( ))

( ( )) ( ) min ( (

( )

( )

Γ Γ Γ Γ

Γ Γ Γ

Γ

Γ

minus + minus ge

ge minus +

isin

isin

sum

sum

0

)))) ( ) minus uk Γ 0

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ




Γ Γk C C

k k

i j

u i j uisin cup

cup minus le( ) ( )

[ ( ( )) ( )] 0













( ) τ Γ Γi

i Nisin isinG

( ) τ Γ Γci

i Nisin isinG



k C C

k k

i jki

kj

k i j k C C i jik

jk

i

i j

i j

u i j u

V V M M X X

V

isin cup

ne isin cup

sum

sum sum

cup minus

= + minus + + +

= +

( ) ( )

( ( ) ( ))

[ ( ( )) ( )]

( ) ( )

Γ Γ

Γ Γ

Γ Γ

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆∆ ∆ ∆Γ Γ

V M Mj

k C Cki

kj

i j

( )

( ) ( )

minus +

gtnotin cup

sum0






Appendix

A1 Proof of Lemma 4






we have



u T u i j T

T u i j u

i

k C iki

i

k C i jki

jii i

i i

( ) ( ( ))

( ( )) ( )

( ) ( )

Γ Γ

Γ Γ

Γ Γ+ ge +

ge minus

isin isinsum sum


T Tij ji = minus



Tji = 0



(A1)

Similarly we have

(A2)






A2 Proof of Lemma 5



u T

v T

iij ji

iki

k N jji

iki

k N

i

( ( ) ( ) ) ( ( ))

( )

( )

Γ Γ

Γ

Γ


= + +

= +

isin

isin

sum

sum ε

ε2

2




v T u T T

u u i j u i j T T

u u i j T

i iki

k N jji

i i iki

k N jji

i i

( ) ( )

( ) ( ( )) ( ( ))

( ) ( ( ))

Γ Γ

Γ Γ Γ

Γ Γ

= + +

= minus + + +

= minus +

isin

isin

sum

sum

jijii

ij ji

iij ji

v i j T

v i j T

( ( ) )

( ( ) )( )

( )

+ge

minus

minus

ΓΓ







X M X Mji

ij

ij

ji = = and

∆ ∆X Xki

kj = = 0

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆

∆ ∆ ∆

u u V V X M X M

X M X M

V V M

i j i jji

ji

ki

k i jki

ij

ij

kj

k i jkj

i jki

k i j

( ) ( )

( ) ( )


+ minus + minus

= + minus

ne

ne

ne

sumsum

sum minusminusnesum

∆Mkj

k i j



A3 Proof of Lemma 6



(A3)









(A4)

u T u i j T

u T u i j T

T T u u i j

klk

l N

klk

l N

kik

kik

ik ikk k

( ) ( ( ))

( ) ( ( ))

( ) ( ( ))

Γ Γ

Γ ΓΓ Γ

+ le + prime


isin isinsum sum



primeTik


k C i jik

i

k C i jik

k k

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0


k C iik

i

k C i jik

k i

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0

u T u i j T

v T v i j T

i

k C iik

i

k C i jik

i i

i i

( ) ( ( ))

( ) ( ( ) )

( ) ( )

Γ Γ

Γ Γ


ge prime

isin isinsum sum

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ








which gives us





which is reduced to



REFERENCES






[ ( ( )) ( )] ( ) ( ( ) ( ))

u i j u T T T Th h


i j

Γ ΓΓ Γ


u i j T u T

u i j T u T

iik

k C

iik

k C

jjk

k C

jjk

i i

j

( ( )) ( )

( ( )) ( )

( ) ( )

( )

Γ Γ

Γ Γ

Γ Γ

Γ

cup minus prime

minus minus

+ cup minus prime

minus minus

isin isin

isin

sum sum

sumkk Cjisinsum

gt( )

Γ

0

primeTij


minus minus


iik

k C

iik

k Ci i

( ( )) ( ) ( ) ( )

Γ ΓΓ Γ

ε2

[ ( ( )) ( )] ( ) ( ) ( )

u i j u T T T Th h


i j

Γ ΓΓ Γ


[ ( ( )) ( )] max ( ) ( ( )) ( ) ( )

u i j u u u i jh h

h i j k C C

k k

i j

Γ Γ Γ ΓΓ Γ

















where


(12)

(13)











ki i

k i

i kki i

k i

( ) ( ) ( ) ( ) t t t t= =ne ne



ik k

k i

i kik k

k i

( ) ( ) ( ) ( ) tminus


q tki i

ki


V t ti i i i i ini


i q ji

qki

s sCk C

kii

( )equiv sum isin

t s ss

s si C

i

C ii

i( )

( )

( ) ( )[ ( ) ( )] =


gt4

3 2 1 4 2 1 20

2


sCi













t tji

ijprime = prime = 0


( ) ( ) ( )

ii ts s

iC ji


8

1 2 4

2

2

α βδ β β δ

sCi

s sC ji ( ) + ge1 2 1 2

Mki X k

i

M s qki i i k


X s q Mki k k i

ik k

ik k( ) ( ) ( ( ))t t t= =βmicro 2

X M X Mi i i iki k

ki i

k i

( ) ( ) [ ( ) ( )]t t t tminus

neminus = minussum

X ki k( )t










∆ ∆ ∆[ ( ) ( )] [ ( ) ( )] ( )


ji i

ki i

k i j

t t t t tminus



1 2444 34441 24 34

t ji

∆Mki i( ) t lt 0

Mij j( )t

M ji i( )t

∆ ∆[ ( ) ( )] [ ( ) ( ) ]X M q qji j

ji i i j i

ij j j




∆ ∆q qji i

ij j( ) ( )t tlt
















primeTij primeTji
















(15)




(i)

(ii)


M Xji i

ij i( ) ( )t t=

∆ ∆[ ( ) ( )] [ ( ) ( )]X M X Mji j

ji i

i

ij i

ij j

j


1 2444 3444 1 2444 3444

t ji ti

j

( ) τΓ Γi


jij t( ) ( )=

τΓi j( ) = 0 τΓ

i

( ) τΓ Γi

i Nisin isinG



V V X X X M M Mii j

i i i i ii j

kk N i

i kk N i

i ii j


u u V X M

V X M T

i j i iki k

k i jki i

k i j

j jkj k

k i jkj j

k i j

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

Γ Γ Γ Γ Γ

Γ Γ Γ

+ = + minus

+ + minus

ne ne

ne ne

sum sum

sum sum

τ τ τ

τ τ

∆ ∆ ∆ ∆ ∆ ∆u u V V M Mi j i jk i j k

ik i j k















( ) τΓ Γi

i Nisin isinG



( ) τΓ Γi


j



comp( ( )) Γ + = 12 1s sC i j ji

comp( ( )) Γ + ge

( ) τ Γ Γci

i Nisin isinG

τΓci ij t( ) =

( ) τ Γ Γci

i Nisin isinG


primeTij



primeTij














î i iki i

k i j

V M ( ) ( )

isin minus

ne

sumargmax t t

t ji i = ˇ t j

i

sCi sCi




























u u i j u u i j T

u i j u u i j

i i

k C i j

k kij

j j

k C i j

k

i

j

( ) ( ( )) max ( ) ( ( ))

( ( )) ( ) min ( (

( )

( )

Γ Γ Γ Γ

Γ Γ Γ

Γ

Γ

minus + minus ge

ge minus +

isin

isin

sum

sum

0

)))) ( ) minus uk Γ 0

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ




Γ Γk C C

k k

i j

u i j uisin cup

cup minus le( ) ( )

[ ( ( )) ( )] 0













( ) τ Γ Γi

i Nisin isinG

( ) τ Γ Γci

i Nisin isinG



k C C

k k

i jki

kj

k i j k C C i jik

jk

i

i j

i j

u i j u

V V M M X X

V

isin cup

ne isin cup

sum

sum sum

cup minus

= + minus + + +

= +

( ) ( )

( ( ) ( ))

[ ( ( )) ( )]

( ) ( )

Γ Γ

Γ Γ

Γ Γ

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆∆ ∆ ∆Γ Γ

V M Mj

k C Cki

kj

i j

( )

( ) ( )

minus +

gtnotin cup

sum0






Appendix

A1 Proof of Lemma 4






we have



u T u i j T

T u i j u

i

k C iki

i

k C i jki

jii i

i i

( ) ( ( ))

( ( )) ( )

( ) ( )

Γ Γ

Γ Γ

Γ Γ+ ge +

ge minus

isin isinsum sum


T Tij ji = minus



Tji = 0



(A1)

Similarly we have

(A2)






A2 Proof of Lemma 5



u T

v T

iij ji

iki

k N jji

iki

k N

i

( ( ) ( ) ) ( ( ))

( )

( )

Γ Γ

Γ

Γ


= + +

= +

isin

isin

sum

sum ε

ε2

2




v T u T T

u u i j u i j T T

u u i j T

i iki

k N jji

i i iki

k N jji

i i

( ) ( )

( ) ( ( )) ( ( ))

( ) ( ( ))

Γ Γ

Γ Γ Γ

Γ Γ

= + +

= minus + + +

= minus +

isin

isin

sum

sum

jijii

ij ji

iij ji

v i j T

v i j T

( ( ) )

( ( ) )( )

( )

+ge

minus

minus

ΓΓ







X M X Mji

ij

ij

ji = = and

∆ ∆X Xki

kj = = 0

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆

∆ ∆ ∆

u u V V X M X M

X M X M

V V M

i j i jji

ji

ki

k i jki

ij

ij

kj

k i jkj

i jki

k i j

( ) ( )

( ) ( )


+ minus + minus

= + minus

ne

ne

ne

sumsum

sum minusminusnesum

∆Mkj

k i j



A3 Proof of Lemma 6



(A3)









(A4)

u T u i j T

u T u i j T

T T u u i j

klk

l N

klk

l N

kik

kik

ik ikk k

( ) ( ( ))

( ) ( ( ))

( ) ( ( ))

Γ Γ

Γ ΓΓ Γ

+ le + prime


isin isinsum sum



primeTik


k C i jik

i

k C i jik

k k

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0


k C iik

i

k C i jik

k i

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0

u T u i j T

v T v i j T

i

k C iik

i

k C i jik

i i

i i

( ) ( ( ))

( ) ( ( ) )

( ) ( )

Γ Γ

Γ Γ


ge prime

isin isinsum sum

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ








which gives us





which is reduced to



REFERENCES






[ ( ( )) ( )] ( ) ( ( ) ( ))

u i j u T T T Th h


i j

Γ ΓΓ Γ


u i j T u T

u i j T u T

iik

k C

iik

k C

jjk

k C

jjk

i i

j

( ( )) ( )

( ( )) ( )

( ) ( )

( )

Γ Γ

Γ Γ

Γ Γ

Γ

cup minus prime

minus minus

+ cup minus prime

minus minus

isin isin

isin

sum sum

sumkk Cjisinsum

gt( )

Γ

0

primeTij


minus minus


iik

k C

iik

k Ci i

( ( )) ( ) ( ) ( )

Γ ΓΓ Γ

ε2

[ ( ( )) ( )] ( ) ( ) ( )

u i j u T T T Th h


i j

Γ ΓΓ Γ


[ ( ( )) ( )] max ( ) ( ( )) ( ) ( )

u i j u u u i jh h

h i j k C C

k k

i j

Γ Γ Γ ΓΓ Γ



























t tji

ijprime = prime = 0


( ) ( ) ( )

ii ts s

iC ji


8

1 2 4

2

2

α βδ β β δ

sCi

s sC ji ( ) + ge1 2 1 2

Mki X k

i

M s qki i i k


X s q Mki k k i

ik k

ik k( ) ( ) ( ( ))t t t= =βmicro 2

X M X Mi i i iki k

ki i

k i

( ) ( ) [ ( ) ( )]t t t tminus

neminus = minussum

X ki k( )t










∆ ∆ ∆[ ( ) ( )] [ ( ) ( )] ( )


ji i

ki i

k i j

t t t t tminus



1 2444 34441 24 34

t ji

∆Mki i( ) t lt 0

Mij j( )t

M ji i( )t

∆ ∆[ ( ) ( )] [ ( ) ( ) ]X M q qji j

ji i i j i

ij j j




∆ ∆q qji i

ij j( ) ( )t tlt
















primeTij primeTji
















(15)




(i)

(ii)


M Xji i

ij i( ) ( )t t=

∆ ∆[ ( ) ( )] [ ( ) ( )]X M X Mji j

ji i

i

ij i

ij j

j


1 2444 3444 1 2444 3444

t ji ti

j

( ) τΓ Γi


jij t( ) ( )=

τΓi j( ) = 0 τΓ

i

( ) τΓ Γi

i Nisin isinG



V V X X X M M Mii j

i i i i ii j

kk N i

i kk N i

i ii j


u u V X M

V X M T

i j i iki k

k i jki i

k i j

j jkj k

k i jkj j

k i j

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

Γ Γ Γ Γ Γ

Γ Γ Γ

+ = + minus

+ + minus

ne ne

ne ne

sum sum

sum sum

τ τ τ

τ τ

∆ ∆ ∆ ∆ ∆ ∆u u V V M Mi j i jk i j k

ik i j k















( ) τΓ Γi

i Nisin isinG



( ) τΓ Γi


j



comp( ( )) Γ + = 12 1s sC i j ji

comp( ( )) Γ + ge

( ) τ Γ Γci

i Nisin isinG

τΓci ij t( ) =

( ) τ Γ Γci

i Nisin isinG


primeTij



primeTij














î i iki i

k i j

V M ( ) ( )

isin minus

ne

sumargmax t t

t ji i = ˇ t j

i

sCi sCi




























u u i j u u i j T

u i j u u i j

i i

k C i j

k kij

j j

k C i j

k

i

j

( ) ( ( )) max ( ) ( ( ))

( ( )) ( ) min ( (

( )

( )

Γ Γ Γ Γ

Γ Γ Γ

Γ

Γ

minus + minus ge

ge minus +

isin

isin

sum

sum

0

)))) ( ) minus uk Γ 0

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ




Γ Γk C C

k k

i j

u i j uisin cup

cup minus le( ) ( )

[ ( ( )) ( )] 0













( ) τ Γ Γi

i Nisin isinG

( ) τ Γ Γci

i Nisin isinG



k C C

k k

i jki

kj

k i j k C C i jik

jk

i

i j

i j

u i j u

V V M M X X

V

isin cup

ne isin cup

sum

sum sum

cup minus

= + minus + + +

= +

( ) ( )

( ( ) ( ))

[ ( ( )) ( )]

( ) ( )

Γ Γ

Γ Γ

Γ Γ

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆∆ ∆ ∆Γ Γ

V M Mj

k C Cki

kj

i j

( )

( ) ( )

minus +

gtnotin cup

sum0






Appendix

A1 Proof of Lemma 4






we have



u T u i j T

T u i j u

i

k C iki

i

k C i jki

jii i

i i

( ) ( ( ))

( ( )) ( )

( ) ( )

Γ Γ

Γ Γ

Γ Γ+ ge +

ge minus

isin isinsum sum


T Tij ji = minus



Tji = 0



(A1)

Similarly we have

(A2)






A2 Proof of Lemma 5



u T

v T

iij ji

iki

k N jji

iki

k N

i

( ( ) ( ) ) ( ( ))

( )

( )

Γ Γ

Γ

Γ


= + +

= +

isin

isin

sum

sum ε

ε2

2




v T u T T

u u i j u i j T T

u u i j T

i iki

k N jji

i i iki

k N jji

i i

( ) ( )

( ) ( ( )) ( ( ))

( ) ( ( ))

Γ Γ

Γ Γ Γ

Γ Γ

= + +

= minus + + +

= minus +

isin

isin

sum

sum

jijii

ij ji

iij ji

v i j T

v i j T

( ( ) )

( ( ) )( )

( )

+ge

minus

minus

ΓΓ







X M X Mji

ij

ij

ji = = and

∆ ∆X Xki

kj = = 0

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆

∆ ∆ ∆

u u V V X M X M

X M X M

V V M

i j i jji

ji

ki

k i jki

ij

ij

kj

k i jkj

i jki

k i j

( ) ( )

( ) ( )


+ minus + minus

= + minus

ne

ne

ne

sumsum

sum minusminusnesum

∆Mkj

k i j



A3 Proof of Lemma 6



(A3)









(A4)

u T u i j T

u T u i j T

T T u u i j

klk

l N

klk

l N

kik

kik

ik ikk k

( ) ( ( ))

( ) ( ( ))

( ) ( ( ))

Γ Γ

Γ ΓΓ Γ

+ le + prime


isin isinsum sum



primeTik


k C i jik

i

k C i jik

k k

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0


k C iik

i

k C i jik

k i

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0

u T u i j T

v T v i j T

i

k C iik

i

k C i jik

i i

i i

( ) ( ( ))

( ) ( ( ) )

( ) ( )

Γ Γ

Γ Γ


ge prime

isin isinsum sum

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ








which gives us





which is reduced to



REFERENCES






[ ( ( )) ( )] ( ) ( ( ) ( ))

u i j u T T T Th h


i j

Γ ΓΓ Γ


u i j T u T

u i j T u T

iik

k C

iik

k C

jjk

k C

jjk

i i

j

( ( )) ( )

( ( )) ( )

( ) ( )

( )

Γ Γ

Γ Γ

Γ Γ

Γ

cup minus prime

minus minus

+ cup minus prime

minus minus

isin isin

isin

sum sum

sumkk Cjisinsum

gt( )

Γ

0

primeTij


minus minus


iik

k C

iik

k Ci i

( ( )) ( ) ( ) ( )

Γ ΓΓ Γ

ε2

[ ( ( )) ( )] ( ) ( ) ( )

u i j u T T T Th h


i j

Γ ΓΓ Γ


[ ( ( )) ( )] max ( ) ( ( )) ( ) ( )

u i j u u u i jh h

h i j k C C

k k

i j

Γ Γ Γ ΓΓ Γ
























∆ ∆ ∆[ ( ) ( )] [ ( ) ( )] ( )


ji i

ki i

k i j

t t t t tminus



1 2444 34441 24 34

t ji

∆Mki i( ) t lt 0

Mij j( )t

M ji i( )t

∆ ∆[ ( ) ( )] [ ( ) ( ) ]X M q qji j

ji i i j i

ij j j




∆ ∆q qji i

ij j( ) ( )t tlt
















primeTij primeTji
















(15)




(i)

(ii)


M Xji i

ij i( ) ( )t t=

∆ ∆[ ( ) ( )] [ ( ) ( )]X M X Mji j

ji i

i

ij i

ij j

j


1 2444 3444 1 2444 3444

t ji ti

j

( ) τΓ Γi


jij t( ) ( )=

τΓi j( ) = 0 τΓ

i

( ) τΓ Γi

i Nisin isinG



V V X X X M M Mii j

i i i i ii j

kk N i

i kk N i

i ii j


u u V X M

V X M T

i j i iki k

k i jki i

k i j

j jkj k

k i jkj j

k i j

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

Γ Γ Γ Γ Γ

Γ Γ Γ

+ = + minus

+ + minus

ne ne

ne ne

sum sum

sum sum

τ τ τ

τ τ

∆ ∆ ∆ ∆ ∆ ∆u u V V M Mi j i jk i j k

ik i j k















( ) τΓ Γi

i Nisin isinG



( ) τΓ Γi


j



comp( ( )) Γ + = 12 1s sC i j ji

comp( ( )) Γ + ge

( ) τ Γ Γci

i Nisin isinG

τΓci ij t( ) =

( ) τ Γ Γci

i Nisin isinG


primeTij



primeTij














î i iki i

k i j

V M ( ) ( )

isin minus

ne

sumargmax t t

t ji i = ˇ t j

i

sCi sCi




























u u i j u u i j T

u i j u u i j

i i

k C i j

k kij

j j

k C i j

k

i

j

( ) ( ( )) max ( ) ( ( ))

( ( )) ( ) min ( (

( )

( )

Γ Γ Γ Γ

Γ Γ Γ

Γ

Γ

minus + minus ge

ge minus +

isin

isin

sum

sum

0

)))) ( ) minus uk Γ 0

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ




Γ Γk C C

k k

i j

u i j uisin cup

cup minus le( ) ( )

[ ( ( )) ( )] 0













( ) τ Γ Γi

i Nisin isinG

( ) τ Γ Γci

i Nisin isinG



k C C

k k

i jki

kj

k i j k C C i jik

jk

i

i j

i j

u i j u

V V M M X X

V

isin cup

ne isin cup

sum

sum sum

cup minus

= + minus + + +

= +

( ) ( )

( ( ) ( ))

[ ( ( )) ( )]

( ) ( )

Γ Γ

Γ Γ

Γ Γ

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆∆ ∆ ∆Γ Γ

V M Mj

k C Cki

kj

i j

( )

( ) ( )

minus +

gtnotin cup

sum0






Appendix

A1 Proof of Lemma 4






we have



u T u i j T

T u i j u

i

k C iki

i

k C i jki

jii i

i i

( ) ( ( ))

( ( )) ( )

( ) ( )

Γ Γ

Γ Γ

Γ Γ+ ge +

ge minus

isin isinsum sum


T Tij ji = minus



Tji = 0



(A1)

Similarly we have

(A2)






A2 Proof of Lemma 5



u T

v T

iij ji

iki

k N jji

iki

k N

i

( ( ) ( ) ) ( ( ))

( )

( )

Γ Γ

Γ

Γ


= + +

= +

isin

isin

sum

sum ε

ε2

2




v T u T T

u u i j u i j T T

u u i j T

i iki

k N jji

i i iki

k N jji

i i

( ) ( )

( ) ( ( )) ( ( ))

( ) ( ( ))

Γ Γ

Γ Γ Γ

Γ Γ

= + +

= minus + + +

= minus +

isin

isin

sum

sum

jijii

ij ji

iij ji

v i j T

v i j T

( ( ) )

( ( ) )( )

( )

+ge

minus

minus

ΓΓ







X M X Mji

ij

ij

ji = = and

∆ ∆X Xki

kj = = 0

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆

∆ ∆ ∆

u u V V X M X M

X M X M

V V M

i j i jji

ji

ki

k i jki

ij

ij

kj

k i jkj

i jki

k i j

( ) ( )

( ) ( )


+ minus + minus

= + minus

ne

ne

ne

sumsum

sum minusminusnesum

∆Mkj

k i j



A3 Proof of Lemma 6



(A3)









(A4)

u T u i j T

u T u i j T

T T u u i j

klk

l N

klk

l N

kik

kik

ik ikk k

( ) ( ( ))

( ) ( ( ))

( ) ( ( ))

Γ Γ

Γ ΓΓ Γ

+ le + prime


isin isinsum sum



primeTik


k C i jik

i

k C i jik

k k

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0


k C iik

i

k C i jik

k i

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0

u T u i j T

v T v i j T

i

k C iik

i

k C i jik

i i

i i

( ) ( ( ))

( ) ( ( ) )

( ) ( )

Γ Γ

Γ Γ


ge prime

isin isinsum sum

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ








which gives us





which is reduced to



REFERENCES






[ ( ( )) ( )] ( ) ( ( ) ( ))

u i j u T T T Th h


i j

Γ ΓΓ Γ


u i j T u T

u i j T u T

iik

k C

iik

k C

jjk

k C

jjk

i i

j

( ( )) ( )

( ( )) ( )

( ) ( )

( )

Γ Γ

Γ Γ

Γ Γ

Γ

cup minus prime

minus minus

+ cup minus prime

minus minus

isin isin

isin

sum sum

sumkk Cjisinsum

gt( )

Γ

0

primeTij


minus minus


iik

k C

iik

k Ci i

( ( )) ( ) ( ) ( )

Γ ΓΓ Γ

ε2

[ ( ( )) ( )] ( ) ( ) ( )

u i j u T T T Th h


i j

Γ ΓΓ Γ


[ ( ( )) ( )] max ( ) ( ( )) ( ) ( )

u i j u u u i jh h

h i j k C C

k k

i j

Γ Γ Γ ΓΓ Γ






























primeTij primeTji
















(15)




(i)

(ii)


M Xji i

ij i( ) ( )t t=

∆ ∆[ ( ) ( )] [ ( ) ( )]X M X Mji j

ji i

i

ij i

ij j

j


1 2444 3444 1 2444 3444

t ji ti

j

( ) τΓ Γi


jij t( ) ( )=

τΓi j( ) = 0 τΓ

i

( ) τΓ Γi

i Nisin isinG



V V X X X M M Mii j

i i i i ii j

kk N i

i kk N i

i ii j


u u V X M

V X M T

i j i iki k

k i jki i

k i j

j jkj k

k i jkj j

k i j

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

Γ Γ Γ Γ Γ

Γ Γ Γ

+ = + minus

+ + minus

ne ne

ne ne

sum sum

sum sum

τ τ τ

τ τ

∆ ∆ ∆ ∆ ∆ ∆u u V V M Mi j i jk i j k

ik i j k















( ) τΓ Γi

i Nisin isinG



( ) τΓ Γi


j



comp( ( )) Γ + = 12 1s sC i j ji

comp( ( )) Γ + ge

( ) τ Γ Γci

i Nisin isinG

τΓci ij t( ) =

( ) τ Γ Γci

i Nisin isinG


primeTij



primeTij














î i iki i

k i j

V M ( ) ( )

isin minus

ne

sumargmax t t

t ji i = ˇ t j

i

sCi sCi




























u u i j u u i j T

u i j u u i j

i i

k C i j

k kij

j j

k C i j

k

i

j

( ) ( ( )) max ( ) ( ( ))

( ( )) ( ) min ( (

( )

( )

Γ Γ Γ Γ

Γ Γ Γ

Γ

Γ

minus + minus ge

ge minus +

isin

isin

sum

sum

0

)))) ( ) minus uk Γ 0

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ




Γ Γk C C

k k

i j

u i j uisin cup

cup minus le( ) ( )

[ ( ( )) ( )] 0













( ) τ Γ Γi

i Nisin isinG

( ) τ Γ Γci

i Nisin isinG



k C C

k k

i jki

kj

k i j k C C i jik

jk

i

i j

i j

u i j u

V V M M X X

V

isin cup

ne isin cup

sum

sum sum

cup minus

= + minus + + +

= +

( ) ( )

( ( ) ( ))

[ ( ( )) ( )]

( ) ( )

Γ Γ

Γ Γ

Γ Γ

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆∆ ∆ ∆Γ Γ

V M Mj

k C Cki

kj

i j

( )

( ) ( )

minus +

gtnotin cup

sum0






Appendix

A1 Proof of Lemma 4






we have



u T u i j T

T u i j u

i

k C iki

i

k C i jki

jii i

i i

( ) ( ( ))

( ( )) ( )

( ) ( )

Γ Γ

Γ Γ

Γ Γ+ ge +

ge minus

isin isinsum sum


T Tij ji = minus



Tji = 0



(A1)

Similarly we have

(A2)






A2 Proof of Lemma 5



u T

v T

iij ji

iki

k N jji

iki

k N

i

( ( ) ( ) ) ( ( ))

( )

( )

Γ Γ

Γ

Γ


= + +

= +

isin

isin

sum

sum ε

ε2

2




v T u T T

u u i j u i j T T

u u i j T

i iki

k N jji

i i iki

k N jji

i i

( ) ( )

( ) ( ( )) ( ( ))

( ) ( ( ))

Γ Γ

Γ Γ Γ

Γ Γ

= + +

= minus + + +

= minus +

isin

isin

sum

sum

jijii

ij ji

iij ji

v i j T

v i j T

( ( ) )

( ( ) )( )

( )

+ge

minus

minus

ΓΓ







X M X Mji

ij

ij

ji = = and

∆ ∆X Xki

kj = = 0

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆

∆ ∆ ∆

u u V V X M X M

X M X M

V V M

i j i jji

ji

ki

k i jki

ij

ij

kj

k i jkj

i jki

k i j

( ) ( )

( ) ( )


+ minus + minus

= + minus

ne

ne

ne

sumsum

sum minusminusnesum

∆Mkj

k i j



A3 Proof of Lemma 6



(A3)









(A4)

u T u i j T

u T u i j T

T T u u i j

klk

l N

klk

l N

kik

kik

ik ikk k

( ) ( ( ))

( ) ( ( ))

( ) ( ( ))

Γ Γ

Γ ΓΓ Γ

+ le + prime


isin isinsum sum



primeTik


k C i jik

i

k C i jik

k k

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0


k C iik

i

k C i jik

k i

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0

u T u i j T

v T v i j T

i

k C iik

i

k C i jik

i i

i i

( ) ( ( ))

( ) ( ( ) )

( ) ( )

Γ Γ

Γ Γ


ge prime

isin isinsum sum

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ








which gives us





which is reduced to



REFERENCES






[ ( ( )) ( )] ( ) ( ( ) ( ))

u i j u T T T Th h


i j

Γ ΓΓ Γ


u i j T u T

u i j T u T

iik

k C

iik

k C

jjk

k C

jjk

i i

j

( ( )) ( )

( ( )) ( )

( ) ( )

( )

Γ Γ

Γ Γ

Γ Γ

Γ

cup minus prime

minus minus

+ cup minus prime

minus minus

isin isin

isin

sum sum

sumkk Cjisinsum

gt( )

Γ

0

primeTij


minus minus


iik

k C

iik

k Ci i

( ( )) ( ) ( ) ( )

Γ ΓΓ Γ

ε2

[ ( ( )) ( )] ( ) ( ) ( )

u i j u T T T Th h


i j

Γ ΓΓ Γ


[ ( ( )) ( )] max ( ) ( ( )) ( ) ( )

u i j u u u i jh h

h i j k C C

k k

i j

Γ Γ Γ ΓΓ Γ
























(15)




(i)

(ii)


M Xji i

ij i( ) ( )t t=

∆ ∆[ ( ) ( )] [ ( ) ( )]X M X Mji j

ji i

i

ij i

ij j

j


1 2444 3444 1 2444 3444

t ji ti

j

( ) τΓ Γi


jij t( ) ( )=

τΓi j( ) = 0 τΓ

i

( ) τΓ Γi

i Nisin isinG



V V X X X M M Mii j

i i i i ii j

kk N i

i kk N i

i ii j


u u V X M

V X M T

i j i iki k

k i jki i

k i j

j jkj k

k i jkj j

k i j

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

Γ Γ Γ Γ Γ

Γ Γ Γ

+ = + minus

+ + minus

ne ne

ne ne

sum sum

sum sum

τ τ τ

τ τ

∆ ∆ ∆ ∆ ∆ ∆u u V V M Mi j i jk i j k

ik i j k















( ) τΓ Γi

i Nisin isinG



( ) τΓ Γi


j



comp( ( )) Γ + = 12 1s sC i j ji

comp( ( )) Γ + ge

( ) τ Γ Γci

i Nisin isinG

τΓci ij t( ) =

( ) τ Γ Γci

i Nisin isinG


primeTij



primeTij














î i iki i

k i j

V M ( ) ( )

isin minus

ne

sumargmax t t

t ji i = ˇ t j

i

sCi sCi




























u u i j u u i j T

u i j u u i j

i i

k C i j

k kij

j j

k C i j

k

i

j

( ) ( ( )) max ( ) ( ( ))

( ( )) ( ) min ( (

( )

( )

Γ Γ Γ Γ

Γ Γ Γ

Γ

Γ

minus + minus ge

ge minus +

isin

isin

sum

sum

0

)))) ( ) minus uk Γ 0

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ




Γ Γk C C

k k

i j

u i j uisin cup

cup minus le( ) ( )

[ ( ( )) ( )] 0













( ) τ Γ Γi

i Nisin isinG

( ) τ Γ Γci

i Nisin isinG



k C C

k k

i jki

kj

k i j k C C i jik

jk

i

i j

i j

u i j u

V V M M X X

V

isin cup

ne isin cup

sum

sum sum

cup minus

= + minus + + +

= +

( ) ( )

( ( ) ( ))

[ ( ( )) ( )]

( ) ( )

Γ Γ

Γ Γ

Γ Γ

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆∆ ∆ ∆Γ Γ

V M Mj

k C Cki

kj

i j

( )

( ) ( )

minus +

gtnotin cup

sum0






Appendix

A1 Proof of Lemma 4






we have



u T u i j T

T u i j u

i

k C iki

i

k C i jki

jii i

i i

( ) ( ( ))

( ( )) ( )

( ) ( )

Γ Γ

Γ Γ

Γ Γ+ ge +

ge minus

isin isinsum sum


T Tij ji = minus



Tji = 0



(A1)

Similarly we have

(A2)






A2 Proof of Lemma 5



u T

v T

iij ji

iki

k N jji

iki

k N

i

( ( ) ( ) ) ( ( ))

( )

( )

Γ Γ

Γ

Γ


= + +

= +

isin

isin

sum

sum ε

ε2

2




v T u T T

u u i j u i j T T

u u i j T

i iki

k N jji

i i iki

k N jji

i i

( ) ( )

( ) ( ( )) ( ( ))

( ) ( ( ))

Γ Γ

Γ Γ Γ

Γ Γ

= + +

= minus + + +

= minus +

isin

isin

sum

sum

jijii

ij ji

iij ji

v i j T

v i j T

( ( ) )

( ( ) )( )

( )

+ge

minus

minus

ΓΓ







X M X Mji

ij

ij

ji = = and

∆ ∆X Xki

kj = = 0

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆

∆ ∆ ∆

u u V V X M X M

X M X M

V V M

i j i jji

ji

ki

k i jki

ij

ij

kj

k i jkj

i jki

k i j

( ) ( )

( ) ( )


+ minus + minus

= + minus

ne

ne

ne

sumsum

sum minusminusnesum

∆Mkj

k i j



A3 Proof of Lemma 6



(A3)









(A4)

u T u i j T

u T u i j T

T T u u i j

klk

l N

klk

l N

kik

kik

ik ikk k

( ) ( ( ))

( ) ( ( ))

( ) ( ( ))

Γ Γ

Γ ΓΓ Γ

+ le + prime


isin isinsum sum



primeTik


k C i jik

i

k C i jik

k k

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0


k C iik

i

k C i jik

k i

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0

u T u i j T

v T v i j T

i

k C iik

i

k C i jik

i i

i i

( ) ( ( ))

( ) ( ( ) )

( ) ( )

Γ Γ

Γ Γ


ge prime

isin isinsum sum

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ








which gives us





which is reduced to



REFERENCES






[ ( ( )) ( )] ( ) ( ( ) ( ))

u i j u T T T Th h


i j

Γ ΓΓ Γ


u i j T u T

u i j T u T

iik

k C

iik

k C

jjk

k C

jjk

i i

j

( ( )) ( )

( ( )) ( )

( ) ( )

( )

Γ Γ

Γ Γ

Γ Γ

Γ

cup minus prime

minus minus

+ cup minus prime

minus minus

isin isin

isin

sum sum

sumkk Cjisinsum

gt( )

Γ

0

primeTij


minus minus


iik

k C

iik

k Ci i

( ( )) ( ) ( ) ( )

Γ ΓΓ Γ

ε2

[ ( ( )) ( )] ( ) ( ) ( )

u i j u T T T Th h


i j

Γ ΓΓ Γ


[ ( ( )) ( )] max ( ) ( ( )) ( ) ( )

u i j u u u i jh h

h i j k C C

k k

i j

Γ Γ Γ ΓΓ Γ




























( ) τΓ Γi

i Nisin isinG



( ) τΓ Γi


j



comp( ( )) Γ + = 12 1s sC i j ji

comp( ( )) Γ + ge

( ) τ Γ Γci

i Nisin isinG

τΓci ij t( ) =

( ) τ Γ Γci

i Nisin isinG


primeTij



primeTij














î i iki i

k i j

V M ( ) ( )

isin minus

ne

sumargmax t t

t ji i = ˇ t j

i

sCi sCi




























u u i j u u i j T

u i j u u i j

i i

k C i j

k kij

j j

k C i j

k

i

j

( ) ( ( )) max ( ) ( ( ))

( ( )) ( ) min ( (

( )

( )

Γ Γ Γ Γ

Γ Γ Γ

Γ

Γ

minus + minus ge

ge minus +

isin

isin

sum

sum

0

)))) ( ) minus uk Γ 0

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ




Γ Γk C C

k k

i j

u i j uisin cup

cup minus le( ) ( )

[ ( ( )) ( )] 0













( ) τ Γ Γi

i Nisin isinG

( ) τ Γ Γci

i Nisin isinG



k C C

k k

i jki

kj

k i j k C C i jik

jk

i

i j

i j

u i j u

V V M M X X

V

isin cup

ne isin cup

sum

sum sum

cup minus

= + minus + + +

= +

( ) ( )

( ( ) ( ))

[ ( ( )) ( )]

( ) ( )

Γ Γ

Γ Γ

Γ Γ

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆∆ ∆ ∆Γ Γ

V M Mj

k C Cki

kj

i j

( )

( ) ( )

minus +

gtnotin cup

sum0






Appendix

A1 Proof of Lemma 4






we have



u T u i j T

T u i j u

i

k C iki

i

k C i jki

jii i

i i

( ) ( ( ))

( ( )) ( )

( ) ( )

Γ Γ

Γ Γ

Γ Γ+ ge +

ge minus

isin isinsum sum


T Tij ji = minus



Tji = 0



(A1)

Similarly we have

(A2)






A2 Proof of Lemma 5



u T

v T

iij ji

iki

k N jji

iki

k N

i

( ( ) ( ) ) ( ( ))

( )

( )

Γ Γ

Γ

Γ


= + +

= +

isin

isin

sum

sum ε

ε2

2




v T u T T

u u i j u i j T T

u u i j T

i iki

k N jji

i i iki

k N jji

i i

( ) ( )

( ) ( ( )) ( ( ))

( ) ( ( ))

Γ Γ

Γ Γ Γ

Γ Γ

= + +

= minus + + +

= minus +

isin

isin

sum

sum

jijii

ij ji

iij ji

v i j T

v i j T

( ( ) )

( ( ) )( )

( )

+ge

minus

minus

ΓΓ







X M X Mji

ij

ij

ji = = and

∆ ∆X Xki

kj = = 0

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆

∆ ∆ ∆

u u V V X M X M

X M X M

V V M

i j i jji

ji

ki

k i jki

ij

ij

kj

k i jkj

i jki

k i j

( ) ( )

( ) ( )


+ minus + minus

= + minus

ne

ne

ne

sumsum

sum minusminusnesum

∆Mkj

k i j



A3 Proof of Lemma 6



(A3)









(A4)

u T u i j T

u T u i j T

T T u u i j

klk

l N

klk

l N

kik

kik

ik ikk k

( ) ( ( ))

( ) ( ( ))

( ) ( ( ))

Γ Γ

Γ ΓΓ Γ

+ le + prime


isin isinsum sum



primeTik


k C i jik

i

k C i jik

k k

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0


k C iik

i

k C i jik

k i

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0

u T u i j T

v T v i j T

i

k C iik

i

k C i jik

i i

i i

( ) ( ( ))

( ) ( ( ) )

( ) ( )

Γ Γ

Γ Γ


ge prime

isin isinsum sum

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ








which gives us





which is reduced to



REFERENCES






[ ( ( )) ( )] ( ) ( ( ) ( ))

u i j u T T T Th h


i j

Γ ΓΓ Γ


u i j T u T

u i j T u T

iik

k C

iik

k C

jjk

k C

jjk

i i

j

( ( )) ( )

( ( )) ( )

( ) ( )

( )

Γ Γ

Γ Γ

Γ Γ

Γ

cup minus prime

minus minus

+ cup minus prime

minus minus

isin isin

isin

sum sum

sumkk Cjisinsum

gt( )

Γ

0

primeTij


minus minus


iik

k C

iik

k Ci i

( ( )) ( ) ( ) ( )

Γ ΓΓ Γ

ε2

[ ( ( )) ( )] ( ) ( ) ( )

u i j u T T T Th h


i j

Γ ΓΓ Γ


[ ( ( )) ( )] max ( ) ( ( )) ( ) ( )

u i j u u u i jh h

h i j k C C

k k

i j

Γ Γ Γ ΓΓ Γ


























î i iki i

k i j

V M ( ) ( )

isin minus

ne

sumargmax t t

t ji i = ˇ t j

i

sCi sCi




























u u i j u u i j T

u i j u u i j

i i

k C i j

k kij

j j

k C i j

k

i

j

( ) ( ( )) max ( ) ( ( ))

( ( )) ( ) min ( (

( )

( )

Γ Γ Γ Γ

Γ Γ Γ

Γ

Γ

minus + minus ge

ge minus +

isin

isin

sum

sum

0

)))) ( ) minus uk Γ 0

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ




Γ Γk C C

k k

i j

u i j uisin cup

cup minus le( ) ( )

[ ( ( )) ( )] 0













( ) τ Γ Γi

i Nisin isinG

( ) τ Γ Γci

i Nisin isinG



k C C

k k

i jki

kj

k i j k C C i jik

jk

i

i j

i j

u i j u

V V M M X X

V

isin cup

ne isin cup

sum

sum sum

cup minus

= + minus + + +

= +

( ) ( )

( ( ) ( ))

[ ( ( )) ( )]

( ) ( )

Γ Γ

Γ Γ

Γ Γ

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆∆ ∆ ∆Γ Γ

V M Mj

k C Cki

kj

i j

( )

( ) ( )

minus +

gtnotin cup

sum0






Appendix

A1 Proof of Lemma 4






we have



u T u i j T

T u i j u

i

k C iki

i

k C i jki

jii i

i i

( ) ( ( ))

( ( )) ( )

( ) ( )

Γ Γ

Γ Γ

Γ Γ+ ge +

ge minus

isin isinsum sum


T Tij ji = minus



Tji = 0



(A1)

Similarly we have

(A2)






A2 Proof of Lemma 5



u T

v T

iij ji

iki

k N jji

iki

k N

i

( ( ) ( ) ) ( ( ))

( )

( )

Γ Γ

Γ

Γ


= + +

= +

isin

isin

sum

sum ε

ε2

2




v T u T T

u u i j u i j T T

u u i j T

i iki

k N jji

i i iki

k N jji

i i

( ) ( )

( ) ( ( )) ( ( ))

( ) ( ( ))

Γ Γ

Γ Γ Γ

Γ Γ

= + +

= minus + + +

= minus +

isin

isin

sum

sum

jijii

ij ji

iij ji

v i j T

v i j T

( ( ) )

( ( ) )( )

( )

+ge

minus

minus

ΓΓ







X M X Mji

ij

ij

ji = = and

∆ ∆X Xki

kj = = 0

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆

∆ ∆ ∆

u u V V X M X M

X M X M

V V M

i j i jji

ji

ki

k i jki

ij

ij

kj

k i jkj

i jki

k i j

( ) ( )

( ) ( )


+ minus + minus

= + minus

ne

ne

ne

sumsum

sum minusminusnesum

∆Mkj

k i j



A3 Proof of Lemma 6



(A3)









(A4)

u T u i j T

u T u i j T

T T u u i j

klk

l N

klk

l N

kik

kik

ik ikk k

( ) ( ( ))

( ) ( ( ))

( ) ( ( ))

Γ Γ

Γ ΓΓ Γ

+ le + prime


isin isinsum sum



primeTik


k C i jik

i

k C i jik

k k

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0


k C iik

i

k C i jik

k i

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0

u T u i j T

v T v i j T

i

k C iik

i

k C i jik

i i

i i

( ) ( ( ))

( ) ( ( ) )

( ) ( )

Γ Γ

Γ Γ


ge prime

isin isinsum sum

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ








which gives us





which is reduced to



REFERENCES






[ ( ( )) ( )] ( ) ( ( ) ( ))

u i j u T T T Th h


i j

Γ ΓΓ Γ


u i j T u T

u i j T u T

iik

k C

iik

k C

jjk

k C

jjk

i i

j

( ( )) ( )

( ( )) ( )

( ) ( )

( )

Γ Γ

Γ Γ

Γ Γ

Γ

cup minus prime

minus minus

+ cup minus prime

minus minus

isin isin

isin

sum sum

sumkk Cjisinsum

gt( )

Γ

0

primeTij


minus minus


iik

k C

iik

k Ci i

( ( )) ( ) ( ) ( )

Γ ΓΓ Γ

ε2

[ ( ( )) ( )] ( ) ( ) ( )

u i j u T T T Th h


i j

Γ ΓΓ Γ


[ ( ( )) ( )] max ( ) ( ( )) ( ) ( )

u i j u u u i jh h

h i j k C C

k k

i j

Γ Γ Γ ΓΓ Γ










































u u i j u u i j T

u i j u u i j

i i

k C i j

k kij

j j

k C i j

k

i

j

( ) ( ( )) max ( ) ( ( ))

( ( )) ( ) min ( (

( )

( )

Γ Γ Γ Γ

Γ Γ Γ

Γ

Γ

minus + minus ge

ge minus +

isin

isin

sum

sum

0

)))) ( ) minus uk Γ 0

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ




Γ Γk C C

k k

i j

u i j uisin cup

cup minus le( ) ( )

[ ( ( )) ( )] 0













( ) τ Γ Γi

i Nisin isinG

( ) τ Γ Γci

i Nisin isinG



k C C

k k

i jki

kj

k i j k C C i jik

jk

i

i j

i j

u i j u

V V M M X X

V

isin cup

ne isin cup

sum

sum sum

cup minus

= + minus + + +

= +

( ) ( )

( ( ) ( ))

[ ( ( )) ( )]

( ) ( )

Γ Γ

Γ Γ

Γ Γ

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆∆ ∆ ∆Γ Γ

V M Mj

k C Cki

kj

i j

( )

( ) ( )

minus +

gtnotin cup

sum0






Appendix

A1 Proof of Lemma 4






we have



u T u i j T

T u i j u

i

k C iki

i

k C i jki

jii i

i i

( ) ( ( ))

( ( )) ( )

( ) ( )

Γ Γ

Γ Γ

Γ Γ+ ge +

ge minus

isin isinsum sum


T Tij ji = minus



Tji = 0



(A1)

Similarly we have

(A2)






A2 Proof of Lemma 5



u T

v T

iij ji

iki

k N jji

iki

k N

i

( ( ) ( ) ) ( ( ))

( )

( )

Γ Γ

Γ

Γ


= + +

= +

isin

isin

sum

sum ε

ε2

2




v T u T T

u u i j u i j T T

u u i j T

i iki

k N jji

i i iki

k N jji

i i

( ) ( )

( ) ( ( )) ( ( ))

( ) ( ( ))

Γ Γ

Γ Γ Γ

Γ Γ

= + +

= minus + + +

= minus +

isin

isin

sum

sum

jijii

ij ji

iij ji

v i j T

v i j T

( ( ) )

( ( ) )( )

( )

+ge

minus

minus

ΓΓ







X M X Mji

ij

ij

ji = = and

∆ ∆X Xki

kj = = 0

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆

∆ ∆ ∆

u u V V X M X M

X M X M

V V M

i j i jji

ji

ki

k i jki

ij

ij

kj

k i jkj

i jki

k i j

( ) ( )

( ) ( )


+ minus + minus

= + minus

ne

ne

ne

sumsum

sum minusminusnesum

∆Mkj

k i j



A3 Proof of Lemma 6



(A3)









(A4)

u T u i j T

u T u i j T

T T u u i j

klk

l N

klk

l N

kik

kik

ik ikk k

( ) ( ( ))

( ) ( ( ))

( ) ( ( ))

Γ Γ

Γ ΓΓ Γ

+ le + prime


isin isinsum sum



primeTik


k C i jik

i

k C i jik

k k

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0


k C iik

i

k C i jik

k i

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0

u T u i j T

v T v i j T

i

k C iik

i

k C i jik

i i

i i

( ) ( ( ))

( ) ( ( ) )

( ) ( )

Γ Γ

Γ Γ


ge prime

isin isinsum sum

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ








which gives us





which is reduced to



REFERENCES






[ ( ( )) ( )] ( ) ( ( ) ( ))

u i j u T T T Th h


i j

Γ ΓΓ Γ


u i j T u T

u i j T u T

iik

k C

iik

k C

jjk

k C

jjk

i i

j

( ( )) ( )

( ( )) ( )

( ) ( )

( )

Γ Γ

Γ Γ

Γ Γ

Γ

cup minus prime

minus minus

+ cup minus prime

minus minus

isin isin

isin

sum sum

sumkk Cjisinsum

gt( )

Γ

0

primeTij


minus minus


iik

k C

iik

k Ci i

( ( )) ( ) ( ) ( )

Γ ΓΓ Γ

ε2

[ ( ( )) ( )] ( ) ( ) ( )

u i j u T T T Th h


i j

Γ ΓΓ Γ


[ ( ( )) ( )] max ( ) ( ( )) ( ) ( )

u i j u u u i jh h

h i j k C C

k k

i j

Γ Γ Γ ΓΓ Γ



























u u i j u u i j T

u i j u u i j

i i

k C i j

k kij

j j

k C i j

k

i

j

( ) ( ( )) max ( ) ( ( ))

( ( )) ( ) min ( (

( )

( )

Γ Γ Γ Γ

Γ Γ Γ

Γ

Γ

minus + minus ge

ge minus +

isin

isin

sum

sum

0

)))) ( ) minus uk Γ 0

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ




Γ Γk C C

k k

i j

u i j uisin cup

cup minus le( ) ( )

[ ( ( )) ( )] 0













( ) τ Γ Γi

i Nisin isinG

( ) τ Γ Γci

i Nisin isinG



k C C

k k

i jki

kj

k i j k C C i jik

jk

i

i j

i j

u i j u

V V M M X X

V

isin cup

ne isin cup

sum

sum sum

cup minus

= + minus + + +

= +

( ) ( )

( ( ) ( ))

[ ( ( )) ( )]

( ) ( )

Γ Γ

Γ Γ

Γ Γ

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆∆ ∆ ∆Γ Γ

V M Mj

k C Cki

kj

i j

( )

( ) ( )

minus +

gtnotin cup

sum0






Appendix

A1 Proof of Lemma 4






we have



u T u i j T

T u i j u

i

k C iki

i

k C i jki

jii i

i i

( ) ( ( ))

( ( )) ( )

( ) ( )

Γ Γ

Γ Γ

Γ Γ+ ge +

ge minus

isin isinsum sum


T Tij ji = minus



Tji = 0



(A1)

Similarly we have

(A2)






A2 Proof of Lemma 5



u T

v T

iij ji

iki

k N jji

iki

k N

i

( ( ) ( ) ) ( ( ))

( )

( )

Γ Γ

Γ

Γ


= + +

= +

isin

isin

sum

sum ε

ε2

2




v T u T T

u u i j u i j T T

u u i j T

i iki

k N jji

i i iki

k N jji

i i

( ) ( )

( ) ( ( )) ( ( ))

( ) ( ( ))

Γ Γ

Γ Γ Γ

Γ Γ

= + +

= minus + + +

= minus +

isin

isin

sum

sum

jijii

ij ji

iij ji

v i j T

v i j T

( ( ) )

( ( ) )( )

( )

+ge

minus

minus

ΓΓ







X M X Mji

ij

ij

ji = = and

∆ ∆X Xki

kj = = 0

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆

∆ ∆ ∆

u u V V X M X M

X M X M

V V M

i j i jji

ji

ki

k i jki

ij

ij

kj

k i jkj

i jki

k i j

( ) ( )

( ) ( )


+ minus + minus

= + minus

ne

ne

ne

sumsum

sum minusminusnesum

∆Mkj

k i j



A3 Proof of Lemma 6



(A3)









(A4)

u T u i j T

u T u i j T

T T u u i j

klk

l N

klk

l N

kik

kik

ik ikk k

( ) ( ( ))

( ) ( ( ))

( ) ( ( ))

Γ Γ

Γ ΓΓ Γ

+ le + prime


isin isinsum sum



primeTik


k C i jik

i

k C i jik

k k

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0


k C iik

i

k C i jik

k i

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0

u T u i j T

v T v i j T

i

k C iik

i

k C i jik

i i

i i

( ) ( ( ))

( ) ( ( ) )

( ) ( )

Γ Γ

Γ Γ


ge prime

isin isinsum sum

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ








which gives us





which is reduced to



REFERENCES






[ ( ( )) ( )] ( ) ( ( ) ( ))

u i j u T T T Th h


i j

Γ ΓΓ Γ


u i j T u T

u i j T u T

iik

k C

iik

k C

jjk

k C

jjk

i i

j

( ( )) ( )

( ( )) ( )

( ) ( )

( )

Γ Γ

Γ Γ

Γ Γ

Γ

cup minus prime

minus minus

+ cup minus prime

minus minus

isin isin

isin

sum sum

sumkk Cjisinsum

gt( )

Γ

0

primeTij


minus minus


iik

k C

iik

k Ci i

( ( )) ( ) ( ) ( )

Γ ΓΓ Γ

ε2

[ ( ( )) ( )] ( ) ( ) ( )

u i j u T T T Th h


i j

Γ ΓΓ Γ


[ ( ( )) ( )] max ( ) ( ( )) ( ) ( )

u i j u u u i jh h

h i j k C C

k k

i j

Γ Γ Γ ΓΓ Γ



























( ) τ Γ Γi

i Nisin isinG

( ) τ Γ Γci

i Nisin isinG



k C C

k k

i jki

kj

k i j k C C i jik

jk

i

i j

i j

u i j u

V V M M X X

V

isin cup

ne isin cup

sum

sum sum

cup minus

= + minus + + +

= +

( ) ( )

( ( ) ( ))

[ ( ( )) ( )]

( ) ( )

Γ Γ

Γ Γ

Γ Γ

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆∆ ∆ ∆Γ Γ

V M Mj

k C Cki

kj

i j

( )

( ) ( )

minus +

gtnotin cup

sum0






Appendix

A1 Proof of Lemma 4






we have



u T u i j T

T u i j u

i

k C iki

i

k C i jki

jii i

i i

( ) ( ( ))

( ( )) ( )

( ) ( )

Γ Γ

Γ Γ

Γ Γ+ ge +

ge minus

isin isinsum sum


T Tij ji = minus



Tji = 0



(A1)

Similarly we have

(A2)






A2 Proof of Lemma 5



u T

v T

iij ji

iki

k N jji

iki

k N

i

( ( ) ( ) ) ( ( ))

( )

( )

Γ Γ

Γ

Γ


= + +

= +

isin

isin

sum

sum ε

ε2

2




v T u T T

u u i j u i j T T

u u i j T

i iki

k N jji

i i iki

k N jji

i i

( ) ( )

( ) ( ( )) ( ( ))

( ) ( ( ))

Γ Γ

Γ Γ Γ

Γ Γ

= + +

= minus + + +

= minus +

isin

isin

sum

sum

jijii

ij ji

iij ji

v i j T

v i j T

( ( ) )

( ( ) )( )

( )

+ge

minus

minus

ΓΓ







X M X Mji

ij

ij

ji = = and

∆ ∆X Xki

kj = = 0

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆

∆ ∆ ∆

u u V V X M X M

X M X M

V V M

i j i jji

ji

ki

k i jki

ij

ij

kj

k i jkj

i jki

k i j

( ) ( )

( ) ( )


+ minus + minus

= + minus

ne

ne

ne

sumsum

sum minusminusnesum

∆Mkj

k i j



A3 Proof of Lemma 6



(A3)









(A4)

u T u i j T

u T u i j T

T T u u i j

klk

l N

klk

l N

kik

kik

ik ikk k

( ) ( ( ))

( ) ( ( ))

( ) ( ( ))

Γ Γ

Γ ΓΓ Γ

+ le + prime


isin isinsum sum



primeTik


k C i jik

i

k C i jik

k k

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0


k C iik

i

k C i jik

k i

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0

u T u i j T

v T v i j T

i

k C iik

i

k C i jik

i i

i i

( ) ( ( ))

( ) ( ( ) )

( ) ( )

Γ Γ

Γ Γ


ge prime

isin isinsum sum

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ








which gives us





which is reduced to



REFERENCES






[ ( ( )) ( )] ( ) ( ( ) ( ))

u i j u T T T Th h


i j

Γ ΓΓ Γ


u i j T u T

u i j T u T

iik

k C

iik

k C

jjk

k C

jjk

i i

j

( ( )) ( )

( ( )) ( )

( ) ( )

( )

Γ Γ

Γ Γ

Γ Γ

Γ

cup minus prime

minus minus

+ cup minus prime

minus minus

isin isin

isin

sum sum

sumkk Cjisinsum

gt( )

Γ

0

primeTij


minus minus


iik

k C

iik

k Ci i

( ( )) ( ) ( ) ( )

Γ ΓΓ Γ

ε2

[ ( ( )) ( )] ( ) ( ) ( )

u i j u T T T Th h


i j

Γ ΓΓ Γ


[ ( ( )) ( )] max ( ) ( ( )) ( ) ( )

u i j u u u i jh h

h i j k C C

k k

i j

Γ Γ Γ ΓΓ Γ




















Appendix

A1 Proof of Lemma 4






we have



u T u i j T

T u i j u

i

k C iki

i

k C i jki

jii i

i i

( ) ( ( ))

( ( )) ( )

( ) ( )

Γ Γ

Γ Γ

Γ Γ+ ge +

ge minus

isin isinsum sum


T Tij ji = minus



Tji = 0



(A1)

Similarly we have

(A2)






A2 Proof of Lemma 5



u T

v T

iij ji

iki

k N jji

iki

k N

i

( ( ) ( ) ) ( ( ))

( )

( )

Γ Γ

Γ

Γ


= + +

= +

isin

isin

sum

sum ε

ε2

2




v T u T T

u u i j u i j T T

u u i j T

i iki

k N jji

i i iki

k N jji

i i

( ) ( )

( ) ( ( )) ( ( ))

( ) ( ( ))

Γ Γ

Γ Γ Γ

Γ Γ

= + +

= minus + + +

= minus +

isin

isin

sum

sum

jijii

ij ji

iij ji

v i j T

v i j T

( ( ) )

( ( ) )( )

( )

+ge

minus

minus

ΓΓ







X M X Mji

ij

ij

ji = = and

∆ ∆X Xki

kj = = 0

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆

∆ ∆ ∆

u u V V X M X M

X M X M

V V M

i j i jji

ji

ki

k i jki

ij

ij

kj

k i jkj

i jki

k i j

( ) ( )

( ) ( )


+ minus + minus

= + minus

ne

ne

ne

sumsum

sum minusminusnesum

∆Mkj

k i j



A3 Proof of Lemma 6



(A3)









(A4)

u T u i j T

u T u i j T

T T u u i j

klk

l N

klk

l N

kik

kik

ik ikk k

( ) ( ( ))

( ) ( ( ))

( ) ( ( ))

Γ Γ

Γ ΓΓ Γ

+ le + prime


isin isinsum sum



primeTik


k C i jik

i

k C i jik

k k

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0


k C iik

i

k C i jik

k i

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0

u T u i j T

v T v i j T

i

k C iik

i

k C i jik

i i

i i

( ) ( ( ))

( ) ( ( ) )

( ) ( )

Γ Γ

Γ Γ


ge prime

isin isinsum sum

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ








which gives us





which is reduced to



REFERENCES






[ ( ( )) ( )] ( ) ( ( ) ( ))

u i j u T T T Th h


i j

Γ ΓΓ Γ


u i j T u T

u i j T u T

iik

k C

iik

k C

jjk

k C

jjk

i i

j

( ( )) ( )

( ( )) ( )

( ) ( )

( )

Γ Γ

Γ Γ

Γ Γ

Γ

cup minus prime

minus minus

+ cup minus prime

minus minus

isin isin

isin

sum sum

sumkk Cjisinsum

gt( )

Γ

0

primeTij


minus minus


iik

k C

iik

k Ci i

( ( )) ( ) ( ) ( )

Γ ΓΓ Γ

ε2

[ ( ( )) ( )] ( ) ( ) ( )

u i j u T T T Th h


i j

Γ ΓΓ Γ


[ ( ( )) ( )] max ( ) ( ( )) ( ) ( )

u i j u u u i jh h

h i j k C C

k k

i j

Γ Γ Γ ΓΓ Γ

















(A1)

Similarly we have

(A2)






A2 Proof of Lemma 5



u T

v T

iij ji

iki

k N jji

iki

k N

i

( ( ) ( ) ) ( ( ))

( )

( )

Γ Γ

Γ

Γ


= + +

= +

isin

isin

sum

sum ε

ε2

2




v T u T T

u u i j u i j T T

u u i j T

i iki

k N jji

i i iki

k N jji

i i

( ) ( )

( ) ( ( )) ( ( ))

( ) ( ( ))

Γ Γ

Γ Γ Γ

Γ Γ

= + +

= minus + + +

= minus +

isin

isin

sum

sum

jijii

ij ji

iij ji

v i j T

v i j T

( ( ) )

( ( ) )( )

( )

+ge

minus

minus

ΓΓ







X M X Mji

ij

ij

ji = = and

∆ ∆X Xki

kj = = 0

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆

∆ ∆ ∆

u u V V X M X M

X M X M

V V M

i j i jji

ji

ki

k i jki

ij

ij

kj

k i jkj

i jki

k i j

( ) ( )

( ) ( )


+ minus + minus

= + minus

ne

ne

ne

sumsum

sum minusminusnesum

∆Mkj

k i j



A3 Proof of Lemma 6



(A3)









(A4)

u T u i j T

u T u i j T

T T u u i j

klk

l N

klk

l N

kik

kik

ik ikk k

( ) ( ( ))

( ) ( ( ))

( ) ( ( ))

Γ Γ

Γ ΓΓ Γ

+ le + prime


isin isinsum sum



primeTik


k C i jik

i

k C i jik

k k

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0


k C iik

i

k C i jik

k i

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0

u T u i j T

v T v i j T

i

k C iik

i

k C i jik

i i

i i

( ) ( ( ))

( ) ( ( ) )

( ) ( )

Γ Γ

Γ Γ


ge prime

isin isinsum sum

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ








which gives us





which is reduced to



REFERENCES






[ ( ( )) ( )] ( ) ( ( ) ( ))

u i j u T T T Th h


i j

Γ ΓΓ Γ


u i j T u T

u i j T u T

iik

k C

iik

k C

jjk

k C

jjk

i i

j

( ( )) ( )

( ( )) ( )

( ) ( )

( )

Γ Γ

Γ Γ

Γ Γ

Γ

cup minus prime

minus minus

+ cup minus prime

minus minus

isin isin

isin

sum sum

sumkk Cjisinsum

gt( )

Γ

0

primeTij


minus minus


iik

k C

iik

k Ci i

( ( )) ( ) ( ) ( )

Γ ΓΓ Γ

ε2

[ ( ( )) ( )] ( ) ( ) ( )

u i j u T T T Th h


i j

Γ ΓΓ Γ


[ ( ( )) ( )] max ( ) ( ( )) ( ) ( )

u i j u u u i jh h

h i j k C C

k k

i j

Γ Γ Γ ΓΓ Γ

















A3 Proof of Lemma 6



(A3)









(A4)

u T u i j T

u T u i j T

T T u u i j

klk

l N

klk

l N

kik

kik

ik ikk k

( ) ( ( ))

( ) ( ( ))

( ) ( ( ))

Γ Γ

Γ ΓΓ Γ

+ le + prime


isin isinsum sum



primeTik


k C i jik

i

k C i jik

k k

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0


k C iik

i

k C i jik

k i

i i

( ) ( ( )) [ max ( ) ( ( )) ]( ) ( )

Γ Γ Γ ΓΓ Γ


sum sum 0

u T u i j T

v T v i j T

i

k C iik

i

k C i jik

i i

i i

( ) ( ( ))

( ) ( ( ) )

( ) ( )

Γ Γ

Γ Γ


ge prime

isin isinsum sum

[ ( ( )) ( )] max ( ) ( ( )) ( ( ) ( ))

u i j u u u i jh h

h i j k C C i j

k k

i j

Γ Γ Γ ΓΓ Γ








which gives us





which is reduced to



REFERENCES






[ ( ( )) ( )] ( ) ( ( ) ( ))

u i j u T T T Th h


i j

Γ ΓΓ Γ


u i j T u T

u i j T u T

iik

k C

iik

k C

jjk

k C

jjk

i i

j

( ( )) ( )

( ( )) ( )

( ) ( )

( )

Γ Γ

Γ Γ

Γ Γ

Γ

cup minus prime

minus minus

+ cup minus prime

minus minus

isin isin

isin

sum sum

sumkk Cjisinsum

gt( )

Γ

0

primeTij


minus minus


iik

k C

iik

k Ci i

( ( )) ( ) ( ) ( )

Γ ΓΓ Γ

ε2

[ ( ( )) ( )] ( ) ( ) ( )

u i j u T T T Th h


i j

Γ ΓΓ Γ


[ ( ( )) ( )] max ( ) ( ( )) ( ) ( )

u i j u u u i jh h

h i j k C C

k k

i j

Γ Γ Γ ΓΓ Γ


















which gives us





which is reduced to



REFERENCES






[ ( ( )) ( )] ( ) ( ( ) ( ))

u i j u T T T Th h


i j

Γ ΓΓ Γ


u i j T u T

u i j T u T

iik

k C

iik

k C

jjk

k C

jjk

i i

j

( ( )) ( )

( ( )) ( )

( ) ( )

( )

Γ Γ

Γ Γ

Γ Γ

Γ

cup minus prime

minus minus

+ cup minus prime

minus minus

isin isin

isin

sum sum

sumkk Cjisinsum

gt( )

Γ

0

primeTij


minus minus


iik

k C

iik

k Ci i

( ( )) ( ) ( ) ( )

Γ ΓΓ Γ

ε2

[ ( ( )) ( )] ( ) ( ) ( )

u i j u T T T Th h


i j

Γ ΓΓ Γ


[ ( ( )) ( )] max ( ) ( ( )) ( ) ( )

u i j u u u i jh h

h i j k C C

k k

i j

Γ Γ Γ ΓΓ Γ




























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