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Characterizations of a Game Theoretical Cost Allocation Method

Otten, G.J.M.

Publication date:1993

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Citation for published version (APA):Otten, G. J. M. (1993). Characterizations of a Game Theoretical Cost Allocation Method. (CentER DiscussionPaper; Vol. 1993-37). CentER.

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Economic Research

No. 9337

Characterizations of a GameTheoretical Cost éllocation Meth~~

by G.-J. Otten

June 1993

ISSN 0924-7815

Characterizations of a Game Theoretical Cost

Allocation Method

Gert-Jau Otten '

Tilburg University

P.O. Box 90153, ~000 LE Tilburg, The Netherlands

March, 1993

Abstract

In the 1930's the Tennessee Valley .Authority developed several methods to

allocate the costs of multipurpose water projects. One of these methods is the

alternate cost avoided method. This paper provides two characterizations of the

alternate cost avoided method, one on a class of cost games with a fixed player

set, the other on a class of cost games with a variable player set using a reduced

game property.

'The author is indebted to P. Borm, T. Driessen, Y. Funaki, S. Tijs and ]. Zarzuelo for helpfuldiscussions and useful suggestions.

~?j~j K.U.B.~ííi~ 81~3L~~TN~cK

~ ~, TILBi~~G

1

1 Introduction

Cost allocation problems occur in many practical situations, where indíviduals work to-gether in a joint project. In these cases the problem arises of allocating the joint coststo the participants in the project in a"fair" way. A mathematical tool to analyse thistype of problems is provided b}' cooperative game theory.

Examples of cost allocation problems studied in a game theoretica] context are the set-ting of airport landing fees (e.g. Littlechild and Owen (1973), Littlechild and Thompson(1977)), the allocation of joint overhead costs of a firm among its different divisions (e.g.Shubik (1962), Jensen (1977), Hamlen et al. (19ï ï)), and the apportioning of costs ofmultipurpose water development projects (e.g. Ransmeier (1942), Suzuki and Nakayama(1976), Loughlin (197ï), Straffin and Heaney (1981), Young et al. (1982)).

Especially the last type of cost allocation problems has a rich history dating backto the 1930's in which the Tenessee Valley Authority (TVA) was established (see Rans-meier 1942, Parker (1943)). The problem TVA engineers were confronted with was theapportioning of costs of projects in the Tennessee River among the different `purposes'to be served (mainly navigation, flood control, and hydro-electric power). TVA engi-neers made several proposals to allocate the costs of projects to these purposes. Almostall these methods begin by allocating the so-called separable cost, to each `participant'(purpose), and then dividing the remaining nonseparable cost.

Two of the methods developed by the TVA are the egalitarian nonseparable cost(ENSC) method, which allocates the nonseparable cost equally among the participants,and the alternate cost avoided (ACA) method, which allocates the nonseparable costamong the participants in proportion to the `cost savings' made by including a participantin the joint project instead of developing a separate project only to serve the purposesof that participant.

A modification of the ACA-method is the separable cost remaining benefit (SCRB)method. This has become the principal method used by civil engineers to allocate thecosts of multipurpose water projects (see e.g. Inter-Agency Committee on Water Re-sources (195b)).

A game theoretical base for the ACA-method was established by Gately (1974). Gatelyproposed a new solution concept for cooperative games based on a player's "propensity

2

to disrupt" the solution. This solution concept has been further generalized by Fischerand Gately (1975), Littlechild and Vaidya (1976) and Charnes et al. (1978). It wasshown by Straffin and Heaney (1981) that the allocation method proposed by Gatelycorresponds precisely to the ACA-method.

The purpose of this paper is to provide an axiomatic characterization of the ACA-method on a certain class of cost games with a fixed player set as well as on a class ofcost games with a variable player set, using a reduced game property. This is the subjectof section 3. First, in section 2 we discuss the cost allocation problem in a formal gametheoretical context, and recall some of the cost allocation methods proposed by the TVA.

2 Game theory and cost allocation problems

To formulate a cost allocation problem in terms of cooperative game theory, it is mod-elled as a cost game ( N, c). Here, .N represents a finite set of participants among whichthe costs of a joint project are allocated. For example, N can be a set of potential cus-tomers of a public facility, the divisions of a firm, or municipalities which share a jointwater system, etc. The elements of :v are called players and subsets of .V are calledcoalitions. For any coalition S C N, the minimal costs of designing a project for thepurposes of S only are denoted by c( S). In particular, c(0) :- 0, where 0 denotes theempty set. The function c : 2N -~ R is called the (jointJ cost function. Let CG`v denotethe set of all cost games with player set ;ti'.

Example 1: As an example of a joint cost game based on a cost allocation problem, weconsider the cost allocation problem for the TVA ten dam system. Here the purposesnavigation, flood control and hydro-electric power are denoted as players 1,2, and 3 re-spectively. Table 1 is adapted from Ransmeier (1942, p. 329).

coalitions S

cost c(S)

00

{1}

163,520

{2}lao,s2s

{3}

250,096

{1,2}301,607

{1,3}

378,821

{2,3}367,370

{1,2,3}

412,584

Taóle 1. The cost game for the TVA ten dam project ( costs in á 1000).

3

Given a cost game ( IV,c), the cost allocation problem now becomes to choose a costallocation in a"fair" manner. A cost allocation for (IV, c) is a vector x E RN such that

~;EN x; - c(N). Here, x, is the cost allocated to player i E N. The TVA engineers pro-posed several cost allocation methods. If ,9~~ is a subset of CGN, then a(cost) allocationmethod on AN is a map f: AN -~ RN, which assigns to every cost game (N, c) E AN acost allocation J(c) E R~~.

Almost all cost allocation methods proposed by the TVA begin by charging every playera minimal cost, called separable cost, which are the additional cost of including the playerin the project already designed for the other players. Thus, for a cost game (N, c), theseparable cosl SC,(c) oJ player í E:1' are formally defined by

SC;(c) :- c(N) - c(.ti'`{i}).

To use methods based on the idea above it is reasonable to make the following twoassumptions on the underlying cost garne.

SC,(c) G c({i}) for all i E.ti',

~ sC;(c) C c(:~') C~ c({i}).~EIV ~EN

(1)

(2)

Conditions ( 1) and (Z) are well-known balancedness conditions for cost games. IfSC;(c) ~ c({i}) for some i E N, then it is not favourable to include player i in thejoint project. Condition ( 2) implies that after each player is charged his minimal coststhere is still a positive amount of cost remaining which should be allocated. Theseremaining cost are called the nonseparaóle cost and are given by

NSC(c) :- c(N) - ~ SC;(c).~EN

The easiest way to allocate the nonseparable cost is to divide these cost equally amongthe players. This method is called the egalitarian nonseparable cost (ENSC) method,and it is one of the first allocation methods proposed by the TVA. Thus, for a cost game(!~',c) the cost allocated to player i E:ti' by the EtiSC-method are

4

ENSC;(c) - SC,(c) f I,1-~,INSC(c).

An alternative allocation method is the alternate cost avoided ACA) method, which wasfirst proposed by Martin Gleaser, a TVA consultant in 1938 ( see Ransmeier (1942)). Bythis method the nonseparable cost are divided in proportion to c({i}) - SC,(c). Hence,

ACA;(c) :- SC;(c)f~~~~{c({j})~SC~(c)NSC(c) for all i E N

The number c({i}) - SC,(c) represents the alternate cost avoided by including player i

in thejoint project.

A modification of the ACA-method is the separable cost remaining benefit ~SCRB)method. If 6(i) is the benefit of the project to player i, then i would not be willingto pay more than min{b(i),c({i})}. The remaining benefit to player i is defined bymin{b(í),c({i})} - SC,(c). The SCRB-method allocates the nonseparable cost propor-tional to the remaining benefits. Since in many situations the benefits exceed the alter-nate costs, the SCRB-method often coincides with the ACA-method.

The mayor drawback of the cost allocation me[hods mentioned above is that they onlytake into account the values of the coalitions with 1, ~a~ - 1 and (.v~ players. In partic-ular, there is no guarantee that the corresponding allocations of these methods are coreelements of the cost game, which means that there might be subcoalitions that have an

incentive to split of from the grand coalition.

From a practical viewpoint however, the advantage of these methods is that in generalthey are much easier to compute than game theoretical solution concepts as the Shapleyvalue ( Shapley ( 1953)), the nucleolus ( Schmeidler ( 1969)) and the cost gap method(Driessen en Tijs (1985), Tijs en Driessen ( 1956)), which take into account the values ofal! coalitions.

Moreover, as is shown in e.g. Suzuki and `akayama (1976), Legros ( 1982) andDriessen and Tijs (1985) there are (large) classes of cost games for which some of thesolution concepts mentioned above coincide with one ( or more) of the game theoreticalsolution concepts.

5

Example 2: For the TVA cost game of example 1 the cost allocations of the ENSC- andACA-method are given in table 2 together with the cost allocations corresponding to thegame theoretical solutions mentioned above. Note that in this case the cost allocationsby the ACA-method and the cost gap method coincide.

ENSC-method

ACA-method

Shapley value

nucleolus

cost gap method

1

119,424

117,476

117,829

116,234

117,476

2

10ï,9ï3

99,157

100,756

93,540

99,157

3

185,187

195,951

193,999

202,810

195,951

Table 2. Cost allocation for the TVA cost game by five methods (cost in ~ 1000).

3 Characterizations of the ACA-method

This section further investigates the ACA-method. Attention is restricted to the class

of cost games (1`', c) for which (1) and (2) hold. This class is denoted by F'N and Fmdenotes the class of cost games with m or more players satisfying (1) and (2).

Geometrically, for a cost game (,V, c) E F" the cost allocation ,9CA(c) is the uniqueelement in the hyperplane {r E Rv~~~,E~r, - c(.N)} which lies on the line segmentwith end points (SC;(c)),E,v and (c({i})),E,~- (see figure 1).

ISCi(C)),E:V ~{r E R~ ~~iEN Z~ - C(l~ )}

figure 1.

6

Let A C F~. Clearly, the ACA-method satisfies individually rationality on A, i.e.,ACA;(c) C c({i}) for all i E .N and all (N,c) E A.

Furthermore, the ACA-method satisfies the symmetry property on A, i.e., for all(N, c) E A and all players i and j that are symmetric in (N, c), i.e., c(SU{i}) - c(SU{~})for all S C 1V`{i, j}, it holds that ACA;(c) - ACA~(c).

The ACA-method also satisfies invnriance w.r.t. strategic equivalence on A, i.e., forall (N,c) E A, all k) 0 and all a E R'N, such that (N,kc f a) E A, we have thatACA(kc ~- a) - kACA(c) -~ a. Here the game (N, kc f a) is defined by (kc f a)(S) :-kc(S) t~,ES a; for all S C:h'.

Another property of the ACA-method on A is weak proportionality which says thatif (N,c) E A is such that SC;(c) - 0 for all i E N, then AC.A(c) is proportional to thevector ( c({i}));EN of individual costs.

This weak proportionality property shows great resemblance to the restricted pro-portionally property of the r-value (Tijs (1981), ( 1987)). Cost games for which each

player's separable cost are zero arise when the increase in the total costs of adding anextra player can be neglected compared to the total cost of the project.

Similar to the characterization of the r-value by Tijs ( 19S ï) one can prove

Theorem 1: The ACA-method is the unique cost allocation method on FN which

satisfies invariance w.r.t. strategic equivalence and weak proportionality.Proof: Suppose that f: F'ti' -a R'~' satisfies the two mentioned properties. Let(.N, c) E F'ti~. It suffices to show that f(c) - AC.-1(c).

Define the game ( N, c) E F,~ by

c(S) :- c(S) -~ SC;(c) for all S C ~~'.~ES

Then SC,(c) - 0 for all i E:~~. From the weak proportionality property it follows that

there exists an a E R such that for all i E iti'

f,(c) - aê({i}) - a(c({ }) - SC,(c)).

From the strategic equivalence property it follows that for all i E:v

7

t~(c) - sc;(c) t I~(~) - sc;(c) f a(c({i}) - sc;(c)).

Using the fact that ~;EN f,(c) - c(N), it easily follows that t(c) - ACA(c). O

The last part of this section provides a characterization of the ACA-method on the classFl using a reduced game property. In the literature several types of reduced games havebeen considered to provide a foundation of game theoretic solution concepts based on theconsistency principle. ~Ve mention, Hart and ~fas-Colell (1989) for the Shapley value,Sobolev (1975), Snijders (1991) for the (pre)nucleolus, Peleg (1986) for the core, andrecently, Driessen (1992) for the r-value. .Also the ENSC-method has been characterizedby means of a reduced game property (lfoulin (1985), Driessen and Funaki (1993)). Fora detailed survey on consistency see e.g. Driessen (1991).

The idea behind consistency is the following. Given a cost game, and a cost allocationfor this game, determined by a cost allocation method, imagine that a coalition decidesto renegotiate the allocation within their subgroup. The new situation is described bya reduced game. A cost allocation method is consistent w.r.t this reduced game if thenew cost allocation within this subgroup is the same as in the original game.Let (.ti', c) be a cost game, k E.~' and s E R'N a cost allocation. The reduced game(N`{k}, ck~r) cor-responding to (.~',c) is defined as follows. For S C:h'`{k}

ck.~( S) :- J c( S) if ~S~ C I

11 C(SU {k}) -rk If 2 C ~S~ c ~.v~ - 1.

It should be noted that the reduced game introduced here coincides with the reducedgame of :líoulin (1985) except for the 1-person coalitions.

The interpretation of this reduced game is as follows. In the reduced game the cost ofa 1-person coalition is the same as in the original game. However, if in the reducedsituation the players want to coopecate in a coaltion S, then player k should be involvedand, therefore, the cost of coalition S in the reduced game is the cost of coalition SU {k}in the original game minus the original cost rk allocated to player k.

Let A C F~ and let m(A) :- min{~.N~ ~(~ti',c) E A}. A cost allocation method f on.9 satisfies the reduced game property on A if for all (:N,c) E A with n ~ m(A) and all

s

k E N it holds that

(i) (N`{k}, ck~~(`)) E A, and

(ii) j;(ck~~(`)) - f,(c) for all i E .N~{k}.

The ACA-method satisfies the reduced game property on the class F3. This is shown in

Lemma 2: The ACA-method satisfies the reduced game property on F3.Proo~ Let (N,c) E F3 with ~N~ 1 4, and let k E N. We first show that the reducedgame (N`{k},ck,ACA(`)) is an element of F3. Herefore note that for all i E N`{k}

~k,ACA(c)( { }) - c({i}) (3)

and since ~ N~ ~ 4 also

SC;(ck,ACA(c}) - c(N) - ACAk(c) - (c(.N`{i}) - ACAk(c)) - SC;(c). (4)

Since (N,c) E F3, it follows that SC,(ck,AC.a(`) ~ ck.ACA(`)({i}) for all i E N`{k}.It remains to show that

~` SC,(ck.ACA(c)) ~ Ck.ACAicI(~ ~ {k}) C ~ Ck,ACA(c)({t}).

iEN[`J{k} iEiti-`{k}

Note that for i E N`{k}

SC,(c) C AC,~1(c) G c({i}).

(5)

Then, using (3), (4), and the fact that ck.ACA(`)(t~'`{k}) -~;E,ti.`{k} ACA;(c) the requiredinequality (5) is easily obtained.

Norv we show that ACA;(Ck,ACA(c)) -.qCA;(c) for all i E a' `{k}.Since ACA;(c) - SC;(c) f a(c({i}) - SC,(c)) for all i E N, where ~ is such that

c(.~ )- ~ SC,(c) f ct ~(c({i} - SC,(c)). (6)~E,Y iE.h'

9

Similarly, using ( 3) and ( 4), we obtain that ,9CA;(ck,.acn(`}) - SC;(c)f(3(c({i})-SG,(c))for all i E N~{k}, where Q is such that

c(N) - ACAk(c) - ~ SC;(c) -}. p ~ (c({i} - SC;(c)).~E.ti~`{k} ~EA'`{k}

Subtracting (7) from (6) we obtain

ACAk(c) - SCk(c) t a(c({k}) - SCk(c)) -~ (a - Q) ~(c({i} - SC;(c)).~E.~'`{k}

Hence,

(a - ~3) ~ (c({i} - SC;(c)) - 0.~E.~'`{k}

(7)

(g)

~~'e now distinguish two cases.

If ~;Etv~{k}(c({i}) - SC,(c)) - 0, then ACA,(c) - SC,(c) for all i E N~ {k}. Since, inthis case,

ck.`'c~~`~(.N ~ {k}) - ~ ACA;(c) - ~ SG,(c) - ~ SC,(ck.acn(~~)~E.ti'`{k} iE,4'`{k} iE.~~`{k}

it easily follows that .9CA,(ck"ac`~~`~) -.9CA,(c) for all i E!V `{k}.

If ~;E,n~`{k}(c({i}) - SC;(c)) ~ 0, then by (a) a - j3 - 0. Hence, ACA;(ck,ACA(c}) -

ACA;(c) for all i E:V `{k}. ~

Example 3 illustrates that the ~CA-method does not satisfy the reduced game property

on the set FZ. This is due to the fact that by reducing a 3-person game to a 2-person

game the separable costs of the players ma}' change.

Example 3: Let N:- {1,2,3} and define (.h',c) as follows. For S C N

2 if {2, 3} ~ Sc(S) -

4 if {2, 3} C S.

10

Clearly, (.~',c) E F~ and .aC.a(c) -(0.'1,2). The reduced game ( {1,2},c3~~e`~t`t) E Fz isgiven by c3.`'c~t`~({1}) - c3..ac`'t`t({2}) - 3 and c'.ac`~(`)({1.2}) - 2.Hence, ACA(c3..acA(`t) - ( 1,1) ~ (0,2) - Í.~1CAi(c),ACAz(c)).

Lemma 3: Let j be a cost allocation method on F3 ~chich satisfies weak proportionality

on F3 `F4 and the reduced game propert} on F3. Then j satisfies weak proportionalityon F3.

Proof Let (.~-,c) E F3 ~~ith ~.`~~ ? 1 be such that SC,(c) - 0 for all i E.ti' and suppose

that j satisfies the ~~eak proportionalit}~ propertc on F3 `Fl~ti'I.Let k E :~~ and let (.~-`{k},cA ~~~~) be the (~ ~~~ - 1)-person reduced game of (.~,c). Then(.ti' `{k},ck~~t`~) E F3 `F~,~-~. Since SC,(c" ~~`~) - 0 for al] i E .~~ `{lL} (cf. ( 4)), thereexists an Q E R such that

Ii(Ckil~l) - aCk.i1~1(J7}) - C1C({t}) for all 2 E .~. `{Á'}'

Since f satisfies the reduced game propert~- on F3 it follo~~s that

f~(~) - f~(cx.~i~~ - oc({i}) for all i E .~-~ {k}.

~'ar~~ing k E ~~ leads to

j(c) - ~~~,{1}~.....~i{,,,ii.a

`o~~~ ~~~e can formulate our maín theorem ~~.hich characterizes the .~C~1-method on Fl.

Theorem 4: The .aC.~-method is the uniyue cost allocation method on Ft ~~-hich satisfies

(i) symmetn~ on F~,

(ii) invaríance ~~~.r.t. strategic equi~~alence on Fl,

(iii) ~~~eak proportionalit}~ on F3 `Fa.

11

(i~') the reduced game propertp on F3.

Proof Clearlt', the ACA-method satisfies (i)-(i~~).Let f be a cost allocation method, defined on Fr, satisfying (i)-(iv). Let (N,c) E Fl.~~-e sho~~~ that j(c) -.4CA(c). Herefore ~ce distinguish three cases.If ~N~ - 1, then f(c) - c({1}) -,-1C.4(c).If ~.~-~ - 3, then (i) and (ii) imply~ that f,(c) - c({i}) ~- i(c(~1') - c({i}) - c(N ~{i})) -

.4C.4,(c) for i - 1, 2.

lf ~.~'~ ~ 3, then theorem 1 and lemnra 3 impl~~ that f(c) - AC.4(c). O

It may' be noted that also the the E`SC-method satisfies sy-mmetry, invariance w.r.tstrategic equivalence and the reduced game property on the set F3. Ho~~~e~.er, this costallocation method does not satisf~ ~~~eak proportionality'.

For a cost game (.`',c) E Fr, the cenler of imputation set (ClSJ ealue is defined by

CIS,(c) :- c({z}) ~- ~-,1-~.~(c(.~') - ~ c({J})) for all i E:~~.J E.~

If in theorem -1 condition (iii) is omitted and condition (i~~) is replaced by' the reducedgame propert} on F, then a characterization of the CIS-~alue on Fr is obtained. It isleft to the reader to sho~~ that the CIS-~-alue is indeed the unique cost allocation methodon Fr ~ehich satisfies s~mmetr~, in~ariance w.r.t. strategic equi~'alence, and the reducedgame property~ on FZ.

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Discussion Paper Series, CentER, Tllburg University, The Netherlands:

(For previous papers please consult previous discussion papers.)

No. Author(s)

9201 M. Verbeek andTh. Nijman

9202 E. Bomhoff

9203 J. Quiggin andP. Wakker

9204 Th. van de Klundertand S. Smulders

9205 E. Siandra

9206 W. Hárdle

9207 M. Verbeek andTh. Nijman

9208 W. H~rdle andA.B. Tsybakov

9209 S. Albaek andP.B. Overgaard

9210 M. Cripps andJ. Thomas

9211 S. Alba;k

9212 T.J.A. Storcken andP.H.M. Ruys

9213 R.M.W.J. Beetsma andF. van der Ploeg

9214 A. van Soest

9215 W. Guth andK. Ritzberger

9216 A. Simonovits

9217 J.-L. Ferceira,I. Gilboa andM. Maschler

9218 P. Borm, H. Keiding,R. Mclean, S. Oortwijnand S. Tijs

Title

Minimum MSE Estimation of a Regression Model withFixed Effects from a Series of Cross Sections

Monetary Poliry and Inflation

The Axiomatic Basis of Anticipated Utility; A Clarification

Strategies for Growth in a Macroeconomic Setting

Money and Specialization in Production

AppGed Nonparametric Models

Incomplete Panels and Selection Bias: A Survey

How Sensitive Are Average Derivatives?

Upstream Pricing and Advertising Signal DownstreamDemand

Reputation and Commitment in Two-Person RepeatedGames

Endogenous Timing in a Game with Incomplete Information

Extensions of Choice Behaviour

Exchange Rate Bands and Optimal Monetary Accommodationunder a Dirty Float

Discrete Choice Models of Family Labour Supply

On Durable Goods Monopolies and the (Anti-) Coase-Conjecture

Indexation of Pensions in Hungary: A Simple Cohort Model

Credible Equilibria in Games with Utilities ChangingDuring the Play

The Compromise Value for NTU-Games

No. Author(s)

9219 J.L. Horowitz and. W. H~rdle

9220 A.L. Bovenberg

9221 S. Smulders andTh. van de Klundert

9222 H. Bester andE. Petrakis

9223 A. van den Nouweland,M. Maschler andS. Tijs

9224 H. Suehiro

9225 H. Suehiro

9226 D. Friedman

9227 E. Bomhoff

9228 P. Borm, G.-J. Ottenand H. Peters

9229 H.G. Bloemen andA. Kapteyn

9230 R. Beetsma andF. van der Ploeg

9231 G. Almekinders andS. Eijffinger

9232 F. Vella andM. Veróeek

9233 P. de Bijl andS. Goyal

9234 J. Angrist andG.Imbens

9235 L. Meijdam,M. van de Venand H. Verbon

9236 H. Houba andA. de Zeeuw

Title

Testing a Parametric Model against a SemiparametricAlternative

Investment-Promoting Policies in Open Economies: TheImportance of Intergenerational and InternationalDistributional Effects

Monopolistic Competition, Product Variety and Growth:Chamberlin vs. Schumpeter

Price Competition and Advertising in Oligopoly

Monotonic Games are Spanning Network Games

A "Mistaken Theories" ReFinement

Robust Selection of Equilibria

Economically Applicable Evolutionary Games

Four Econometric Fashions and the Kalman FilterAlternative - A Simulation Study

Core Implementation in Modified Strong and Coalition ProofNash Equilibria

The Joint Estimation of a Non-Linear Labour Supply Functionand a Wage Equation UsingSimulated Response Probabilities

Does Inequality Cause Inflation? - The Political Economy ofInflation, Taxation and Government Debt

Daily Bundesbank and Federal Reserve Interventions- Do they Affect the Level and Unexpected Volatility of theDM~S-Rate?

Estimating the [mpact of Endogenous Union Choice onWages Using Panel Data

Technological Change in Markets with Network Externalities

Average Causal Response with Variable Treatment Intensity

Strategic Decision Making and the Dynamics of GovernmentDebt

Strategic Bargaining for the Control of a Dynamic System inState-Space Form

No. Author(s)

9237 A. Cameron andP. Trivedi

9238 J.-S. Pischke

9239 H. Bloemen

9240 F. Drost andTh. Nijman

9241 R. Gilles, P. Ruysand J. Shou

9242 P. Kort

9243 A.L. Bovenberg andF. van der Ploeg

9244 W.G. Gale andJ.K. Scholz

9245 A. Bera and P. Ng

9246 R.T. Baillie,C.F. Chung andM.A. Tieslau

9247 M.A. Tieslau,P. Schmidtand R.T. Baillíe

9248 K. W~rneryd

9249 H. Huizinga

9250 H.G. Bloemen

9251 S. Eijffinger andE. Schaling

9252 A.L. Bovenberg andR.A. de Mooij

9253 A. Lusardi

9254 R. Beetsma

Title

Tests of tndependence in Parametric Models: WithApplications and Wustrations

Individual Income, Incomplete Information, and AggregateConsumption

A Model of Labour Supply with Job Offer Restrictions

Temporal Aggregation of GARCH Processes

Coalition Formation in Large Network Ewnomies

The Effects of Marketable Pollution Permits on the Firm'sOptimal Investment Policies

Environmental Policy, Public Finance and the Labour Marketin a Second-Best World

IRAs and Household Saving

Robust Tests for Heteroskedasticiry and AutocorrelationUsing Score Function

The Long Memory and Variability of Inflation: AReappraisal of the Friedman Hypothesis

A Generalized Method of Moments Estimator for Long-Memory Processes

Partisanship as Information

The Welfare Effects of Individual Retirement Accounts

Job Search Theory, Labour Supply and UnemploymentDuration

Central Bank Independence: Searching for the Philosophers'Stone

Environmental Taxation and Labor-Market Distortions

Permanent Income, Current Income and Consumption:Evidence from Panel Data

Imperfect Credibility of the Band and Risk Premia in theEuropean Monetary System

No. Author(s)

9301 N. Kahana andS. Nitzan

9302 W. Góth andS. Nitzan

9303 D. Karotkin andS. Nitzan

9304 A. Lusardi

9305 W. Giith

9306 B. Peleg andS. Tijs

9307 G. Imbens andA. Lancaster

9308 T. Ellingsen andK. Wárneryd

9309 H. Bester

9310 T. Callan andA. van Soest

9311 M. Pradhan andA. van Soest

9312 Th. tiijman andE. Sentana

9313 K. W~rneryd

9314 O.P.Attanasio andM. Browning

9315 F. C. Drost andB. J. M. Werker

9316 H. Hamers,P. Borm andS. Tijs

9317 W. Guth

9318 M.J.G. van Eijs

Title

Credibility and Duration of Political Contests and the Extentof Rent Dissipation

Are Moral Objedions to Free Riding Evolutionarily Stable?

Some Peculiarities of Group Decision Making in Teams

Euler Equations in Micro Data: Merging Data from TwoSamples

A Simple Just~cation of Quantity Competition and theCournot-Oligopoly Solution

The Consistency Principle For Games in Strategic Form

Case Control Studies with Contaminated Controls

Foreign Direct Investment and the PoGtical Economy ofProtection

Price Commitment in Search Markets

Female Labour Supply in Farm Households: Farm andOff-Farm Participation

Formal and Informal Sector Employment in Urban Areas ofBolivia

Marginalization and Contemporaneous Aggregation inMultivariate GARCH Processes

Communicatíon, Complexity, and Evolutionary Stability

Consumption over the Life Cycle and over the BusinessCycle

A Note on Robinson's Test of Independence

On Games Corresponding to Sequencing Situationswith Ready Times

On Ultimatum Bargaining Experiments - A Personal Review

On the Determination of the Control Parameters of theOptimal Can-order PoGry

9319 S. Hurkens Multi-sided Pre-play Communication by Burning Money

No. Author(s)

9320 J.J.G. Lemmen and~ S.C.W. Eijffinger

9321 A.L. Bovenberg andS. Smulders

9322 K.-E. W~rneryd

9323 D. Talman,Y. Yamamoto andZ. Yang

9324 H. Huizinga

9325 S.C.W. Eijffinger andE. Schaling

9326 T.C. To

9327 J.P.J.F. Scheepens

9328 T.C. To

9329 F. de Jong, T. Nijmanand A. Róell

9330 H. Huizinga

9331 H. Huizinga

9332 V. Feltkamp, A. Koster,A. van den Nouweland,P. Borm and S. Tijs

9333 B. Lauterbach andU. Ben-Zion

9334 B. Melenberg andA. van Soest

9335 A.L. Bovenberg andF. van der Ploeg

9336 E. Schaling

9337 G.-J.Otten

Title

The Quantity Approach to Financial Integration: TheFeldstein-Horioka Criterion Revisited

Environmental Quality and Pollution-saving TechnologicalChange in a Two-sector Endogenous Growth Model

The Will to Save Money: an Essay on Economic Psychology

The (2""' - 2)-Ray Algorithm: A New Variable DimensionSimplicial Algorithm For Computing Economic Equilibria onS" x R"

The Financing and Taxation of U.S. Direct InvestmentAbroad

Central Bank Independence: Theory and Evidence

Infant Industry Protection with Learning-by-Doing

Bankruptry Litigation and Crptimal Debt Contracts

Tariffs, Rent Extraction and Manipulation of Competition

A Comparison of the Cost of Tradíng French Shares on theParis Bourse and on SEAQ International

The Welfare Effects of Individual Ret'uement Accounts

Time Preference and International Tax Competition

Linear Production with Transport of Products, Resources andTechnology

Panic Behavior and the Performance of C'ucuit Breakers:Emp'uical Evidence

Semi-parametric Estimation of the Sample Selection Model

Green Policies and Public Finance in a Small Open Economy

On the Economic Independence of the Central Bank and thePersistence of Inflation

Characterizations of a Game Theoretical Cost AllocationMethod

ii W V YV ~I ~N~ ~ N ~I i