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Cooperative games
Pierre DehezCORE, University of Louvain
(Center for Operations Research et Econometrics)
Content
1. Games in characteristic function form
2. Imputations and the core
3. The Shapley value
4. Cost sharing
5. Measuring power in decision games
6. Beyond the core
2
The problem is the following: A group of individuals considers the possibility of cooperating on a common project.
To cooperate means that each member of the group accepts to act in the interest of the group ...
… by adopting strategies aiming at the maximization of the collective welfare
Question: How to share the results of this cooperation ?
5
Example: sharing a euro
Two persons are proposed a euro provided they agree on the way to split it.
Question: How to split it ?
The immediate answer is clearly an equal splitting: 50 cents for each.
This is the fair solution.
6
Example: porters
Imagine three porters in a train station. They are asked to move four (heavy) suitcases for 6 €.
The four suitcases must be moved together and two porters can do it.
Question: How to share the 6 € ?
Here too, the egalitarian solution seems fair: 2 € for each.
7
Example: auction between one seller and two buyers
Each trader has a reservation price.
The object will naturally be sold to the buyer with the largest reservation price.
Questions:
- At which price?
- Should the buyer with the lowest reservation be compensated?
8
Example: sharing a crop
Imagine a landlord and m (identical) workers, and a technology described by a production function y = F(s) where s is the number of workers.
With the landlord's agreement and all workers on the land, the crop is F(m).
Question: How to share the crop between the landlord and the workers?
Workers receiving the same amount is fair but how much should be allocated to the landlord?
9
The four examples are defined in terms of a commodity which is used as a "money": euros or kilos of wheat…
Quantities of these commodities can be added and "monetary" transfers between players can be organized.
These situations are (cooperative) games with transferable utility or games with side payments: TU games.
We shall see that coalitions of player will play a central role in the analysis of TU-games.
10
Let N = {1,…,n} denote the set of players, n 2.
A coalition is a subset of players S N.
In particular, {i} and N are also considered as coalitions. N is the "grand coalition".
If we add the empty set, there are 2n coalitions and to each coalition S N, is associated a real number, its worth:
with the convention v() = 0.
( )v S
12
This defines a set function, called "characteristic" function:
where v(S) is measures the minimum gain that coalition S can generate if it forms, whatever the other players do.
v(S) is expressed in units of a "money" that is some perfectly divisible commodity (euros, tons de wheat, ...).
The set G of all set functions on a set N is a vector space of dimension 2n-1 that can be identified as
: 2Nv
, ( ) for all ,v w G v w G
2 1n
13
If a coalition S N forms, v(S) can be defined on the basis of the two player noncooperative game where the players are S and N\S: v(S) is then the "security level":
the members of S act to maximize the gain of the coalition assuming that the excluded members act
against S.
This is the MaxiMin associated to prudent strategies.
v(N) is the maximum gain obtained by the "grand" coalition when all players coordinate their actions.
14
Notation:
v(i) = v({i}), v(ij) = v({i,j}), v(ijk) = v({i,j,k}), …
Si = S{i}, S\i = S\{i}
|A| is the cardinal of the set A and n = |N|, s = |S|, t = |T|,…
1for a vector ( ,..., ) and a subset {1,..., }
we write ( )n
ii S
x x x S n
x S x
15
In sharing a euro, N = {1,2} and the characteristic function is given by:
v(1) = v(2) = 0
v(12) = 1
Alone, a player gets nothing. Together they receive 1 €.
16
In the porters' game, N = {1,2,3} and the characteristic function is given by:
v(1) = v(2) = v(3) = 0
v(12) = v(13) = v(23) = 6
v(123) = 6
Alone, a porter cannot move the 4 suitcases, but two or three can do it.
17
In the auction game, if S contains the seller and at least one buyer, v(S) is the sum of the gains of the seller and of the buyer whose reservation price is highest:
v(1) = v(2) = v(3) = 0
v(12) = (p – p1) + (p2 – p) = (p2 – p1)
v(13) = (p – p1) + (p3 – p) = (p3 – p1)
v(23) = 0
v(123) = (p – p1) + (p3 – p) = (p3 – p1)
where p is the price at which the transaction takes place. It does not appear because gains are added!
seller: 1buyers: 2 and 3
p1 ≤ p2 ≤ p3
18
In the crop game, there are n = m + 1 players:
v(S) = 0 if S does not include the landlord
v(S) = F(s – 1) if S includes the landlord (he/she does not work)
In particular, v(i) = 0 for all i and v(N) = F(m).
We suppose that F is increasing with F(0) = 0, not more at this stage.
19
Merging coalitions should be beneficial or, at least, not detrimental:
This is the property of super-additivity.
From now on, a game is a pair (N,v) where N is a set of players and v is a super-additive set function on N.
( ) ( ) ( )S T v S T v S v T
20
Super-additivity implies in particular the following inequalities:
Hence cooperation does not induce a loss.
In case of a strict inequality, there is a potential gain for all and the game is said to be essential:
All the examples we have seen so far are super-additive and essential games.
1
( ) ( ) for all ( ) ( )n
i S i
v i v S S v i v N
1
( ) ( )n
i
v i v N
21
The importance of a player is not only measured by his/her contribution to the "grand coalition" v(N) – v(N\i).
One has to take also into account his/her (marginal) contributions to all those coalitions to which he/she belongs to:
In particular, Cmi(i) = v(i).
By super-additivity we have:
( ) ( ) ( \ )iCm S v S v S i
22
( ) ( \ ) ( ) ( ) ( ) ifiv S v S i v i Cm S v i i S
Two players i et j are substitutable in a game (N,v) if their marginal contributions are identical:
for all coalitions S containing i et j.
Alternatively, i et j are substitutes in (N,v) if:
for all coalitions S containing i et j.
( ) ( \ ) ( ) ( \ )v S v S i v S v S j
( \ ) ( \ )v S i v S j
23
For instance, in the 3-player case, players 2 et 3 are substitutes if and only if:
v(2) = v(3)
v(12) = v(13)
Indeed we have:
v(23) – v(2) = v(23) – v(3)
v(123) – v(13) = v(123) – v(12)
24
Player i is a null player in a game (N,v) if he/she contributes to no coalition:
for all coalition S containing i.
In a 3-player game, player 1 is a null player if:
v(1) = 0
v(12) = v(2)
v(13) = v(3)
v(123) = v(23)
( ) ( \ ) 0v S v S i
25
A game is monotonic if enlarging a coalition is never detrimental, in which case the marginal contributions are nonnegative:
Proposition 1.1 A game such that v(S) 0 for all S N is monotonic.
This is an immediate consequence of super-additivity:
( ) ( )S T v S v T
( ) ( / ) ( ) ( ) ( )S T v S v T S v T v S v T
( ) 0iCm S
27
A game is symmetric if the worth of any coalition depends only upon the number of its members, independently of their identity: all players are substitutable.
It means that there exists a function f such that:
Proposition 1.2 A symmetric game is monotone if the function f that defines it is non-decreasing.
( ) ( ).v S f s
( ) ( ) ( ) ( )S T s t f s f t v S v T
28
Example: garbage game
Each player has a garbage bag he/she would like to dispose of in a neighbor's garden.
v(S) = – (n – s) for all S N,
v(N) = – n
i.e. the "disutility" is proportional to the numbers of bags.
This game is super-additive and symmetric but it is not monotonic.
29
A game (N,v) is convex if for all coalitions S et T:
Clearly, convexity implies super-additivity and the set of convex game is a sub-cone of the cone of superadditive games.
Proposition 1.3 A game (N,v) is convex if and only if marginal contributions are non-decreasing:
for all i, S, T such that i S T.
( ) ( ) ( ) ( )v S T v S v T v S T
( ) ( )i iCm S Cm T
31
The crop game is convex if and only if returns to scale are constant or increasing (non-decreasing marginal productivity).
Consider a player i and two distinct coalitions S and T such that i S T:
1if :
( ) ( \ ) ( 1) ( 2) ( 1) ( 2) ( ) ( \ )
i S
v S v S i F s F s F t F t v T v T i
1if \ :
( ) ( \ ) 0
( ) ( \ ) ( 1) ( 2) 0
i T S
v S v S i
v T v T i F t F t
32
1if :
( ) ( \1) ( 1) 0
( ) ( \1) ( 1)
( 1)
i
v S v S F s
v T v T F t F s
Notice that the hypothesis of nondecreasing marginal productivity has only been used only in the 1st case. The other three cases rely on the non-decreasingness and nonnegativeness of F.
1if :
( ) ( \ ) ( ) ( \ )
0
i T
v S v S i v T v T i
33
The auction game with p = (0,200,300) is not convex:
v(12) + v(13) – v(1) = 500 > v(123) = 300
Alternatively, in terms of marginal contributions:
v(13) – v(1) = 300 > v(123) – v(12) = 100
The porter's game is not convex:
v(12) + v(13) – v(1) = 12 > v(123) = 6
Alternatively, in terms of marginal contributions:
v(12) – v(1) = 6 > v(123) – v(13) = 0
34
A simple game is defined by a binary characteristic function:
v(S) {0,1} with
The normalized porters' game is a simple (and symmetric) game:
v(S) = 1 if and only if s 2
it is obtained by dividing by 6 all coalitions' worth.
( ) 0 and ( ) 1
( ) 1 ( \ ) 0
v i v N
v S v N S
35
Given a TU-game (N,v), the problem is to distribute the result of the cooperation as measured by v(N).
Cooperation means that each player acts in the interest of the collectivity so as to obtain the largest "social output".
The immediate and natural question then concerns the remuneration of the players for their participation to the collective project.
Formally, the problem is to find n numbers x1,...,xn such that:
37
1
( )n
ii
x v N
There are two minimal conditions to impose on a distribution of the social output:
(i) collective rationality (Pareto efficiency)
all v(N) should be distributed: no waste
(ii) individual rationality (incentives)
no player should get less that what he/she can get alone
1
( )n
ii
x v N
39
( )ix v i i N
An imputation is an allocation satisfying these two conditions:
Super-additivity ensures that the set of imputations is nonempty:
The set of imputations is a polyhedron of dimension n – 1 if the game is essential. It is actually a regular simplex.
1
( ) ( ) ( , )n
i
v i v N I N v
40
( , ) { | ( ) ( ) and ( ) for all }niI N v x x N v N x v i i N
If the game is inessential the imputation set reduces to the allocation that gives each player his/her worth:
If the game is essential
and there are imputations (x1,...,xn) which are better for all members of S:
( )ix v i i N
1
( ) ( )n
i
v i v N
41
( , ) { (1),..., ( )}I N v v v n
In the porters' game, an imputation is defined by three nonnegative numbers x1, x2, x3 summing to 6:
x1, x2, x3 0
x1 + x2 + x3 = 6
In the auction game, an imputation is defined by three nonnegative numbers x1, x2, x3 summing to 300:
x1, x2, x3 0
x1 + x2 + x3 = 300
42
x2
x1
imputations line for n = 2:xi v(i) i = 1,2x1 + x2 = v(N)
43
( ) (2), (2)v N v v
(1), ( ) (1)v v N v
0 v(N)
v(N)
v(1)
v(2)
v(N) – v(2) – v(3)x1
x2 x3
(v(1),v(2),v(3))
imputations' triangle for n = 3:xi v(i) for i = 1,2,3x1 + x2 + x3 = v(N)
44
v(N) – v(1) – v(3) v(N) – v(1) – v(2)
x2 = v(2)x 3
= v
(3)
x1 = v(1)
45
( ) (2) (3), (2), (3)v N v v v v
(1), ( ) (1) (3), (3)v v N v v v (1), (2), ( ) (1) (2)v v v N v v
imputations' triangle for n = 3:xi v(i) for i = 1,2,3x1 + x2 + x3 = v(N)
Extending individual rationality to coalitions leads to the following condition:
no coalition should be allocated less than what it could get by itself
Indeed if coalition S would be in a position to object against the allocation x = (x1,…,xn)…
( ) ( ) pour toutx S v S S N
( ) ( )x S v S
( ) ii S
x S x
49
… because it could then offer more to each of to its members, whatever the players outside S do:
( ) ( ) there exists ( | ) such that:
( ) ( ) and for all
i
i i
x S v S y i S
y S v S y x i S
50
( , ) ( , )C N v I N v
The set of allocations against which no coalition can formulate an objection defines the core (Gillies, 1953):
i.e. core allocations are "socially" stable.
Core allocations are imputations:(just take S = {i} and S = N)
Geometrically, the core is an intersection of half-spaces. Hence it is a convex polyhedron whose dimension does not exceed n – 1.
( , ) { | ( ) ( ) and ( ) ( ) for all }nC N v x x N v N x S v S S N
51
In the porters' game, the core is defined by:
x1, x2, x3 0x1 + x2 + x3 = 6x1 + x2 6x1 + x3 6x2 + x3 6
No allocation does satisfy all seven conditions: they are not compatible.
the core of the porter's game is empty: C(N,v) =
x3 = 0 ... x2 = 0 ... x1 = 0
52
Whatever is the allocation proposed, there will be a coalition to object.
In particular, 2-player coalitions will refuse the egalitarian allocation (2,2,2): they can get 4€ instead of 6€.
Coalition {1,2} can indeed get 6€ and propose the allocation (3,3,0) against which coalition {1,3} will object by proposing
(4,0,2), an allocation which will be rejected by coalition {2,3}, …
Endless… 53
Proposition 2.1 Core allocations x C(N,v) satisfy the following inequalities:
Indeed, if x C(N,v), we have
Hence, in particular, core allocations x satisfy
( ) ( ) ( \ ) for alliv i x v N v N i i N
( ) ( ) ( ) ( \ ) for allv S x S v N v N S S N
54
( ) ( ) ( ) ( \ ) ( ) ( \ )v N x N x S x N S x S v N S
The core of the auction game is defined by:
x1, x2, x3 0x1 + x2 + x3 = p3 – p1 x1 + x2 p2 – p1
x1 + x3 p3 – p1
x2 + x3 0 (redundant)
First observation: (p3 – p1, 0, 0) belongs to the core.
Second observation: x2 = 0 in all core allocations.
x2 = 0
55
i.e. - buyers cannot object against the fact that the seller gets all!
- any allocation giving a positive amount to the firstbuyer is not stable.
The allocation (p2 – p1, 0, p3 – p2) is also in the core. It is the allocation which is most favorable to the second buyer.
56
To summarize, the core of the auction game is defined by:
We observe that the maximum the second buyer can expect is equal to the price difference p3 – p2. The minimum the seller can expect is the lowest price p2.
if the buyers' prices would be equal, say p3, the core then reduces to the allocation (p3 – p1, 0, 0) !
1 3 2 3( , ) {( ,0, ) | }C N v p p p p p p p
57
Indeed, the core conditions are then given by:
x1, x2, x3 0
x1 + x2 + x3 = p3 – p1
x1 + x2 p3 – p1
x1 + x3 p3 – p1
x2 = x3 = 0 x1 = p3 – p1
3 1( , ) {( ,0,0)}C N v p p
58
Core of the crop game with 2 workers and constant return to scale: m = 2 and F(s) = s
x1, x2, x3 0
x1 + x2 + x3 = 2x1 + x2 x1 + x3
x2 + x3 0 (redundant)
First observation: (2, 0, 0) and (0, , ) belong to the core
Second observation: the maximum a worker can expect is
landlord: 1travailleurs: 2 et 3
v(i) = 0v(12) = v(13) = v(23) = 0v(123) = 2
59
We first observe that the extreme allocation (F(m), 0, …,0) always belongs to the core.
Furthermore, the maximum a worker can expect within the core is determined by the marginal productivity of the "last" worker. For all i 1, we have:
This is a consequence of Proposition 2.1. Indeed we have:
( ) ( \ ) ( ) ( 1)ix v N v N i F m F m
( ) ( 1)ix F m F m
60
Example: 4-player gloves market (perfect complements)
Consider four players and the following initial distribution of gloves :
(R, R, L, LL)
The worth of a coalition is equal to the number of pairs it can form.
With two left gloves, player 4 seems to be in an advantageous position...
61
There are 15 coalitions and the characteristic function is defined by:
v(i) = 0 for all i N
v(12) = v(34) = 0
v(13) = v(14) = v(23) = v(24) = 1
v(123) = v(134) = v(234) = 1
v(124) = v(N) = 2
The core consists of a unique allocation: (1, 1, 0, 0) !
(R, R, L, LL)
62
(1, 1, 0, 0) is indeed the unique allocation satisfying the core conditions :
x1, x2, x3, x4 0
x1 + x2 + x3 + x4 = 2
x1 + x3 1
x1 + x4 1
x2 + x3 1
x2 + x4 1
x1 + x2 + x4 2
(ignoring redundant inequalities)
x3 = 0
x1, x2 1 x4 = 0
(R, R, L, LL)
63
Example: 3-player gloves market
Initial distribution of gloves:
(RR, L, LL)
The characteristic function is defined by:
v(1) = v(2) = v(3) = v(23) = 0
v(12) = 1
v(13) = v(N) = 2
64
The core conditions are given by:
x1, x2, x3 0,
x1 + x2 + x3 = 2
x1 + x2 1
x1 + x3 2
In the absence of transfers between players, only two allocations actually satisfy these conditions:
(1, 0, 1) and (2, 0, 0)
(RR, L, LL)
( , ) {(1 ,0,1 ) | 0 1}C N v z z z
65
Game where some v(i) are nonzero can be modified by a translation in such a way that individual values are zero:
starting from a game (N,v), we define the game (N,u) by:
( ) ( ) ( )i S
u S v S v i S N
By construction, u(i) = 0 for all i N.
66
If (y1,…,yn) belongs to the core of the game (N,u), the allocation (x1,…,xn) defined by
belongs to the core of the game (N,v). Indeed:
( )i ix y v i
( ) ( ) ( ) ( ) ( ) ( )i S i S
x S y S v i u S v i v S S N
( ) ( )y S u S S N
( ) ( )x S v S S N
67
translation invariance
x1
x2 x3
0c
x1 + x3 = v(N) – c
fixing the coordinate of one player,we get a line segment in the triangle which is the locus of allocations distributing what is left between theother two players
x2 = c, 0 c v(N)
v(N)
v(N) v(N)
68
(v(N),0,0)
(0,v(N),0) (0,0,v(N))
the segment divides the triangle in two parts
x2 cx1 + x3 v(N) - c
(v(N) – c, c, 0)
(0, c, v(N) – c)
x2 cx1 + x3 v(N) - c
x2 = cx1 + x3 = v(N) - c the imputations (v(N) – c, c, 0)
and (0, c, v(N) – c) are the twoextreme points of the line segment
70
(v(N),0,0)
(0,v(N),0) (0,0,v(N))
the center of gravity of the triangle is the egalitarian allocation x defined byxi = v(N)/3 (i = 1,2,3)
71
(v(N),0,0)
(0,v(N),0)
x1 + x2 = v(1,2)
x2 + x3 = v(2,3)
(0,0,v(N))
x1 + x3 = v(1,3)
x1 +
x3 =
v(N)
x2 =
0
x 1 +
x 2 =
v(N
)
x 3 =
0
x2 + x3 = v(N) x1 = 0
nonempty core
72
(v(N),0,0)
(0,v(N),0)
x1 + x2 = v(1,2)
x2 + x3 = v(2,3)
(0,0,v(N))
x1 + x3 = v(1,3)x
1 + x
3 = v(N
)
x2 =
0
x 1 +
x 2 =
v(N
)
x 3 =
0
x2 + x3 = v(N) x1 = 0
core reduced to a single point
73
(v(N),0,0)
(0,v(N),0)
x1 + x2 = v(1,2)
x2 + x3 = v(2,3)
(0,0,v(N))
x1 + x3 = v(1,3)x
1 + x
3 = v(N
)
x2 =
0
x 1 +
x 2 =
v(N
)
x 3 =
0
x2 + x3 = v(N) x1 = 0
empty core
74
(15,0,0)
(0,15,0)
x1 + x2 = 10
(0,0,15)
x1 + x3 = 10
Crop game with decreasing returns
(5,5,5)
(10,0,5)(10,5,0)
(0,10,5) (0,5,10)75
v(i) = 0v(12) = v(13) = 10v(23) = 0v(123) = 15
(300,0,0)
(0,300,0)
x1 + x2 = 200
(0,0,300)
x1 + x3 = 300
(200,0,100)
Auction game:v(i) = 0v(12) = 200v(13) = 300v(23) = 0v(123) = 300
76
Super-additivity is not sufficient (nor necessary) to ensure non-emptiness of the core. There is a general theorem (due to Shapley and to Bondareva) that gives necessary and sufficient conditions under which the core is nonempty.
For n = 3 and assuming super-additivity and v(i) = 0 for all i, these conditions reduce to a single inequality:
v(12) + v(13) + v(23) ≤ 2 v(N)
Indeed, adding the three inequalities xi + xj v(ij) for all i j, we get:
2(x1 + x2 + x3) v(12) + v(13) + v(23)
79
Proposition 2.2 A game (N,v) whose core is nonempty satisfies the following inequalities:
Indeed, if x C(N,v) we have for all S N:
( ) ( \ ) ( ) pour toutv S v N S v N S N
( ) ( )and ( ) ( \ ) ( ) ( \ )( \ ) ( \ )
( ) ( \ ) ( ) ( )
v S x Sv S v N S x S x N S
v N S x N S
v S v N S x N v N
80
There are classes of games whose cores are empty. The core of a inessential game is nonempty: it reduces to the allocation which gives to each player his/her value: xi = v(i) for all i.
The largest class of games whose cores are nonempty are the convex games.
Proposition 2.3 The core of a convex game is nonempty.
This is a result due to Shapley (1971).
81
Proposition 2.4 The core of a simple game is nonempty if and only if there are veto players who then share the worth of the game.
Proof Consider a decision game (N,v). Let V N denote the set of players with a veto right. We proceed in three steps.
(i) V = C(N,v) =
If V = , then v(N\i) = 1 for all i N. Let x be a core allocation. Combining x(N) = 1 with x(N\i) v(N\i) = 1, we get x = 0, a contradiction.
82
(ii) V C(N,v)
Consider the allocation x defined by
where the xi's for i V are choosen in such a way that x(N) = 1, belongs to the core.
Indeed v(S) = 1 if and only if V S. Hence x(S) ≤ v(S) for all S N.
83
0 for all
0 for alli
i
x i V
x i V
(iii) x C(N,v) xi = 0 for all i V
Consider a core allocation x such that xi > 0 for some i V. Then, using the fact that xi 0 for all i N, we get:
a contradiction: the coalition of veto players can improve upon the allocation x.
84
1 ( )ii V
x v V
The core is a set which may be empty or contain many allocations: they are "stable". The notion of core does not include any value judgement on allocations.
A game with an empty core means that the situation it describes is such that it is impossible to distribute the "social output" without facing objections from some coalitions. A compromise based on accepted principles is then necessary.
A compromise is also necessary when there are many stable allocations or when core allocations are not fair.
The Shapley value is such a compromise.
87
What can a player expect from playing a cooperative game ?
In the porters' game, a possible neutral mechanism consists in ordering the players at random and let them take the suitcases sequentially:
being the first is too early: two suitcases left
being second means 6 € in the pocket
being last is too late: no more suitcases
Each porter has a probability 1/3 of being 2nd
each porter may expect to receive 2 €
88
(0,6,0)
1(6 6 0) 2
6ix
1each branch has the same probability:
6
(3,1,2)
(3,2,1)
(6,0,0)
(1,2,3) (0,6,0)
(1,3,2) (0,0,6)
(2,1,3) (6,0,0)
(2,3,1) (0,0,6)
89
In this case, the egalitarian solution is the natural allocation:
the three porters are in a symmetric position:no one is more "important" than an other
In situations where players are identical ("substitutable"), fairness requires an identical treatment:
each player must have the same expectation
Let's apply this random mechanism to the auction game.
90
1
2
3
1 1100(200 300 300 300) 183
6 61 200
200 336 61 500
(300 100 100) 836 6
x
x
x
(3,1,2)
(3,2,1)
(1,2,3) (0, p2 – p1, p3 – p2) = (0, 200, 100)
(1,3,2)
(2,1,3)
(2,3,1)
(0,0, p3 – p1) = (0, 0, 300)
(p2 – p1, 0, p3 – p2) = (200, 0, 100)
(p3 – p1 ,0, 0,) = (300, 0, 0)
(p3 – p1, 0, 0) = (300, 0, 0)
(p3 – p1, 0, 0) = (300, 0, 0)p = (0, 200, 300)
91
1 2 3
123 0 200 100
132 0 0 300
213 200 0 100
231 300 0 0
312 300 0 0
321 300 0 0
1/6 1100 200 500
550 100 250( , , ) (183,33,83)
3 3 3x
v(i) = 0v(12) = 200v(13) = 300v(23) = 0v(123) = 300
each row corresponds to a permutation
each column corresponds to a player
92
The object is sold at a price
p = 183
which means a gain equal to 116 for the second buyer…
... who gives 33 to the first and keeps 83 for him/herself.
What happens if the two buyers have the same reservation price (say 300) ?
93
1 2 3
123 0 300 0
132 0 0 300
213 300 0 0
231 300 0 0
312 300 0 0
321 300 0 0
1/6 1200 300 300
(200,50,50)x
v(i) = 0v(12) = 300v(13) = 300v(23) = 0v(123) = 300
94
The object is sold at a price
p = 200
and the two buyers have get the same, 50.
... we just don't know which one gets the object !
They are anyway indifferent.
It could be allocated by flipping a coin in which case the buyer who gets the object compensates the other buyer.
95
The procedure is the following...
- players are ordered at random,
- they enter in a room following the given order,
- when a player gets in, he/she receives his/her marginal contribution to the set of players already present.
Each player expects to receive his/her average "marginal contribution", all orders having the same probability.
We denote by n the set of n! orderings (permutations) of the n players.
97
In this way, v(N) is entirely allocated.
Indeed, if the players are ordered according to the permutation = (i1,…,in) n we have successively:
v(i1) – v()
+ [v(i1,i2) – v(i1)]
+ [v(i1,i2,i3) – v(i1,i2)]
+ …
+ [v(i1,i2,…,in-1) – v(i1,i2,…,in-2)]
+ [v(i1,i2,…,in) – v(i1,i2,…,in-1)] = v(N)
98
To each permutation = (i1,…,in) we associate the vector of marginal contributions () defined by:
1 1 1
1 1 1
( ) ( ) ( ) ( )
( ) ( ,..., ) ( ,..., ) ( 2,..., 1)
( ) ( ) ( \ )
k
n
i
i k k
i n
v i v v i
v i i v i i k n
v N v N i
There are n! vectors, not necessarily distinct.
Proposition 3.1 A game is convex if and only if its core is a (Shapley, 1971) convex polyhedron whose vertices are the
marginal contribution vectors.
99
Crop game: constant returns to scale
1 2 3
123 0
132 0
213 0
231 2 0 0
312 0
321 2 0 0
1/6 6 3 3 ( , , )2 2
x
the crop is shared equally between the landlord and the workers who receive 1/2 of the marginal product
v(i) = 0v(12) = v(13) = v(23) = 0v(123) = 2
100
the share of the landlord is greater than the share of the workers who receive 2/3 of the marginal product
1 2 3
123 0 /2
132 0 /2
213 0 /2
231 3/2 0 0
312 /2 0
321 3/2 0 0
1/6 5 2 25
( , , )6 3 3
x
v(i) = 0v(12) = v(13) = v(23) = 0v(123) = 3/2
Crop game: decreasing returns to scale
101
1 2 3
123 0 3/2
132 0 3/2
213 0 3/2
231 5/2 0 0
312 3/2 0
321 5/2 0 0
1/6 7 4 4
v(i) = 0v(12) = v(13) = v(23) = 0v(123) = 5/2
7 2 2( , , )
6 3 3x
Crop game: increasing returns to scale
the share of the landlord is smaller than the share of the workers who receive 4/9 of the marginal product
102
1 2 3
123 0 6 0
132 0 0 6
213 6 0 0
231 0 0 6
312 6 0 0
321 0 6 0
1 2 3
123 0 10 10
132 0 10 10
213 10 0 10
231 20 0 0
312 10 10 0
321 20 0 0
1 2 3
123 0 10 5
132 0 5 10
213 10 0 5
231 15 0 0
312 10 5 0
321 15 0 0
6 0 0
0 6 0
0 0 6
0 10 10
10 0 10
10 10 0
20 0 0
0 10 5
0 5 10
10 0 5
10 5 0
15 0 0
porter game crop game with decreasing returns crop game with decreasing returns
103
This procedures allocates to player i his/her average marginal contribution:
1( )
!n
i ixn
It defines a rule SV which, to any game (N,v) associates an allocation, called "value":
SV : (N,v) (x1,...,xn) = SV(N,v)
It is precisely the value introduced by Loyd Shapley in 1953. He has proposed a set of criteria (axioms) that a rule should meet and that actually characterize uniquely the above rule.
104
Efficiency (Eff): the rule is such that
for all and all ( ) : ( , ) ( )ii N
N v G N N v v N
This is a requirement of collective rationality:
only rules which distribute the entirevalue of the game are considered
106
Symmetry (SYM): the rule is such that
if players i and j are substitutable in a game (N,v), they should be allocated the same amount:
This is a fairness requirement:
"equal treatment of equals"
( , ) ( , )i jN v N v
107
In a symmetric game, all players are substitutable. As aconsequence, any rule satisfying efficiency and symmetry allocates a same amount to each player:
( )( , ) ( , )i j
v NN v N v
n
The porters' game is symmetric:
6( , ) 2 for 1,2,3
3i N v i
108
Equal division (ED) is the simplest rule which satisfies efficiency and symmetry:
( )( , ) 1,...,i
v NED N v i n
n
Is this egalitarian solution always fair ?
The way players contribute to a game may be different.
In particular, should a player who never contributes to a coalition be compensated ?
109
Null player (NULL): the rule is such that
if player i is null in a game (N,v), he/she should receive nothing within that game:
With this axiom, null players can be neglected:
if L is the set of null players in a game (N,v), a rule satisfying A3 can be restricted to the reduced game
(N\L,v') where v'(S) = v(S\L).
( , ) 0i N v
110
Let's modify the porters' game by assuming that player 1 can carry only one suitcase. The characteristic function is then given by:
v(1) = v(2) = v(3) = 0
v(12) = v(13) = 0
v(23) = v(123) = 6
Applying any rule satisfying the three axioms we get:
1 3 26
2( , ) 0 et ( , ) ( , ) 3N v N v N v
player 1 is null
111
This example shows that equal division does not satisfy the null player axiom.
An other rule consists in taking only into account marginal contributions to the grand coalition, and to share the resulting surplus (or deficit) equally among the players:
1
1( , ) [ ( ) ( \ )] ( ) [ ( ) ( \ )]
surplus or déficit
n
ij
AC N v v N v N i v N v N v N jn
112
This rule satisfies efficiency by construction.
It also satisfies symmetry: two substitutes players have identical marginal contributions to the grand coalition.
However it does not satisfy the null player axiom.
Applied to the modified porters' game, this rule results in an absurd outcome:
1
2 3
( , ) 0 2 2
( , ) ( , ) 6 2 4
N v
N v N v
113
There are actually many rules satisfying the above three axioms.
One may for instance modify the rules ED and AC by proceeding in two steps: first allocate zero to the null players and then apply the rule to the remaining player set.
Applied to the modified porters' game the two modified rules lead to the same outcome: (0,3,3).
114
We need additional axioms to reduce the possible rules. Actually, one additional axiom will reduce the set of possible rules to a single rule.
We have seen that games defined on a given set of players can be added to define a new game:
the sum of the games (N,v1) and (N,v2) is the game (N,v) whose characteristic function v is defined by:
1 2( ) ( ) ( )v S v S v S
115
Consider the games (N,v1) and (N,v2) defined by:
v1(i) = 0 v2(i) = 0
v1(12) = v1(13) = 0 v2(12) = v2(23) = 0
v1(23) = 6 v2(13) = 6
v1(123) = 6 v2(123) = 6
v(i) = 0
v(12) = 0
v(13) = v(23) = 6
v (123) = 12
v = v1 + v2
116
Additivity (ADD): the rule is such that
the value of the sum of two games is the sum of the values:
This is a independence axiom:
facing differents games, players evaluate them independently
1 2 1 2 1( , ) ( , ) ( , ) ,...,i i iN v v N v N v i n
117
The modified rules ED and AC do not satisfy additivity. This can be seen by applying them to the games v1, v2 and v = v1 + v2. The modified ED rule gives:
1
2
1 2
1 2
( , ) (0,3,3)
( , ) (3,0,3)
( , ) ( , ) (3,3,6)
( , ) (4, 4,4) (3,3,6)
N v
N v
N v N v
N v v
118
Shapley (1953) has proved the following remarquable proposition:
Proposition 3.2 There is one and only one rule that satisfies (Shapley, 1953) simultaneously EFF, SYM, NULL and ADD.
That unique rule is the Shapley value. Indeed, it is easily verified that the rule defined by the average of the marginal contribution vectors satisfies all four axioms.
There are actually other axiomatizations of the Shapley value.
The most remarquable is due to Young (1985).119
A rule is a marginalist if what it allocates to a player in a game depends only upon his/her marginal contributions:
Proposition 3.3 The Shapley value is the unique marginalist (Young, 1985) rule that satisfies EFF and SYM.
1 1 2 2
1 2
( ) ( \ ) ( ) ( \ )
implies
( , ) ( , )i i
v S v S i v S v S i S N
N v N v
120
Alternative definition
Proposition 3.4 The Shapley value allocates to each player a weighted sum of his/her marginal
contributions:
where the weights n(s) depend only on the size of the coalitions:
( , ) ( ) [ ( ) ( \ )]i nS N
SV N v s v S v S i
( 1)!( )!( )
!n
s n ss
n
121
Proof Let denote the coalition formed by player i and the players who preecede him in the permutation . In particular, if i is last and if i is first.
For a given coalition S containing i, there are (s – 1)!(n – s)! permutations such that
1( , ) [ ( ) ( \ )]
!
1( 1)!( )![ ( ) ( \ )]
!
n
i i i
S N
SV N v v S v S in
s n s v S v S in
iS
{ }iS i iS N
.iS S
122
The coefficients 's are at first sight surprising: they give more weight to small or large coalitions and less weight to coalitions of intermediary size.
But there are less coalitions of small or large size than coalitions of intermediary size …
The number de coalitions of a given size containing a given player is the number de combinations of n – 1 players in groups of size s – 1:
11
1
1
( )!( )
( )!( )!s
n n
ns C
n s s
!
!( )!kn
nC
k n k
123
Hence the sum of the n(s) is equal to 1 if we limit ourself to coalitions containing a given player:
the Shapley value is a weighted sum that is uniform with respect to the size of coalitions
1( )nS NS i
s
1Indeed we have: ( ) ( ) pour tout , 0 .n ns s s s n
n
124
2 2
3 3
4 4
5 5
6 6
1 12 ( , ) (1,1)
2 2
1 1 13 ( , , ) (1,2,1)
3 6 3
1 1 1 14 ( , , , ) (1,3,3,1)
4 12 12 4
1 1 1 1 15 ( , , , , ) (1,4,6,4,1)
5 20 30 20 5
1 1 1 1 1 16 ( , , , , , ) (1,5,10,10,5,1)
6 30 60 60 30 6
n
n
n
n
n
125
Auction game
1 2 3 (s)
1 0 0 0 1/3
2 0 0 0 1/3
3 0 0 0 1/3
12 200 200 0 1/6
13 300 0 300 1/6
23 0 0 0 1/6
123 300 0 100 1/3
1
2
3
1 1 1 1100200 300 300 183
6 6 3 61
200 3361 1 500
300 100 836 3 6
x
x
x
126
v(i) = 0v(12) = 200v(13) = 300v(23) = 0v(123) = 300
Crop game: constant returns
1 2 3 (s)
1 0 0 0 1/3
2 0 0 0 1/3
3 0 0 0 1/3
12 a a 0 1/6
13 a 0 a 1/6
23 0 0 0 1/6
123 2a a a 1/3
1
2
3
2 12
6 31 1
6 3 21 1
6 3 2
x a a a
ax a a
ax a a
v(i) = 0v(12) = av(13) = av(23) = 0v(123) = 2a
127
By construction, the Shapley value satisfies the four axioms.
Individual rationality is not included in the axioms and the Shapley value may not be individually rational when applied to a game which is not superadditive.
The following proposition is an immediate consequence of super-additivity:
Proposition 3.5 The Shapley value defines an imputation that is reasonable in the sense of Milnor.
129
Proof By superadditivity, marginal contributions vectors are individually rational allocations and Milnor-reasonable and the Shapley value is an average of these vectors.
The Shapley value does not necessarily define a core allocation:
in the auction game, the Shapley value compensates the buyer with the smallest reservation price.
However for some classes of games, the Shapley value defines a core allocation. This is the case of convex games.
130
We know that the core of a convex game is a polyhedron whose vertices are the marginal contribution vectors.
The Shapley value is a convex combination of the marginal contribution vectors. The following proposition then follows:
Proposition 3.6 The Shapley value of a convex game defines a core allocation.
Remark: The n! vectors all appear in the computation of the value: if two vectors are identical, they are taken into account twice.
131
Example: bankruptcy game
n players (creditors)
E > 0 value to be distributed
d1,...,dn 0 the claims 1
n
ii
d E
( ) 0, ( \ )v S Max E d N S
( )v N E
v(S) is the maximum S can expect to receive if all other creditors have been compensated possibly partly
( ) [0, ( ) ]
where ( ) 0
v S Max d S K
K d N E
or
132
Case where n = 3, with E = 600 and d = (150, 300, 450):
v(1) = v(2) = 0
v(3) = v(12) = 150
v(13) = 300
v(23) = 450
v(123) = 600
The bankruptcy game is convex: its core is nonempty andcontains the Shapley value.
( ) 0, ( \ )v S Max E d N S
133
Convexity:
( ) ( ) ( ) ( )
( ( ) ( ) 2 ) ( ( ) ) ( )
( ( ) ) ( ) 0
si ( ) et ( )
(0 ( ) ) ( ( ) ) 0 ( \ ) 0
si ( ) et ( )
0 0 (0, ( ) ) 0 0
si ( ) et ( )
v S v T v S T v S T
d S d T K d S T K v S T
d S T K v S T
d S K d T K
d T K d S T K d S T
d S K d T K
Max d S T K
d S K d T K
( ) ( ) ( ) ( )d S d T d S T d S T
( ) [0, ( ) ]v S Max d S K
134
1 2 3
123 0 150 450
132 0 300 300
213 150 0 450
231 150 0 450
312 150 300 150
321 150 300 150
x 1/6 600 1050 1950
1
2
3
600100
61050
1756
1950325
6
x
x
x
losses = (50, 125, 125)
(150)
(300)
(450)
v(1) = v(2) = 0v(3) = v(12) = 150v(13) = 300 v(23) = 450v(123) = 600
Shapley value
135
1 2 3 (s)
1 0 0 0 1/3
2 0 0 0 1/3
3 0 0 150 1/3
12 150 150 0 1/6
13 150 0 300 1/6
23 0 300 450 1/6
123 150 300 450 1/3
1
2
3
300 150100
6 3450 300
1756 3
750 600325
6 3
x
x
x
v(1) = v(2) = 0v(3) = v(12) = 150v(13) = 300 v(23) = 450v(123) = 600
136
(600,0,0)
(0,600,0)
x3 = 150
x1 = 150
(0,0,600)
x2 = 300
(150,0,450)(150,300,150)
(0,300,300) (0,150,450)
x3 = 4500 x1 150
0 x2 300150 x3 450
v(1) = v(2) = 0v(3) = v(12) = 150v(13) = 300 v(23) = 450v(123) = 600
imputations' triangle
(100,175,325)
imputations and core
137
Observations:
- there are 4 distinct marginal contributions vectors:
(150,300,150), (150,0,450), (0,300,300), (0,150,450)
- they are the vertices of the core: it is a characteristic of convex games
- the Shapley value is somewhere in the center of the core, an other characteristic of convex games.
138
The Talmud
A man dies and his three wives have each a claim on his estate, following past promises. The value of the estate falls short of the total of the claims. Here is what a Mishnah suggests:
The solution is the nucleolus in all 3 cases.
d1=100 d2=200 d3=300
E=100 33.3 33.3 33.3
E=200 50 75 75
E=300 50 100 150
Shapley valueNucleolus(equal division)
Shapley value
Nucleolus
139
The Shapley value for the three Talmudic cases
In the case where E = 100, the game is symmetric: v(S) = 0 for all S N and the allocation is equal division, as suggested by the Talmud.
In the two other cases, we have the following outcomes:
E = 200 (33,3, 83,3, 83,3)
E = 300 (50, 100, 150)
140
They result from the following tables:
1 2 3
123 0 0 200
132 0 200 0
213 0 0 200
231 100 0 100
312 0 200 0
321 100 100 0
x 1/6 200 500 500
1 2 3
123 0 0 300
132 0 200 100
213 0 0 300
231 100 0 200
312 100 200 0
321 100 200 0
x 1/6 300 600 900
141
1 2 3
123 0 10 15
132 0 15 10
213 10 0 15
231 25 0 0
312 10 15 0
321 25 0 0
1/6 70 40 40
35 20 20( , , )
3 3 3x
Crop game: increasing returns ( = 10)
5 distincts marginal contributions vectors
v(i) = 0v(12) = 10v(13) = 10v(23) = 0v(123) = 25
142
(25,0,0)
(0,25,0) (0,0,25)
Core of the crop game with increasing returns
(10,0,15)(10,15,0)
(0,15,10) (0,10,15)
the core is the convexenvelope of the marginalcontribution vectors
35 20 20( , , )
3 3 3a convex game
143
1 2 3
123 0 10 5
132 0 5 10
213 10 0 5
231 15 0 0
312 10 5 0
321 15 0 0
1/6 50 20 2025 10 10
( , , )3 3 3
x
Crop game: decreasing returns ( = 10)
v(i) = 0v(12) = 10v(13) = 10v(23) = 0v(123) = 15
5 distincts marginal contributions vectors
144
(15,0,0)
(0,15,0) (0,0,15)
Core of the crop game with decreasing returns
(5,5,5)
(10,0,5)(10,5,0)
(0,10,5) (0,5,10)
25 10 10( , , )
3 3 3
2 320
3x x
145
(15,0,0)
(0,15,0) (0,0,15)
(10,0,5)(10,5,0)
(0,10,5) (0,5,10)
The core is a subset of the convex envelope of the marginal contribution vectors called the "Weber set"
146
1 2 3
123 0 10 2
132 0 2 10
213 10 0 2
231 12 0 0
312 10 2 0
321 12 0 0
1/6 44 14 1422 7 7
( , , ) noyau3 3 3
x
Case where the Shapley value does not belong to the core
v(i) = 0v(12) = 10v(13) = 10v(23) = 0v(123) = 12
147
(12,0,0)
(0,12,0) (0,0,12)
(2,2,10)
(10,0,2)(10,2,0)
(0,10,2) (0,2,10)
22 7 7( , , )
3 3 3
2 320
3x x
148
Crop game: general case
Workers are substitutes: they get the same wage and we need only to compute what the value allocates to the landlord. In a given permutation, only the position of the landlord counts.
if the landlord is in position k, he gets F(k-1)
and there are m + 1 positions possibles
1
11 1
1 11
1 1( , ) ( ) ( )
m m
k k
SV N v F k F km m
149
Crop game: case of an excess labor supply (k < m)
0 k m x
y( ) if 0
( ) if
y F x x k
F k x k
v(S) = 0 if 1 S
v(S) = F(s – 1) if 1 S and s k + 1
v(S) = F(k) if 1 S and s > k + 1
F(k)
155
The core reduces to the allocation which gives all to the landlord: C(N,v) = {(F(k), 0,…, 0)}. Indeed we have seen that the maximum a worker can hope to get is equal to the marginal product that is zero here!
Alternatively:
( ) ( )
( \ ) ( \ ) ( )
( ) ( ) 0 0j j
x N F k
x N j v N j F k
x F k F k x
156
The Shapley value allocates a rent
to the landlord. What is left is uniformly distributed to the workers. If F(k) = k, we have:
There is equality between the rent r and the wage bill mw iff k = m r = mw = k/2.
1
1( ) ( ) ( )
1
k
s
r F s m k F km
1
1 ( 1) 2 1( ) ( )
1 1 2 1 2
k
s
k k k m kr s m k k m k k
m m m
157
The wage of an individual worker is given by:
2 1 1 1
1 2 2 1
k m k k kw k
m m m
0 k m x
y
L
W
k
158
Production game: y = f(x) (one output – one input, with a fixed cost)
1
1 1
( ) 0 if 0
3( ) if
f x x
x x
10 x
y
1 2 3input endowments: 1, 2, 3 ( 3)n
(1) (1) 0 (12) (3) 6
(2) (2) 3 (13) (4) 9
(3) (3) 6 (23) (5) 12 (123) (6) 15
v f v f
v f v f
v f v f v f
159
(0,15,0) (0,0,15)
(15,0,0)
x3 = 9x2 = 6
x1 = 3 (3,6,6) (3,3,9)
(0,6,9)
0 x1 3
3 x2 66 x3 9
x3 = 6x2 = 3v(1) = 0
v(2) = 3v(3) = v(12) = 6v(13) = 9 v(23) = 12v(123) = 15
x1 + x2 = 6
x1 + x2 = 9
x1 + x3 = 12
x1 + x3 = 9
x2 + x3 = 12
core
161
1 2 3
123 0 6 9
132 0 6 9
213 3 3 9
231 3 3 9
312 3 6 6
321 3 6 6
1/6 12 30 48 (2,5,8)x
v(1) = 0v(2) = 3v(3) = v(12) = 6v(13) = 9 v(23) = 12v(123) = 15
Shapley value
The core is the convex hull of the marginal contribution vectors the game is convex.
162
Modified game:
This is a symmetric game and the Shapley value is therefore the egalitarian allocation:
(N,u) = (2,2,2)
(1) 0 (12) 3
(2) 0 (13) 3
(3) 0 (23) 3 (123) 6
u u
u u
u u u
( ) ( ) ( )i S
u S v S v i
163
(6,0,0)
(0,6,0) (0,0,6)
x2 + x3 = 3
x1 + x2 = 3x1 + x3 = 3
(3,0,3)(3,3,0)
(0,3,3)
core of the modified game
164
1
1
If ( ,..., ) is an imputation of the game ( , ), the imputation ( ,..., ) corresponding to the game ( , )is given by:
( )
n
n
i i
y y N ux x N v
x y v i
We obtain in this way the Shapley value of the original game:
(N,v) = (2,2,2) + (0,3,6) = (2,5,8)
which is at the center of the core.
165
1 2 3 4
13 1 0 1 014 1 0 0 123 0 1 1 024 0 1 0 1
123 0 0 1 0134 1 0 0 0234 0 1 0 0124 1 1 0 2
1234 1 1 0 1
Gloves market: (R,R,L,LL)
4
1 1 1 1( , , , )4 12 12 4
7 7 3 7( , ) ( , , , )
12 12 12 12N v
( , ) ( , )N v C N v
( , ) {(1,1,0,0)}C N v
166
1 2 3
12 1 1 013 2 0 2
123 2 0 1
Gloves market: (RR,L,LL)
3
1 1 1( , , )3 6 3
7 4( , ) ( , , )
6 6
1
6N v
( , ) ( , )N v C N v
0( , ) {(1 , ,1 ) | 0 1}C N v z z z
167
The formation of cartels or syndicates, viewed as agreements between players who decide to act as a single player can be analyzed using the Shapley value .
In the porters' game, we immediately see that any two players have an interest to form a cartel cartel, but this does not tell you which cartel will form !
If players 1 et 2 get together, the game reduces to a 2-player game where player 3 is null:
v*(12) = 6 et v*(3) = 0
v*(12,3) = 6
169
In the auction game, the buyers have an interest to form a cartel. The game becomes a 2-player game:
v*(i) = 0 pour i = 1 et 23v*(1,23) = 300
which calls naturally for an equal division between the seller and the cartel of buyers.
In the case where the buyers have the same reservation price, we have:
the buyers get 150 when in a cartel while the Shapley value allocates them 100
170
In the crop game, if workers form a syndicate, the game becomes a 2-player symmetric game whose solution is:
to be compared to the allocation derived from the Shapley value:
1 23(2)
2
Fx x
1
2 3
(1) (2)
32 (2) (1)
3
F Fx
F Fx x
171
The difference is given by:
It is positive if and only if returns to scale are decreasing:
F(2) < 2F(1).
workers have an incentive to form a syndicate if returns to scale are decreasing.
(2) 2 (2) (1) 1 (2)(1)
2 3 3 2
F F F FF
172
A set N = {1,…,n} of players have a common project that costs c(N).
The (minimum) cost c(S) of realizing this project to the benefit of the members of any coalition S N is also known.
This defines a cost function c: S N c(S) with c() = 0.
It is a characteristic function and the couple (N,c) defines a cooperative game with transferable utility, here interpreted as a cost sharing game.
175
Example: water distribution
Three towns consider building of a water distribution system linked to a source. The costs of connecting each town to the source and to the other towns are given by:
Town 1 Town 2 Town 3
Source 18 21 27Town 1 15 12Town 2 24
S 2
3
18
21
27
15
24
12
1
176
Looking at coalitions of towns, the (minimum) costs are given by:
c(1) = 18 c(12) = 33
c(2) = 21 c(13) = 30
c(3) = 27 c(23) = 45
Together with c(123) = 45, this defines the cost game (N,c).
S
1
2
3
18
21
27
15
24
12
178
Subadditivity:
Cost functions defining cost games are assumed to be subadditive :
Subadditivity is typically verified by cost function derived from problems involving networks: the water distribution example is subadditive.
( ) ( ) ( )S T c S T c S c T
179
Subadditivity implies that there is no loss in cooperating:
A cost game is essential if
The water distribution example is essential.
The cost function of the water distribution example is actually concave.
180
( ) ( )i N
c i c N
( ) ( )i S
c i c N
A cost game is concave if
Concavity implies subadditivity.
Equivalently, a cost game is concave if the marginal costs are non-increasing:
( ) ( ) ( ) ( ) for all ,c S T c S c T c S T S T N
for all , , such that :
( ) ( \ ) ( ) ( \ )
i S T i S T
c S c S i c T c T i
181
In a given cost sharing situation (N,c), the surplus generated by cooperation is measured by the difference
The natural surplus sharing game (N,v) associated to a cost game (N,c), is then defined by:
Proposition 4.1 The surplus game associated to an essential – subadditive – concave cost game is essential – superadditive – convex.
( ) ( )i N
c i c N
182
( ) ( ) ( )i S
v S c i c S
Subadditivity of the cost function implies superadditivity of the surplus function:
The surplus game (N,v) associated to an essential cost game (N,c) is essential. Indeed v(i) = 0 for all i and
Concavity of the cost function implies convexity of the surplus function:
183
( ) ( ) ( ) ( ) ( ) ( )S T v S T v S v T c S c T c S T
( ) ( ) ( ) 0i N
c N c i v N
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )v S T v S v T v S T c S c T c S T c S T
Question: how to share c(N) between the players?
A first approach consists in imposing conditions on cost allocations y = (y1, y2,..., yn):
individual rationality(incentive compatibility)
collective rationality(Pareto efficiency)
core (social stability)
( ) ( )y N c N
( ) for alliy c i i N
( ) ( ) for ally S c S S N
185
Proposition 4.2 Core conditions y satisfy the following inequalities:
Proof The left hand inequalities is part of core's definition. sing core's inequalities, we get:
Hence no coalition pays less than its "additionnal cost": there are no cross-subsidies.
( \ ) ( \ )
( ) ( )
y N S c N S
y N c N
( ) ( ) ( \ )y S c N c N S
186
( ) ( \ ) ( ) ( ) for allc N c N S y S c S S N
In particular, core allocations y satisfy the inequalities:
i.e. players pay at least their marginal cost to the grand coalition.
Surplus allocations x and cost allocations y are related by the following equations:
( ) ( \ ) ( )ic N c N i y c i i N
187
( )i ix y c i ( )i iy c i x
( )i ix c i y
In terms of surplus sharing, we have successively:
( ) ( ) ( ) ( ) ( ) ( )ii N i N
y N c N c i x c N x N v N
( ) ( ) ( ) 0 ( )i i iy c i c i x c i x v i
( ) ( ) ( ) ( ) ( ) ( )ii S i S
y S c S c i x c S x S v S
( )i ix y c i
188
(0,45,0)
(45,0,0)
y3 = 12
y2 = 15
(0,0,45)
y3 = 27y2 = 21
y1 = 18
y1 18 y2 21 y3 27y1 + y2 33 y1 + y3 30y2 + y3 45y1 + y2 + y3 = 45
0 y1 18 15 y2 2112 y3 27y1 + y2 + y3 = 45
Water distribution
(-3,21,27) 189
the yellow is the imputation set
We need to define a sharing rule that associates to each cost game an allocation y = (N,c)
There exist various such rules. The most "popular" are the Shapley value and the nucleolus. They can be obtained equivalently from the cost game or indirectly from the surplus game.
The nucleolus defines a core allocation, if the core is nonempty, while the cost allocation derived from the Shapley value may not be stable.
191
To compute the Shapley value, we construct the vectors of marginal costs that, to each permutation = (i1,...,in) n associates the imputation t() defined by:
The Shapley value of a cost game (N,c) is defined as the average marginal cost vector:
1 1 1
1 1 1 2 1)
( ) ( ) ( ) ( )
( ) ( ,..., ) ( ,..., ) ( ,...,
( ) ( ) ( \ )k
n
i
i k k
i n
t c i c c i
t c i i c i i k n
t c N c N i
11
( , ) ( ) ,...,!
n
i iSV N c t i nn
192
Given the surplus game (N,v) associated to the cost game (N,c), the marginal contribution vector () and the marginal cost vector t() associated to permutation n are related by the equations
Hence, the value of the cost game (N,c) can be obtained from the value of the surplus game (N,v):
1( , ) ( ) ( , ) ,...,i iSV N c c i SV N v i n
193
1( ) ( ) ( ) ,...,i it c i i n
Water distribution: cost game
c(1) = 18 c(12) = 33c(2) = 21 c(13) = 30c(3) = 27 c(23) = 45c(123) = 45
1 2 3
123 18 15 12
132 18 15 12
213 12 21 12
231 0 21 24
312 3 15 27
321 0 18 27
1/6 51 105 114
( , ) (8.5,17.5,19)SV N c
194
Water distribution: surplus game
v(1) = 0 v(12) = 6v(2) = 0 v(13) = 15v(3) = 0 v(23) = 3v(123) = 21
1 2 3
123 0 6 15
132 0 6 15
213 6 0 15
231 18 0 3
312 15 6 0
321 18 3 0
1/6 57 21 48
( , ) (9.5, 3.5, 8)SV N v
195
( , ) ( , ) (9.5, 3.5, 8) (8.5,17.5,19) (18, 21, 27)SV N v SV N C
1 2 3
123 20 15 15
132 20 5 25
213 10 25 15
231 5 25 20
312 15 5 30
321 5 15 30
c(1) = 20 c(12) = 35c(2) = 25 c(13) = 45c(3) = 30 c(23) = 45c(123) = 50
5 ≤ y1 ≤ 20 5 ≤ y2 ≤ 2515 ≤ y3 ≤ 30
x1/6 75 90 135
12.5 15 22.5
(20,15,15)
(10,25,15)
(5,25,20) (5,15,30)
(15,5,30)
(20,5,25)
y3 = 15
y3 = 30
(50,0,0)
(0,50,0) (0,0,50)
y2 = 5
y2 = 25
y1 = 20
y1 = 5
(12.5,15,22.5)
6 distinct vectors
196
An airport game is defined by individual costs ci 0 and the associated cost function is given by:
Airport games are monotone increasing and concave (hence subadditive). They are essential if costs are not all equal.
Without loss of generality, players are ordered in terms of their costs:
( ) i S ic S Max c
198
1 20 ... ( )n nc c c c N c
Concavity: let S et T be two coalitions with values:
where cS cT (without loss of generality).
( ) et ( )S Tc S c c T c
( ) ( ) ( ) ( ) ( ) 0Tc S c T c S T c S T c c S T
( ) Max ( , )
( ) Min ( , )T S S
T S T
c S T c c c
c S T c c c
199
The natural cost allocation is the following:
11
1 2 12
11 2 1
1 21 2 11
1
1 1
1 2
...
...
...
...
k kk
n nn n n
cy
nc c c
yn n
c cc c cy
n n n k
c cc c cy c c
n n
it is the Shapley value !
200
In the case where n = 3:
1
2
3
(1)
(2) (12)
(3) (13) (23) (123)
c c
c c c
c c c c c
11
1 2 1 2 12
1 2 1 2 13 3 2 3
3
3 2 2 6
3 2 2 6
cy
c c c c cy
c c c c cy c c c
201
1 2 3
123 c1 c2 – c1 c3 – c2
132 c1 0 c3 – c1
213 0 c2 c3 – c2
231 0 c2 c3 – c2
312 0 0 c3
321 0 0 c3
x 1/6 2c1 3c2 – c1 6c3 – 3c2 – c1
1 2 1 2 13( , ) ( , , )
3 2 6 2 6
c c c c cSV N c c
4 distinct vectors
202
y1 c1 y2 c2 y3 c3
y1 + y2 c2 y1 + y3 c3
y2 + y3 c3
y1 + y2 + y3 = c3
(c3,0,0)
(0,c3,0) (0,0,c3)
0 y1 c1 0 y2 c2 c3 – c2 y3 c3
y1 + y2 + y3 = c3
y2 = c2y3 = c3 – c2
(0,c2,c3-c2)
(c1,c2-c1,c3-c2) (c1,0,c3-c1)y1 = c1
203
The associated surplus game is given by:
( ) Maxi i S ii S
v S c c
1
2
1 2
(1) (2) (3) 0
(12) (13)
(23)
(123)
v v v
v v c
v c
v c c
Remark The last two players are substitutable, a property that is independent of the number of players.
204
1 2 3
123 0 c1 c2
132 0 c2 c1
213 c1 0 c2
231 c1 0 c2
312 c1 c2 0
321 c1 c2 0
x 1/6 4c1 c1 + 3c2 c1 + 3c2
1 1 2 1 22( , ) ( , , )
3 6 2 6 2
c c c c cSV N v 4 distinct vectors
205
1 2 3( , ) ( , ) ( , , )SV N v SV N C c c c
(c1 + c2,0,0)
x1 = c1
x2 = c2x3 = c2
x1 0 x2 0x3 0x1 + x2 c1 x1 + x3 c1
x2 + x3 c2
x1 + x2 + x3 = c1 + c2
0 x1 c1 0 x2 c2 0 x3 c2
y1 + y2 + y3 = c3
(0,c1 + c2,0) (0,0,c1 + c2)
(c1,c2,0) (c1,0,c2)
(0,c2,c1) (0,c1,c2)
206
The Shapley value of the airport game (N,c) can be written compactly as:
SV(N,c) = B.c
where B is the n x n triangular matrix defined by:
207
1if
( 1)( )
1for all
1
ij
ii
b j in j n j
b in i
The matrix B has a simple recursive structure. For n = 4, it is given by:
The matrices are overlapping, starting from the lower right element 1. For instance, if n = 5 the first column starts with 1/5, followed by –1/20...
208
1/ 4 0 0 0
1/12 1/ 3 0 0
1/12 1/ 6 1/ 2 0
1/12 1/ 6 1/ 2 1
Initial motivation: REACH program (Registration, Evaluation, Authorisation and Restriction of Chemicals) imposed on EU Chemical Industry:
about 30.000 substances and
100 parameters per substance
Submission process: started in 2008, it will extend until 2018.
Cooperation between firms is encouraged both in terms of data acquisition and data sharing: there is an exchange forum for each substance (SIEF) whose role is to facilitate the exchange of data among firms.
210
Looking at a particular parameter, for a given substance, there are three cases:
1. that data is freely accessible2. no firm has that data: it has to be produced3. that data is held by some but not all firms
In case 1, the data is freely accessible and there is no property rights: no need for compensations
In case 2, no firm has the data: it must be acquired jointly at a cost that has to be shared among the firms
equal splitting is the natural rule in this case.
211
We shall consider only case 3 that opens the possibility of compensations among firms.
Consider first the simplest case involving 2 firms where only one holds the data (say firm 1).
The data has a value that corresponds to the cost of duplicating it, say d > 0.
What would be a fair compensation: how much should firm 2 be asked to pay to firm 1 within a joint submission ?
212
If that data was held by no player, they should each pay d/2.
So firm 2 could be asked to pay d/2 to firm 1. Equivalently, both firm pay d/2 and firm 1 gets d back.
With 3 firms, only firm 1 holding the data, the same argument suggests that the other firms should each pay d/3 to firm 1.
In the situation where only firm 3 does not hold the data, firm 3 should still pay d/3, an amount divided equally between firm 1 and 2: each receives d/6.
213
In case of n firms, the data being held by t firms, 0 < t ≤ n, the n – t firms without the data each pay d/n each.
The resulting allocation is then given by:
0 if owns the data
if does not
i
i
d dy i
n t
dy i
n
It turns out to be the Shapley value of some appropriate game.
214
A data sharing situation is described by:
N = {1,...,n} the set of players
M0 = {1,…,m} the set of available data
dh > 0 the cost of reproducing data h
Mi M0 the dataset held by player i
01with
n
iiM M
215
The cost for a coalition S if the cost of completing its dataset:
where
This defines a cost game called data game.
0
0\
( )S S
h hh M M h M
C S d v d
0
0
is the data set held by coalition
is the value of the complete dataset
S ii S
hh M
M M S
v d
( ) 0C N
216
Example: N = {1,2,3} and M0 = {1,2,3,4}
M1 = {1,3} d1 = 90 v0 = 210
M2 = {1,2} d2 = 20
M3 = {3,4} d3 = 40
d4 = 60
c(1) = d2 + d4 = 80 c(1,2) = d4 = 60
c(2) = d3 + d4 = 100 c(1,3) = d2 = 20
c(3) = d1 + d2 = 110 c(2,3) = c(1,2,3) = 0
217
Proposition 4.4 The cost function c defining a data game (N,c) is:
decreasing: S T c(T) c(S)
subadditive: S T = c(S) + c(T) c(ST)
essential if Mi M0 for some i: i c(i) > c(N)
218
Proof
essential:
subadditive: if S T =
non-increasing: S T MS MT and
0 01 1
for some ( ) 0 ( )i
n n
i hi i h M
M M i c i nv d c N
0
0
( ) ( ) 2
( ) ( )S T
S T
h hh M h M
hh M M
c S c T v d d
c S T v d c S T
219
( ) ( ) 0S T
h hh M h Mc T c S d d
Case of a single data
The "elementary" data game (N,ch) associated to data h:
where Th denotes the set of players holding data h.
( ) if
0 if
h h h S
h S
c S d S T h M
S T h M
220
By working data by data, any data game can be written as a sum of elementary data games
\
( ) ( )S
h hh M h M M
c S d c S
( ) if
0 if
h h S
S
c S d h M
h M
221
Because c(N) = 0, we are facing a pure compensation problem.
We consider imputations:
where yi > 0 means that i pays yi
yi < 0 means that i receives – yi
1
( , ) | 0 and ( ) for alln
ni i
i
I N c y y y c i i N
222
Proposition 4.5 The core of a data game is then by:
where
is the value of the data player i is alone to hold and is the set of data held by single players
the core of a data game depends only on the data held by single players
and | for somei
i h i i jh M
v d M h M h M j i
( , ) { | 0 and for all }ni i i
i N
C N c y y y v i
( \ )iv c N i
223
iM
the core is nonempty: it always contains the nocompensation allocation 0 = (0,0,…0)
the maximum amount a player can expect to receive is the value of the data he/she is alone to hold
no player can expect to be compensated for data he/she is not alone to hold
if all data are each owned by at least 2 players, the core reduces to {0}
( , ) { | 0 and for all }ni i i
i N
C N C y y y v i N
224
Proof If y C(N,c) we have
If instead y satisfies core's definition, we have:
Because y(N) = 0, we then have:
( \ ) ( \ ) i i iy N i c N i v y v
225
\
( \ ) ( ) for allii N S
y N S v C S S
0\ \
( ) ( ) for allS
i hi N S h M M
y S v d C S S
226
The core of a data game
is a regular simplex:
adding the vector to core allocation and dividing by result in the standard unit simplex
1( ,..., )nv v
0 iv v
( , ) { | ( ) 0 and for all }ni iC N C y y N y v i N
{ | 1 and 0 for all }nn i i
i N
y y y i N
The core is full dimensional if and its vertices are given by:
10 1 2
21 0 2
1 2 0
( , ,..., )
( , ,..., )
...
( , ,..., )
n
n
nn
v v v v
v v v v
v v v v
for someiM i
0 1 2 3( , , )v v v v
1 0 2 3( , , )v v v v 1 2 0 3( , , )v v v v
3 3y v2 2y v
1 1y v
Core of a 3-player data game
2y
1y
3y
228
Looking at the core of the elementary game (N,Ch) associated to data h:
either it reduces to {0} because data h is held by more than one player
or it is a full dimensional regular simplex because only one player holds data h
229
(0,0,0)
( , ,0)h hd d ( , 0, )h hd d
3 0y 2 0y
1 hy d
Core of a 3-player elementary data game where player 1 is alone to hold data h
1
2 3 0
hv d
v v
230
The Shapley value of an elementary game
ch (S) = 0 si S Th
ch (S) = dh si S Th =
where th is the number of players in Th.
1( , ) ( )[ ( ) ( )] hh i h n h h
di T N c c i c
n
1( , ) h h h
h i h hh h
n t d di T N c d
t n n t
231
By additivity
- the cost of the complete dataset is uniformly allocated among all players
- the cost of each data is uniformly redistributed to the players holding it.
0
( , ) ( , )i i hh M
N c N c
0 1( , ) ,...,i
hi
h M h
v dN c i n
n t
0
0 hh M
v d
232
Example d = (90, 20, 40, 60)
M1 = {1,3}, M2 = {1,2}, M3 = {3,4}
Here y belongs to the core.
0210
703
v
n
0( , )i
hi
h M h
v dN c
n t
1
2
3
90 4070 5
2 290
70 20 52
4070 60 10
2
y
y
y
233
Axiomatization of the value on the set of data games
The Shapley value is uniquely determined by 4 axioms: efficiency, symmetry, null player and additivity.
There are no null players in data games.
Keeping efficiency, symmetry and additivity, one possible additional axiom could be:
for all data sharing situations (M,d) = (M1,...,Mn,d1,...,dn) such that Mi = or M0:
234
0( , )i i
vM M d
n
A set N = {1,…,n} of agents and a set M = {1,…,m} of indivisible objects (say houses) to be allocated, one for each agent (m n).
Data: a "utility" matrix U = [ui(h) | i N, h M]
Here ui(h) is the reservation price of agent i for house h (i.e. the maximum price i is willing to pay for house h).
It is expressed in monetary terms. It is the value that agent i attach to house h.
236
Side payments being allowed, the associated TU-game is given by:
where F is the set of all functions f: N M that associate a house to each player.
c(S) is the cost of the houses that are optimally allocated to its members. (N,c) is the assignment game studied by Shapley and Shubik (1972).
It is a concave (hence subadditive) cost game.
( ) ( ( ))f F ii Sc S Max u f i
237
Example
u1 u2 u3
a 3 9 9
b 12 6 6
c 9 6 3
c(1) = 12c(2) = 9c(3) = 9c(12) = 21c(13) = 21c(23) = 15c(123) = 27
1 2 3
123 12 9 6
132 12 6 9
213 12 9 6
231 12 9 6
312 12 6 9
321 12 6 9
1/6
72 45 45
SV(N,c) = (12, 7.5, 7.5)optimal allocation = (12,6,9):1 receives house b2 receives house c3 receives house a
transfers: (12,6,9) – (12,7.5,7.5) = (0, –1.5, 1.5)
players 2 and 3 are substitutes
238
(0,27,0) (0,0,27)
(27,0,0)
x3 = 9
x2 = 6
x1 = 12
(12,6,9)(12,9,6)
x1 = 12
6 x2 96 x3 9 x3 = 6
x2 = 9v(1) = 12v(2) = 9v(3) = 9v(12) = 21v(13) = 21 v(23) = 15v(123) = 27
core
(12,7.5,7.5)
set ofimputations
(9,9,9)
(12,6,9)(12,9,6)
optimal allocation before transfers
239
The functioning of most institutions relies on groups of decision makers facing choices and there are rules specifying how decisions are taken.
Rules may be as simple as unanimity or simple majority, or they may be more or less complex, like for instance decisions within the UN Security Council or EU Council of Ministers.
The question is to measure the "power" that each decision maker has given the rules. How much power has a permanent member of the UN Security Council or a given country within the EU Council of Ministers.
241
A decision games is defined by:
players = decision makers i N = {1,...,n}
rules = a collection W of winning coalitions
A paire (N,W) defines a decision game.
There are no restrictions; all coalitions are a priori possible.
A coalition minimal winning if removing any of its members makes it loosing:
243
if and only if and \S M S W S i W
Simple 3-player decision games:
unanimity: W = { {1,2,3} }
simple majority: W = { {1,2}, {1,3}, {2,3}, {1,2,3} }
simple majority + veto: W = { {1,2}, {1,3}, {1,2,3} }
dictatorship: W = { {1}, {1,2}, {1,3}, {1,2,3} }
244
Minimals winning coalitions:
unanimity: M = {1,2,3}
simple majority: M = {1,2}, {1,3}, {2,3}
simple majority + veto: M = {1,2}, {1,3}
dictatorship: M = {1}
245
Assumptions
D1 the grand coalition is always winning: N W
D2 two disjoint coalitions cannot be simultaneously winning:
Consequence: if a coalition is winning, its complement is necessarily loosing:
246
andS W S T T W
\S W N S W
D3 enlarging a winning coalition keeps it winning :
the set of all winning coalitions can be obtained from the set of minimal winning coalitions by adding players
247
andS W S T T W
Weighted majority games form a particular class of decision games.
Decision maker i is characterized by a weight wi 0 and a coalition is winning if and only if its weight is not below some given quota Q:
where the quota and the weights satisfy the inequalities:
In this way, assumptions D1 et D2 are verified.
iff ( ) : ii S
S W w S w Q
1( ) ( )
2w N Q w N
249
Assumption D3 is automatically satisfied: adding a player to a winning coalition does not decrease its weight.
Apparent power of players are given by their relative weights:
By construction:
1
where ( )( )
ni
i jj
ww N w
w N
1
0 1 for all
1
i
n
ii
i
250
EU Council of Ministers
Distribution of votes (EU-6 à EU-15)
Quota = minimum number of votes (environ 70%):
12/17 in 1958 54/76 in 1986 62/87 in 1994
Fr De It Be Nl Lu UK Dk Irl Gr Sp Pt Se Fi Au
1958 4 4 4 2 2 1
1986 10 10 10 5 5 2 10 3 3 5 8 5
1994 10 10 10 5 5 2 10 3 3 5 8 5 4 3 4
251
A voting game is said to be weighted if it is equivalent to a weighted majority game coalitions winnings:
starting from a voting game (N,W), there exist weights w1,…,wn and quota Q such that:
i.e. reproduces the same winning coalitions.
iff ( )S W w S Q
252
In situations involving three decision makers:
- unanimity and simple majority (qualified majority too) are by definition of weighted majority games where each decision makers have the same weight: wi = 1 for all i and
for unanimity: Q = n
for simple majority: 1 if is even2
1if is odd
2
nQ n
nQ n
253
- the weighted majority game defined by Q = 1, w1 = 1 and w2 = w3 = 0 is equivalent to the situation where 1 is a dictator.
- the situation where decision maker 1 has a veto right is also equivalent to a weighted majority game:
- assign weight 1 to decision makers 2 et 3
- assign weight x to decision maker 1
- choose a quota Q
such that {1,2}, {1,3} et {1,2,3} are winning.
254
The largest loosing coalition is {2,3} with a weight equal to 2.
The smallest winning coalition est {1,2} with a weight equal to x + 1.
The following inequalities must be verified:
2 < Q x + 1
It works for x = 2 and Q = 3.
255
The UN Security Council has 15 members: 5 permanent with veto right and 10 non-permanent The quota is 9. It is weighted:
- assign weight 1 to non-permanent members
- assign weight x to permanent members
- choose a quota Q
so as to reproduce the same winning coalitions.
The largest loosing coalition is of the form {4p, 10np} with a weight equal to 4x + 10.
256
The smallest winning coalition is of the form {5p, 4np} with a weight equal to 5x + 4.
The following inequalities must be verified:
4x + 10 < Q 5x + 4
It works for x = 7 and Q = 39.
257
Example: Apex game (partial veto)
This is a 5-player decision game defined by;
S W if and only if either 1 S and |S| 2
or |S| 4
This is a weighted decision game:
Q = 4, w1 = 3 and wi = 1 (i = 2,...,5)
258
Assign weight 1 to the players 2 to 5 and weight x au player 1.
the largest loosing coalition is of the type {i,j,k} where i,j,k {2,3,4,5} and its weight is equal to 3
the smallest winning coalition is of the form {1,i } where i {2,3,4,5} and its weight is equal to x + 1
The following inequalities must be verified:
3 < Q x + 1
It works for x = 3 and Q = 4.
259
A decision game can be written as a simple game (N,v) with:
Assumptions D1, D2 et D3 imply that this simple game is super-additive and monotone.
It is essential if and only if there is no dictator. Indeed, v(i) = 0 for all i in the absence of a dictator and v(N) = 1.
261
( ) 1 v S if and only if S W
Concepts introduced to solve TU-games can therefore be applied to these games, in particular the Shapley value.
Remember that the core of a simple game is nonempty if and only if there are veto players (Proposition 2.5).
Transposing the question of allocating v(N) between players, the question is to measure how decision power is distributed:
where xi is interpreted as a (relative) measure of power of decision maker i or as his/her share in the"cake" resulting from the decision taken.
262
11
( ,..., ) such that ( ) 1 ( 100%)n
n ii
x x x v N
We look for a measure of the power of each decision maker that results from the decision rules, assuming that all coalitions can form and without taking into account the nature of the proposition put to vote nor the preferences of the decision makers.
On can measure the power of a political party in a parliamentary system by identifying parties to single decision makers (assuming party discipline!):
each party has a weight equal to its number of seats.
263
In simple or qualified majority (that includes unanimity), decision makers are equal and they have therefore the same power.
Within a coalition, the power of a decision maker is linked to his/her capacity to make loosing coalitions winning by joining them.
A decision maker is decisive (or key) in a winning coalition if that coalition is loosing without him/her:
264
and \S W i S S i W
A coalition is minimal winning if all its members are decisive.
A decision maker has a veto right if he/she is decisive in all winning coalitions.
A dictator is decisive in all coalitions.
Two decision makers are substitutable if they are decisive in the same coalitions:
for all such that , : \ \S W i j S S i W S j W
265
Alternatively:
In unanimity and in simple/qualified majority games, decision makers are all substitutable.
A decision maker who is never decisive is null:
A null decision maker has no power. It was the case of Luxembourg in EU-6.
or[ \ and \ ] [ \ and \ ]S i W S j W S i W S j W
\ for all S i W S W
266
In a weighted majority game, two decision makers with identical weights are substitutable...
though two decision makers may be substitutable while having different weights !
In the game defined by w = (10, 20, 30, 40) and Q = 51 (simple majority), we have:
W = {24, 34, 123, 124, 134, 234, 1234}
2 and 3 are substitutable: they are decisive in coalition {123} and they are not decisive in coalitions {234} and {1234}.
267
We need a method – a rule – to compute power in any given voting game: a power index.
Banzhaf (1965)
... proposed to simply compute the number of coalitions in which decision makers are decisive
Banzhaf power index (BI)
Banzhaf (normalized) power index (NBI)
269
Shapley et Shubik (1954)
… proposed to apply the Shapley value to the associated simple game:
Shapley-Shubik index (SSI)
( ) 1 if is winning
( ) 0if is loosing
v S S
v S S
270
Marginal contributions are equal to 0 or 1:
v(S) – v(S\i) = 0 if S is loosing
v(S) – v(S\i) = 0 if S and S\i are winnings
v(S) – v(S\i) = 1 if S is winning and S\i is loosing
a decision maker is decisive in a coalition if and only if
his/her marginal contribution to that coalition is equal to 1.
271
The number of coalitions in which decision maker i is decisive is given by the sum of his/her marginal contributions:
This is the "raw" Banzhaf index from which two indices can be defined.
( , ) [ ( ) ( \ )]iS N
N v v S v S i
272
The Banzhaf index
and the "normalized" Banzhaf index
where
1 1
1 1( , ) [ ( ) ( \ )] ( , )
2 2i in nS N
BI N v v S v S i N v
1
( , ) ( , )n
ii
N v N v
1 ( , )( , ) [ ( ) ( \ )]
( , ) ( , )i
iS N
N vNBI N v v S v S i
N v N v
273
1 2 3
1 0 0 0 1/3
2 0 0 0 1/3
3 0 0 0 1/3
12 1 1 0 1/6
13 1 0 1 1/6
23 0 0 0 1/6
123 1 0 0 1/3
3 1 1
SSI1 = 2/6 + 1/3 = 2/3SSI2 = SSI3 = 1/6
simple majority with veto
BNI1 = 3/5BNI2 = BNI3 = 1/5
Decision maker 1 is decisive in 3 coalitions while the other two are decisive in only one coalition.
274
BI1 = 3/4BI2 = BI3 = 1/4
1 2 3
123 0 1 0
132 0 0 1
213 1 0 0
231 1 0 0
312 1 0 0
321 1 0 0
1/6 4 1 12 1 1
( , , )3 6 6
SSI
Alternative computation of the Shapley-Shubik index:
275
Apex game (n = 5)
1 S and |S| 2 S W iff or
|S| 4
Coalitions in which 1 is decisive:
12,13,14,15 4 5(2) = 1/5123,124,125,134,135,145 6 5(3) = 1/51234,1235,1245,1345 4 5(4) = 1/5
SSI1 = 3/5 SSIi = 1/10 (i = 2,...,5)
5 = (1/5,1/20,1/30,1/20,1/5)
SSI = (60,10,10,10,10) en %
276
Decision maker 1 is decisive in 14 coalitions and the other decision makers (substitutes) are decisives in 2 coalitions.
For example, decision maker 2 is decisive in 2 coalitions: 12 et 2345.
NBI1 = 14/22 and NBIi = 2/22 (i = 2,...,5)
NBI = (64, 9, 9, 9, 9) %
to be compared to SSI = (60, 10, 10, 10, 10) %
277
Comparing the two normalized indices:
we observe that they differ in the way they weight marginal contributions: in Banzhaf, the weights do not depend upon coalition size.
1[ ( ) ( \ )]
( , )iS N
NBI v S v S iN v
( ) [ ( ) ( \ )]i nS N
SSI s v S v S i
278
Interpreting Shapley-Shubik index and Banzhaf index as expectations:
we observe that in the Banzhaf index, probabilities are independent of coalition size.
1/2n-1 is the probability that a coalition containing a given player forms while n(s) is the probability that a coalition of size s containing a given player forms.
1
1[ ( ) ( \ )]
2i nS N
BI v S v S i
( ) [ ( ) ( \ )]i nS N
SSI s v S v S i
279
NBI and SSI satisfy the first three axioms of Shapley:
Efficiency:
they sum up to 1 (= the worth of the game v(N))
Symmetry:
substitutable decision makers have a same power
Null player:
null decision makers has no power
280
Security Council
A permanent member i is decisive in the coalitions
{4 P, k NP, i} where k {4, 5, 6, …, 9, 10}
A non-permanent member i is decisive in the coalitions
{5 P, 3 NP, i}
The number of these coalitions is given by the number of combinaisons of 3 elements among 9.
39 15
9! 8!6!(9) 0.002
3!6! 15!NPSSI C
!
! ( )!km
mC
k m k
1( )!( )!
( )!n
s n ss
n
281
The following equation must be satisfied:
5 SSIP + 10 SSINP = 1.
Hence:
SSIP 0.196
To compute the normalized Banzhaf index, we must compute the number of coalitions in which a permanent membre is decisive. They are of the type {4 P, k NP, i} where k = 4,…,10.
4 5 6 7 8 9 1010 10 10 10 10 10 10 848C C C C C C C
282
The number of times a decision maker is decisive is:
8485 + 8410 = 5080.
NBINP = 84/5080 0.016 0.002
NBIP = 848/5080 0.167 0.196
or, in %
NBINP 1.6 0.2
NBIP 16.7 19.6
98 according to SSIratio P/NP:
10 according to NBI
283
Quota games
A quota game with n players is a weighted majority game defined by a vector of relative weight1,…,n such that a coalition S is winning iff
For = (10, 20, 30, 40) in % we have:
W = {24, 34, 123, 124, 134, 234, 1234}
Decision makers 2 et 3 are substitutable: they are decisive in only one coalition of which they are members:{123}.
1
2ii S
284
1 2 3 4 4
24 0 1 0 1 1/12
34 0 0 1 1 1/12
123 1 1 1 0 1/12
124 0 1 0 1 1/12
134 0 0 1 1 1/12
234 0 0 0 1 1/12
1234 0 0 0 0 1/4
4=(1/4,1/12,1/12,1/4)
1 3 3 5, , ,
12 12 12 12SSI NBI
285
On large party and small parties:
(40, 20, 20, 20) % SSI = (50, 17, 17, 17) %
Two large parties and small parties:
(40, 40, 20) % SSI = NBI = (33, 33, 33) %
(35, 35, 20, 10) % SSI = NBI = (33, 33, 33, 0) %
(30, 30, 10, 10, 10, 10) %
SSI = (30, 30, 10, 10, 10, 10) !
NBI = (28.5, 28.5, 11, 11, 11, 11) %286
Party 1 is decisive in the coalitions:
1 2
6 1 2
12 1
12 {3,4,5,6} 4
12 , {3,4,5,6} 6 ( , ) ( , ) 16
1 , , {3,4,5,6} 4
13456 1
1 1 1 1 1 1 1 4 6 4 1 3( ) ( , , , , , )
6 30 60 60 30 6 30 60 60 60 30 10
3 3 1 1 1 1( , , , , , )10 10 10 10 10 10
i i
ij i j N v N v
ijk i j k
s SSI SSI
SSI
287
Party 3 is decisive in the coalitions:
13 , {4,5,6} 3
23 , {4,5,6} 3
( , ) 6 {3,4,5,6}
( , ) 2 16 4 6 56
8 8 3 3 3 3( , , , , , )28 28 28 28 28 28
i
ij i j
ij i j
N v i
N v
NBI
288
Der Deutsche Bundestag 1994-Today
Seats CDU/CSU SPD FDP GRÜNE LINKE fraktionslos
1994 294 252 47 49 30 0 672
1998 245 298 43 47 36 0 669
2002 248 251 47 55 0 2 603
2005 223 222 61 51 53 2 612
Today 239 146 93 68 76 0 622
NBI CDU/CSU SPD FDP GRÜNE LINKE fraktionslos
1994 0.50 0.17 0.17 0.17 0.00 0.00 100
1998 0.13 0.55 0.13 0.13 0.05 0.00 100
2002 0.33 0.33 0.00 0.33 0.00 0.00 100
2005 0.30 0.30 0.13 0.13 0.13 0.00 100
Today 0.50 0.17 0.17 0.00 0.17 0.00 100
289
Der Deutsche Bundestag 1994-Today
Seats in % CDU/CSU SPD FDP GRÜNE LINKE fraktionslos
1994 44 37 7 7 4 0 100
1998 37 44 6 7 5 0 100
2002 41 42 8 9 0 0 100
2005 36 36 10 8 9 0 100
Today 38 23 15 11 12 0 100
NBI in % CDU/CSU SPD FDP GRÜNE LINKE fraktionslos
1994 50 17 17 17 0 0 100
1998 13 55 13 13 5 0 100
2002 33 33 0 33 0 0 100
2005 30 30 13 13 13 0 100
Today 50 17 17 0 17 0 100
290
The actual coalitions are underlined.
We observe that the last elections result in a situation similar to 1994 in terms of coalition (CDU-FDP) and power distribution. This time the Grüne have no power. This party is now a null player while SPD, FDP and Linke are substitutes. Hence any power index which gives the same power to substitutable voters (symmetry) and no power to null voters must end up with the same power distribution. In particular, Banzhaf (normalized) and Shapley-Shubik indices coincide: BNI = SSI.
291
1958 1986 1994
France 4 10 10
Germany 4 10 10
Italy 4 10 10
Belgium 2 5 5
Netherland 2 5 5
Luxembourg 1 2 2
England 10 10
Denmark 3 3
Ireland 3 3
Greece 5 5
Spain 8 8
Portugal 5 5
Sweden 4
Finland 3
Austria 4
Total 17 76 87
Quota 12 54 62
EU-6 EU-12 EU-15
23.8 12.9 11.2
23.8 12.9 11.2
23.8 12.9 11.2
14;3 6.6 5.8
14.3 6.6 5.8
0.0 1.8 2.2
12.9 11.2
4.6 3.6
4.6 3.6
6.6 5.8
10.9 9.2
6.6 5.8
4.8
3.6
4.8
EU-6 EU-12 EU-15
23.3 13.42 11.7
23.3 13.42 11.7
23.3 13.42 11.7
15.0 6.37 5.5
15.0 6.37 5.5
0.0 1.2 2.0
13.42 11.7
4.26 3.5
4.26 3.5
6.37 5.5
11.12 9.5
6.37 5.5
4.5
3.5
4.5
Banzhaf Shapley-Shubik
292
The core of a cooperative game with transferable utility (N,v) is the set of imputations x = (x1,…,xn) against which no coalition can object:
This defines a set that may be empty or contain a large number of imputations.
Several concepts have been proposed in relation to the core: the bargaining set, the stable sets, the least-core and the nucleolus .
( ) ( ) pour toutx S v S S N
294
The idea behind the notion of bargaining set is to limit the possibilities of objection, by only considering "credible" objections.
We follow here the definition proposed by Mas Colell (198x).
An objection to an allocation x = (x1,…,xn) is formed of a coalition S and an allocation y such that:
( ) ( )
pour touti i
y S v S
y x i S
296
An objection (S,y) to an allocation x faces a counter-objection (T,z) if:
An objection is credible if it has no counter-objection.
The bargaining set is the set of imputations against which there are no credible objection.
( ) ( )
for all
for all \
i i
i i
z T v T
z y i T S
z x i T S
T
S
TS
T\S
297
In the porter game, there are no credible objections against the egalitarian allocation x = (2, 2, 2) :
- a single player or the grand coalition cannot object.
- all objection (S,y) where S = {1,2} and y = (a, 6 – a, 0)face an objection (T,z) where T = {1,3}, z = (b, 0, 6 – b) for 2 < a < 4 and a < b < 4.
For instance the objection (S,y) where S = {1,2} and y = (3, 3, 0) may face the conter-objection (T,z) where T = {1,3} and z = (3.5, 0, 2.5).
298
Stable sets have been introduced by von Neumann and Morgenstern (1944). It used to be called "solution".
Given a TU-game (N,v) and two allocations x and y:
dominates via coalition if
for all
( ) ( )
i i
y x S
y x i S
y S v S
Sy x
dominates if for some Sy x y x S N y x
300
Proposition 6.1 An imputation x belongs to the core if and only if it is never dominated.
Proof
If is in the core and is dominated by , we have:
( ) ( ) ( )
contredicting the inequality ( ) ( ).
x y
v S x S y S
y S v S
301
If is not in the core:
( ) ( ) for some
Then the allocation defined by:
1( ) ( ) if
0 otherwise
is such that and ( ) ( )
i i
i i S
x
x S v S S N
y
y x v S x S i Ss
y x y S v S y x
302
The idea of the von Neumann et Morgenstern solution is to exclude allocations that are dominated by allocations that are themself dominated.
A set K of imputations est stable if K is a set of all imputations that are non-dominated by imputations in K.
Proposition 6.2 There may be several stable sets and the core is a subset of all stable sets.
Proposition 6.3 In the case of a convex game, the core is the unique stable set.
303
Proposition 6.4 A set of imputations K is stable if and only if for all imputations x and y,
internal stability
external stability
(i) , Sx y K y x
(ii) if , there exists such thaty K x K x y
304
Proof Denote
internal stability:
external stability:
(I is the set of imputations)
( ) set of imputations dominated by
( ) set of imputations dominated by some
Dom x x
Dom K x K
( ) ( )x K
Dom K Dom x
\ ( )K I Dom K
\ ( )I Dom K K \ ( )I Dom K K
305
Two observations:
(i) no imputation dominates another imputation via a single player
Consider a 0-normalized game (N,v) and two imputations x and y such that for some i N.
Then xi ≤ v(i) = 0 and xi = 0 implies xi = 0. This is in contradiction with yi 0.
ix y
306
(ii) no imputation dominates another imputation via the grand coalition
Consider two imputations x and y such that
Then x(N) > y(N). This is in contradiction with the equality x(N) = y(N) = v(N).
Proposition 6.5 In 2-player games, the set of all imputations is the only stable set and it coincides with the core.
.Nx y
307
Example 3-player simple majority game:
The following set is a solution:
It is the "symmetric" solution.
(1) (2) (3) 0
(12) (13) (23) (123) 1
v v v
v v v v
1 1 1 1 1 1, ,0 , ,0, , 0, ,
2 2 2 2 2 2K
308
Indeed, domination can only occur through 2-player coalitions. As a consequence, internal stability holds.
External stability holds as well. Consider any imputation x K such that for instance
It is dominated by (1/2, 0, 1/2) .
If instead it is dominated either by (1/2, 0, 1/2) or by (1/2, 1/2, 0).
309
1 2 1
11 and
2x x x
1 2 1x x
310
(1,0,0)
(0,1,0) (0,0,1)
x2 = 1/2 x3 = 1/2
x1 = 1/2
every imputation notin K is dominated by an imputation in K
1 1 1 1 1 1, ,0 , ,0, , 0, ,
2 2 2 2 2 2K
311
(1,0,0)
(0,1,0) (0,0,1)
x2 = a2
x3 = a3
x1 = a1
31 2 3{ | 1, 0 for all }j i iA x x x x x a i j
A1
A3A2
a\i N iy A a y
imputations dominated by an interior imputation
From this we can conclude that no singleton can be a solution.
Furthermore, if K is a stable set, the line segment joining any two points in K must be parallel to one side of the imputation triangle. Otherwise one would dominate the other.
The following figures show:
- if a line segment parallel to one side of the imputation triangle is a solution, it must join two of its sides
- the interior vertices of some triangle cannot be a solution
312
314
(1,0,0)
(0,1,0) (0,0,1)
a b
imputations dominated by imputations on the line segment [a,b] not joiningtwo sides of the triangle
315
(1,0,0)
(0,1,0) (0,0,1)
a
(1,0,0)
(0,1,0) (0,0,1)
b
(1,0,0)
(0,1,0) (0,0,1)
c
the interior vertices of some triangle cannot be a solution
The next figures show that the only other solutions are line segments of the following type:
These are the "discriminatory" solutions.
1
2
3
( ) {( , ,1 ) | 0 1 }
( ) {( , ,1 ) | 0 1 }
( ) {( ,1 , ) | 0 1 }
1with 0
2
K x x x
K x x x
K x x x
316
We define the excess associated to an allocation x and a coalition S by:
In the words of Maschler, Peleg and Shapley (1979) who have introduced the notion of least core:
"It represents the gain (or loss if negative) to the coalition S if its members depart from an agreement that yields x in order to form their own coalition."
( , ) ( ) ( )e x S v S x S
320
The core is then equivalently defined as the set of imputations for which no excess is positive.
The –core is defined for some > 0 by the set of imputations x such that no excess is larger than :
The inequalities defining the –core can be written as:
The core corresponds to = 0.
( ) ( )x S v S
321
( , ) ( , ) | ( , ) for all ,C N v x I N v e x S S N S N
The least-core is the intersection of all nonempty –cores.
Equivalently it is defined by the smallest for which the –core is nonempty:
The –cores have all dimension n–1 or less (if nonempty), except for the least-core which has dimension n–2 or less.
322
*
( , ),
( , ) ( , )
where * ( , ).x I N v S NS N
LC N v C N v
Min Max e S x
The idea of the least core is to minimize the largest excess. This defines a set of imputations, a subset of the core if nonempty.
Schmeidler has proposed a procedure that goes further to eventually retain a unique imputation.
To each imputation, we associate the vectorformed by the excesses placed in a decreasing order.
Imputations are then compared lexicographically in terms of the ordered vectors to which they are associated.
323
To each x I(N,v) we associate the list of m = 2n - 2 proper coalitions (all coalitions except and N) ordered in terms of excesses and the corresponding vector of excesses:
with
We then retain the imputations x* I(N,v) such that:
324
1,..., ( ) mmx S S x
( ) ( , ) and ( ) ( ) for all .i i i jx e S x x x i j
( *) ( ) for all ( , )Lx x x I N v
Proposition 6.6 Given any game (N,v), this procedure leads (Schmeidler, 1969) to one and only one imputation.
The resulting imputation is called the nucleolus. It defines a rulethat to any game (N,v) associates an imputation NUC(N,v).
Proposition 6.7 As a rule, the nucleolus satisfies efficiency, symmetry and null player
The nucleolus does not satisfy additivity.
325
The nucleolus is included in any nonempty –core. It is therefore also an element of the least core which can be alternatively defined by:
In the case where the least core reduces to a single imputation, that imputation defines the nucleolus.
326
*( , ) ( , )LC N v C N v
( , ) 1where * ( ).x I N vMin x
Example: auction game (p1 = 0)
v(1) = v(2) = v(3) = v(23) = 0
v(12) = p2
v(13) = v(123) = p3
The core is the set of allocations of the form
The nucleolus being contained in the core, it has this form and the parameter p suffices to identify the imputations.
3 2 3( ,0, ) wherep p p p p p
327
For each p, we order the excesses in a decreasing way:
3
2
( , ) if {1}
0 if {2} and {13}
if {3} and {23}
if {12}
e p S p S
S S
p p S S
p p S
2 3 3 2
3 3 2 3
2 3
(0, 0, , , , ) if [ , ]
(0, 0, , , , ) if [ , ]
where2
p p p p p p p p p p
p p p p p p p p p p
p pp
328
From this we conclude that the interval [p2, p3] defines the least core (that coincides with the core).
Furthermore, the mid-point of interval [p2, p3] defines the nucleolus:
i.e. (250, 0, 50) in the case where p3 = 300 and p2 = 200.
3 2 3 2,0,2 2
p p p pNUC
330
Example: 3-player airport game
The associated surplus game is given by:
v(1) = v(2) = v(3) = 0v(12) = v(13) = c1
v(23) = c2
v(123) = c1 + c2
Players 2 et 3 are substitutable the nucleolus is of the form:
1 21 2 3 1 2, where 0
2
c c qx q x x q c c
331
1 2
1 2
1
( , ) if {1}
( )if {2} et {3}
2
if {12} et {13}2
if {23}
e q S q S
q c cS S
c c qS S
q c S
1 21 1Solution: ,
2 3
c cx q Max c
332
c1 + c2
- q
q0
– (c1+c2)
q – c1
– c1
2c2 > 3c1
1 2
2c c
1
2c
1 2
2c c
1 2( )2
q c c
1 22
c c q
1
2c
123
cc
333
1c
c1 + c2
- q
q0
– (c1+c2)
q – c1
– c1
2c2 = 3c1
11
22 3c
cc
1 2
2c c
1 2
2c c
1 22
c c q
1 2( )2
q c c
1
2c
334
c1 + c2
- q
q0
– (c1+c2)
q – c1
– c1 2c2 < 3c1
1 2
2c c
1 2
2c c
1 22
c c q
1 2( )2
q c c 1
23
cc
1
2c
1
2c
335
The least-core is a singleton. It is therefore also the nucleolus:
nucleolus of the cost game:
2 2 21 1 2
1 2 1 2 11 2
2 2( , ) , , if 3 2
3 3 3
, , if 3 22 2 4 2 4
c c cNUC N v c c c
c c c c cc c
2 2 23 1 2
1 2 1 2 13 1 2
2( , ) , , if 3 2
3 3 3
, , if 3 22 2 4 2 4
c c cNUC N c c c c
c c c c cc c c
( , ) ( , )i i iNUC N c NUC N v c
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Example: apex games
The core of the apex game is empty: no player has a veto right.
The nucleolus being symmetric, the problem can be reduced to a single variable, say w, the share of player 1.
For each w [0,1] and each coalition S, S and S N, we first compute the excess of S at w
337
Depending upon the coalition, we have:
The locus of maximum excesses can easily be identified graphically: only the first and the third actually matter and the nucleolus is defined by their intersections.
338
5( , ) (1 ) if 1 and 2 4
4
( , ) (1 ) if 1 and 1 34
( , ) if {2,3,4,5}
( , ) if {1}
se w S w S S
se w S w S S
e w S w S
e w S w S
339
e
w10 w* = 3/7
-1
- w
3(1-w)/4
3/4 w
1/2
1/4
- 3/4
- 1/2
- 1/4
3/7
3 1 1 1 1( , , , , )7 7 7 7 7
NUC
The nucleolus coincides with the least core which is the -core for = 3/7. Indeed, v(S) – x(S) 3/7 for all S N with equality for S = {2,3,4,5}.
Comparison with the SSI and NBI:
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3 1 1 1 1( , , , , ) 0.43,0.14,0.14,0.14,0.147 7 7 7 7
NUC
3 1 1 1 1( , , , , ) (0.60,0.10,0.10,0.10,0.10)5 10 10 10 10
SSI
7 1 1 1 1( , , , , ) (0.64,0.09,0.09,0.09,0.09)11 11 11 11 11
NBI
Example: data games
We have seen that the core of a data game is is a regular simplex given by:
As a consequence, the core's center of gravity is simply the average of its vertices:
It defines the least core and, thereby, the nucleolus.
( , ) { | ( ) 0 and for all }ni iC N C y y N y v i N
341
0( , )i i
vNUC N C v
n
0 1 2 3( , , )v v v v
1 0 2 3( , , )v v v v 1 2 0 3( , , )v v v v
3 3y v 2 2y v
1 1y v
( , )N C
The nucleolus of a 3-player data game
0( , )3i i
vNUC N C v
342
Comparing the two allocation rules
we observe that they coincide if and only if each data is held by a single player:
0( , )i
hi
h M h
v dN c
n tSV
0( , )i
i hh M
vNUC N c d
n
01 for all and for allh i it h M M M i N
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Outside the partition case, the Shapley value may not be in the core… but the nucleolus may be unfair as the following example illustrates.
Consider a n-player situation, n 3, where only the last two players hold data and their datasets differ by a single data:
1
2
{1,..., }
{2,..., }
1,...,
n
n
i
M m
M m
M i n
1
0
( ) 0 if
( ) if and 1
( ) if and 1
C S n S
C S d n S n S
C S v n S n S
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Core allocations impose that only player n may be compensated:
0 yi d1 for all i = 1,…, n – 1
The nucleolus goes further by imposing that the n – 1 other players contribute a same amount:
1
1
( 1,... 1)
( 1)
i
n
dy i n
n
dy n
n
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The Shapley value instead is given by:
the first n – 2 players contribute more
player n – 1 contributes less and may be compensated
the last player gets a higher compensation
0
0 1 11 0
2 2
0 11 0 1 1
2
( 1,... 2)
2 2
2 2 2 2
20
2 2 2
i
m mh
n hh h
mh
n nh
vy i n
n
v d n d d ny v d
n n n n
v d n dy d v y d
n n
346
Special case: the case where th = 1 for all h M0 is the partition case:
Mi Mj = for all i j.
It fits patent and copyright pooling aiming for instance at developing new products or standard technologies.
We shall see that:
- the resulting cost game is concave
- the Shapley value and the nucleolus coincide.
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In the partition case:
where ci = c(i) is the cost of the data that player i is missing.
0( ) (1 ) ii S
c S s d c
0 0\
0 0\
:
( ) (1 )
i i i
S i
i h h h ih M M h M h M
h ih M M i S
hh M i S
c d d d d d c
c S dd d s d c
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The associated surplus game
is given by:
It is a symmetric game (all players are substitutes) and the Shapley value and the nucleolus coincide:
01( ) ( )v S s d
( ) ( ) ( )i S
v S c i c S
0
( ) ( 1)( , ) ( , )i i
v N nSV N v NUC N v d
n n
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The compensations derived from the Shapley value or the nucleolus are then given by:
A player is compensated if and only if the value of the data he/she owns exceeds the per capita value of the complete database.
00 0
0
( , ) ( ) ( , )
( 1)( )
( , )i
i i
i i
h ih M
SV N c c i SV N v
n vc d v c
n n
vd NUC N c
n
350
Exemple N = {1,2,3,4} and M0 = {1,2,3,4}
M1 = d1 = 6M2 = {1} d2 = 10M3 = {2} d3 = 4M4 = {3,4} d4 = 12
y1 = 8 – 0 = 8
y2 = 8 – 6 = 2
y3 = 8 – 10 = – 2
y4 = 8 – 4 – 12 = – 8
032
84
v
n
0( , ) ( , )i
i i hh M
vSV N c NUC N c d
n
351
In the partition case, data games are concave.
It is more easy to check that the associated surplus game is convex:
0
0 0
( ) ( ) ( ) ( )
[( 1) ( 1) ( | | 1) (| | 1)] 0
if
[( 1) ( 1) ( 1)] 0
if
v S v T v S T v S T
s t s t S T S T v
S T
s t s t v v
S T
352
By concavity, the core of a partition data game is the polyhedron whose vertices are the marginal cost vectors. Using Proposition 4.5, they are given by:
353
10 1 2 1 2 0 0
21 0 2 1 0 2 0
1 2 0 1 0 2 0
( , , ... , ) ( , , ... , )
( , , ... , ) ( , , ... , )
...
( , , ... , ) ( , , ... , )
n n
n n
nn n
v v v v c c v c v
v v v v c v c c v
v v v v c v c v c
1 2 0 3 0( , , )c c v c v
1 0 2 3 0( , , )c v c c v
3 3y v 2 2y v
1 1y v
Core of a 3-player partition data game
354
1 0 2 0 3( , , )c v c v c
(c1 – v0, c2 – v0, c3, c4 – v0)
(c1 – v0, c2, c3 – v0, c4 – v0)
(c1, c2 – v0, c3 – v0, c4 – v0)
(c1 – v0, c2 – v0, c3 – v0, c4)
y3 = c3 – v0
y2 = c2 – v0y4 = c4 – v0
y1 = c1 – v0
y1 + y
4 = c1 + c
4 – v0
y 1 + y
2 = c
1 + c
2 – v
0
y2 + y
3 = c2 + c
3 – v0 y 3 +
y4 =
c3 +
c4 –
v 0
y 2 + y 4 = c 2 + c 4 – v 0
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Core of a 4-player partition data game
Some references
Luce R.D. and H. Raiffa Games and decisions, Wiley 1957.
Meyerson R.B. Game theory. Analysis of conflict, Harvard University Press, 1991.
Moulin H. Cooperative microeconomics, Princeton University Press, 1995.
Osborne M.J. An introduction to game theory, Oxford University Press, 2004.
Osborne M.J. and A Rubinstein, A course in game theory, MIT Press, 1994.
Peeters H. Game theory: a multi-leveled approach, Springer Verlag, 2008.
Peleg B. and P. Sudhölter, Introduction to the theory of cooperative games, Kluwer, 2003.
Shubik M. Game theory in the social sciences, MIT Press, 1982.
Shubik M. A game-theoretic approach to political economy, MIT Press, 1984.
von Neumann J. and O. Morgenstern, Theory of games and economic behavior, Princeton University Press, 1944.
Young, H.P. Cost allocation: methods, principles, applications, North-Holland, 1985.
Young, H.P. (ed.) Fair allocation, American Mathematical Society, 1985. 356
Some interesting web sites:
theory of games (cooperative and non-cooperative):
www.citg.unige.it/siti_internet_web.html (in italien)www.econ.canterbury.ac.nz/personal_pages/paul_walker (historical)arielrubinstein.tau.ac.ilwww.economics.utoronto.ca/osborne/igt (web site of his book)
cooperative games:
www.econ.usu.edu/acaplan/tugames.htm (a nice software for 3-person games)
webs.uvigo.es/matematicas/campus_vigo/profesores/mmiras/TUGlabWeb/TUGlab.html(a matlab toolbox for 2 to 4 player TU-games)
power indices:
powerslave.val.utu.fi www.warwick.ac.uk/~ecaae (online computation)
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