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Cooperative games Pierre Dehez CORE, University of Louvain (Center for Operations Research et Econometrics) pierre.dehez@uclouvain.be

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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

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Chapter 1 Characteristic function games 3

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1.1 Transferable utility 4

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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

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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

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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

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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

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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

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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

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1.2 Characteristic function 11

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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 2 n coalitions and to each coalition S N, is associated a real number, its worth: with the convention v( ) = 0. 12

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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 2 n-1 that can be identified as 13

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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

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Notation: v(i) = v({i}), v(ij) = v({i,j}), v(ijk) = v({i,j,k}), … S i = S {i}, S\i = S\{i} |A| is the cardinal of the set A and n = |N|, s = |S|, t = |T|,… 15

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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

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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

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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 – p 1 ) + (p 2 – p) = (p 2 – p 1 ) v(13) = (p – p 1 ) + (p 3 – p) = (p 3 – p 1 ) v(23) = 0 v(123) = (p – p 1 ) + (p 3 – p) = (p 3 – p 1 ) where p is the price at which the transaction takes place. It does not appear because gains are added! seller: 1 buyers: 2 and 3 p 1 ≤ p 2 ≤ p 3 18

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In the crop game, there are n = m + 1 players: v(S) = 0if 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

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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. 20

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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. 21

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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, Cm i (i) = v(i). By super-additivity we have: 22

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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. 23

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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

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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) 25

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1.3 Particular classes of games 26

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A game is monotonic if enlarging a coalition is never detrimental, in which case the marginal contributions are nonnegative: Proposition 1.1A game such that v(S) 0 for all S N is monotonic. This is an immediate consequence of super-additivity: 27

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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 1.2A symmetric game is monotone if the function f that defines it is non-decreasing. 28

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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

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s n 0 1 –n–n f(s)f(s) –(n–1) n–1 –1 v(S) = f(s) 30

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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 1.3A game (N,v) is convex if and only if marginal contributions are non-decreasing: for all i, S, T such that i S T. 31

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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: 32

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Notice that the hypothesis of nondecreasing marginal productivity has only been used only in the 1 st case. The other three cases rely on the non-decreasingness and nonnegativeness of F. 33

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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

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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. 35

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Chapter 2 Imputations and the core 36

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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 x 1,...,x n such that: 37

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2.1 Imputations 38

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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 39

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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. 40

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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 (x 1,...,x n ) which are better for all members of S: 41

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In the porters' game, an imputation is defined by three nonnegative numbers x 1, x 2, x 3 summing to 6: x 1, x 2, x 3 0 x 1 + x 2 + x 3 = 6 In the auction game, an imputation is defined by three nonnegative numbers x 1, x 2, x 3 summing to 300: x 1, x 2, x 3 0 x 1 + x 2 + x 3 = 300 42

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x2x2 x1x1 imputations line for n = 2: x i v(i) i = 1,2 x 1 + x 2 = v(N) 43 0 v(N)v(N) v(N)v(N) v(1) v(2)

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v(N) – v(2) – v(3) x1x1 x2x2 x3x3 (v(1),v(2),v(3)) imputations' triangle for n = 3: x i v(i) for i = 1,2,3 x 1 + x 2 + x 3 = v(N) 44 v(N) – v(1) – v(3) v(N) – v(1) – v(2) x 2 = v(2) x 3 = v(3) x 1 = v(1)

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45 imputations' triangle for n = 3: x i v(i) for i = 1,2,3 x 1 + x 2 + x 3 = v(N)

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x1x1 x2x2 x3x3 0 x 2 = 0 x 3 = 0 x 1 = 0 v(N)v(N) v(N)v(N) v(N)v(N) 46 imputations' triangle when v(i) = 0 for all i

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47 imputations' triangle when v(i) = 0 for all i

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2.2 The core 48

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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 = (x 1,…,x n )… 49

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… because it could then offer more to each of to its members, whatever the players outside S do: 50

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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. 51

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In the porters' game, the core is defined by: x 1, x 2, x 3 0 x 1 + x 2 + x 3 = 6 x 1 + x 2 6 x 1 + x 3 6 x 2 + x 3 6 No allocation does satisfy all seven conditions: they are not compatible. the core of the porter's game is empty: C(N,v) = x 3 = 0... x 2 = 0... x 1 = 0 52

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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

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Proposition 2.1Core allocations x C(N,v) satisfy the following inequalities: Indeed, if x C(N,v), we have Hence, in particular, core allocations x satisfy 54

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The core of the auction game is defined by: x 1, x 2, x 3 0 x 1 + x 2 + x 3 = p 3 – p 1 x 1 + x 2 p 2 – p 1 x 1 + x 3 p 3 – p 1 x 2 + x 3 0 (redundant) First observation: (p 3 – p 1, 0, 0) belongs to the core. Second observation: x 2 = 0 in all core allocations. x 2 = 0 55

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i.e.-buyers cannot object against the fact that the seller gets all! -any allocation giving a positive amount to the first buyer is not stable. The allocation (p 2 – p 1, 0, p 3 – p 2 ) is also in the core. It is the allocation which is most favorable to the second buyer. 56

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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 p 3 – p 2. The minimum the seller can expect is the lowest price p 2. if the buyers' prices would be equal, say p 3, the core then reduces to the allocation (p 3 – p 1, 0, 0) ! 57

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Indeed, the core conditions are then given by: x 1, x 2, x 3 0 x 1 + x 2 + x 3 = p 3 – p 1 x 1 + x 2 p 3 – p 1 x 1 + x 3 p 3 – p 1 x 2 = x 3 = 0 x 1 = p 3 – p 1 58

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Core of the crop game with 2 workers and constant return to scale: m = 2 and F(s) = s x 1, x 2, x 3 0 x 1 + x 2 + x 3 = 2 x 1 + x 2 x 1 + x 3 x 2 + x 3 0 (redundant) First observation: (2 , 0, 0) and (0, , ) belong to the core Second observation: the maximum a worker can expect is landlord: 1 travailleurs: 2 et 3 v(i) = 0 v(12) = v(13) = v(23) = 0 v(123) = 2 59

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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: 60

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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

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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

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(1, 1, 0, 0) is indeed the unique allocation satisfying the core conditions : x 1, x 2, x 3, x 4 0 x 1 + x 2 + x 3 + x 4 = 2 x 1 + x 3 1 x 1 + x 4 1 x 2 + x 3 1 x 2 + x 4 1 x 1 + x 2 + x 4 2 (ignoring redundant inequalities) x 3 = 0 x 1, x 2 1 x 4 = 0 (R, R, L, LL) 63

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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

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The core conditions are given by: x 1, x 2, x 3 0, x 1 + x 2 + x 3 = 2 x 1 + x 2 1 x 1 + x 3 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) 65

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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: By construction, u(i) = 0 for all i N. 66

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If (y 1,…,y n ) belongs to the core of the game (N,u), the allocation (x 1,…,x n ) defined by belongs to the core of the game (N,v). Indeed: 67 translation invariance

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x1x1 x2x2 x3x3 0 c x 1 + x 3 = 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 the other two players x 2 = c, 0 c v(N) v(N)v(N) v(N)v(N) v(N)v(N) 68

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(v(N),0,0) (0,v(N),0) (0,0,v(N)) x 1 + x 3 = v(N) – c x 1 + x 3 = v(N) x 2 = c x 2 = 0 69

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(v(N),0,0) (0,v(N),0) (0,0,v(N)) the segment divides the triangle in two parts x 2 c x 1 + x 3 v(N) - c (v(N) – c, c, 0) (0, c, v(N) – c) x 2 c x 1 + x 3 v(N) - c x 2 = c x 1 + x 3 = v(N) - c the imputations (v(N) – c, c, 0) and (0, c, v(N) – c) are the two extreme points of the line segment 70

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(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 by x i = v(N)/3 (i = 1,2,3) 71

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(v(N),0,0) (0,v(N),0) x 1 + x 2 = v(1,2) x 2 + x 3 = v(2,3) (0,0,v(N)) x 1 + x 3 = v(1,3) x 1 + x 3 = v(N) x 2 = 0 x 1 + x 2 = v(N) x 3 = 0 x 2 + x 3 = v(N) x 1 = 0 nonempty core 72

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(v(N),0,0) (0,v(N),0) x 1 + x 2 = v(1,2) x 2 + x 3 = v(2,3) (0,0,v(N)) x 1 + x 3 = v(1,3) x 1 + x 3 = v(N) x 2 = 0 x 1 + x 2 = v(N) x 3 = 0 x 2 + x 3 = v(N) x 1 = 0 core reduced to a single point 73

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(v(N),0,0) (0,v(N),0) x 1 + x 2 = v(1,2) x 2 + x 3 = v(2,3) (0,0,v(N)) x 1 + x 3 = v(1,3) x 1 + x 3 = v(N) x 2 = 0 x 1 + x 2 = v(N) x 3 = 0 x 2 + x 3 = v(N) x 1 = 0 empty core 74

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(15,0,0) (0,15,0) x 1 + x 2 = 10 (0,0,15) x 1 + x 3 = 10 Crop game with decreasing returns (5,5,5) (10,0,5) (10,5,0) (0,10,5)(0,5,10) 75 v(i) = 0 v(12) = v(13) = 10 v(23) = 0 v(123) = 15

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(300,0,0) (0,300,0) x 1 + x 2 = 200 (0,0,300) x 1 + x 3 = 300 (200,0,100) Auction game: v(i) = 0 v(12) = 200 v(13) = 300 v(23) = 0 v(123) = 300 76

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(2,0,0) (0,2,0) x 1 + x 2 = 1 (0,0,2) x 1 + x 3 = 2 (1,0,1) Gloves market: (RR, L, LL) 77

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2.3 Nonemptiness of the core 78

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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 x i + x j v(ij) for all i j, we get: 2(x 1 + x 2 + x 3 ) v(12) + v(13) + v(23) 79

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Proposition 2.2A game (N,v) whose core is nonempty satisfies the following inequalities: Indeed, if x C(N,v) we have for all S N: 80

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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: x i = v(i) for all i. The largest class of games whose cores are nonempty are the convex games. Proposition 2.3The core of a convex game is nonempty. This is a result due to Shapley (1971). 81

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Proposition 2.4The 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

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(ii) V C(N,v) Consider the allocation x defined by where the x i '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

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(iii) x C(N,v) x i = 0 for all i V Consider a core allocation x such that x i > 0 for some i V. Then, using the fact that x i 0 for all i N, we get: a contradiction: the coalition of veto players can improve upon the allocation x. 84

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Chapter 3 The Shapley value 85

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3.1 Looking for a fair compromise 86

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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

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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 2 nd each porter may expect to receive 2 € 88

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(0,6,0) (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

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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

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(3,1,2) (3,2,1) (1,2,3) (0, p 2 – p 1, p 3 – p 2 ) = (0, 200, 100) (1,3,2) (2,1,3) (2,3,1) (0,0, p 3 – p 1 ) = (0, 0, 300) (p 2 – p 1, 0, p 3 – p 2 ) = (200, 0, 100) (p 3 – p 1,0, 0,) = (300, 0, 0) (p 3 – p 1, 0, 0) = (300, 0, 0) p = (0, 200, 300) 91

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123 1230200100 13200300 2132000100 23130000 31230000 32130000 1/6 1100200500 v(i) = 0 v(12) = 200 v(13) = 300 v(23) = 0 v(123) = 300 each row corresponds to a permutation each column corresponds to a player 92

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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

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123 12303000 13200300 21330000 23130000 31230000 32130000 1/6 1200300 v(i) = 0 v(12) = 300 v(13) = 300 v(23) = 0 v(123) = 300 94

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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

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3.2 The Shapley value 96

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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

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In this way, v(N) is entirely allocated. Indeed, if the players are ordered according to the permutation = (i 1,…,i n ) n we have successively: v(i 1 ) – v( ) + [v(i 1,i 2 ) – v(i 1 )] + [v(i 1,i 2,i 3 ) – v(i 1,i 2 )] + … + [v(i 1,i 2,…,i n-1 ) – v(i 1,i 2,…,i n-2 )] + [v(i 1,i 2,…,i n ) – v(i 1,i 2,…,i n-1 )] = v(N) 98

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To each permutation = (i 1,…,i n ) we associate the vector of marginal contributions ( ) defined by: There are n! vectors, not necessarily distinct. Proposition 3.1A game is convex if and only if its core is a (Shapley, 1971) convex polyhedron whose vertices are the marginal contribution vectors. 99

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Crop game: constant returns to scale 123 1230 1320 213 0 231 22 00 312 0 321 22 00 1/666 33 33 the crop is shared equally between the landlord and the workers who receive 1/2 of the marginal product v(i) = 0 v(12) = v(13) = v(23) = 0 v(123) = 2 100

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the share of the landlord is greater than the share of the workers who receive 2/3 of the marginal product 123 1230 /2 1320 /2/2 213 0 /2 231 3 /2 00 312 /2 0 321 3 /2 00 1/655 22 22 v(i) = 0 v(12) = v(13) = v(23) = 0 v(123) = 3 /2 Crop game: decreasing returns to scale 101

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123 1230 3 /2 1320 3/23/2 213 0 3 /2 231 5 /2 00 312 3 /2 0 321 5 /2 00 1/677 44 44 v(i) = 0 v(12) = v(13) = v(23) = 0 v(123) = 5 /2 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

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123 123060 132006 213600 231006 312600 321060 123123010 132010 213100 2312000 31210 0 3212000 123 1230105 1320510 2131005 2311500 3121050 3211500 600 060 006 010 0 0 2000 0105 05 05 50 1500 porter game crop game with decreasing returns 103

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This procedures allocates to player i his/her average marginal contribution: It defines a rule SV which, to any game (N,v) associates an allocation, called "value": SV : (N,v) (x 1,...,x n ) = 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

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3.3 Axiomatization of the value 105

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Efficiency (Eff): the rule is such that This is a requirement of collective rationality: only rules which distribute the entire value of the game are considered 106

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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" 107

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In a symmetric game, all players are substitutable. As a consequence, any rule satisfying efficiency and symmetry allocates a same amount to each player: The porters' game is symmetric: 108

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Equal division (ED) is the simplest rule which satisfies efficiency and symmetry: 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

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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). 110

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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: player 1 is null 111

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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: 112

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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: 113

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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

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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,v 1 ) and (N,v 2 ) is the game (N,v) whose characteristic function v is defined by: 115

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Consider the games (N,v 1 ) and (N,v 2 ) defined by: v 1 (i) = 0 v 2 (i) = 0 v 1 (12) = v 1 (13) = 0v 2 (12) = v 2 (23) = 0 v 1 (23) = 6v 2 (13) = 6 v 1 (123) = 6v 2 (123) = 6 v(i) = 0 v(12) = 0 v(13) = v(23) = 6 v (123) = 12 v = v 1 + v 2 116

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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 117

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The modified rules ED and AC do not satisfy additivity. This can be seen by applying them to the games v 1, v 2 and v = v 1 + v 2. The modified ED rule gives: 118

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Shapley (1953) has proved the following remarquable proposition: Proposition 3.2There 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

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A rule is a marginalist if what it allocates to a player in a game depends only upon his/her marginal contributions: Proposition 3.3The Shapley value is the unique marginalist (Young, 1985) rule that satisfies EFF and SYM. 120

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Alternative definition Proposition 3.4The 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: 121

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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 122

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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 : 123

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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 124

125
125

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Auction game 123 (s)(s) 10001/3 2000 3000 12200 01/6 133000 1/6 230001/6 12330001001/3 126 v(i) = 0 v(12) = 200 v(13) = 300 v(23) = 0 v(123) = 300

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Crop game: constant returns 123 (s)(s) 10001/3 2000 3000 12aa01/6 13a0a1/6 230001/6 1232a2aaa1/3 v(i) = 0 v(12) = a v(13) = a v(23) = 0 v(123) = 2a 127

128
3.4 Properties of the value 128

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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 3.5The Shapley value defines an imputation that is reasonable in the sense of Milnor. 129

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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

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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 3.6The 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

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Example: bankruptcy game n players (creditors) E > 0 value to be distributed d 1,...,d n 0 the claims v(S) is the maximum S can expect to receive if all other creditors have been compensated possibly partly or 132

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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 and contains the Shapley value. 133

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Convexity: 134

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123 1230150450 1320300 2131500450 2311500450 312150300150 321150300150 x 1/660010501950 losses = (50, 125, 125) (150) (300) (450) v(1) = v(2) = 0 v(3) = v(12) = 150 v(13) = 300 v(23) = 450 v(123) = 600 Shapley value 135

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123 (s)(s) 10001/3 2000 3001501/3 12150 01/6 1315003001/6 2303004501/6 1231503004501/3 v(1) = v(2) = 0 v(3) = v(12) = 150 v(13) = 300 v(23) = 450 v(123) = 600 136

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(600,0,0) (0,600,0) x 3 = 150 x 1 = 150 (0,0,600) x 2 = 300 (150,0,450) (150,300,150) (0,300,300) (0,150,450) x 3 = 450 0 x 1 150 0 x 2 300 150 x 3 450 v(1) = v(2) = 0 v(3) = v(12) = 150 v(13) = 300 v(23) = 450 v(123) = 600 imputations' triangle (100,175,325) imputations and core 137

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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

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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. d 1 =100d 2 =200d 3 =300 E=10033.3 E=2005075 E=30050100150 Shapley value Nucleolus (equal division) Shapley value Nucleolus 139

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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

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They result from the following tables: 123 12300200 13202000 21300200 2311000 31202000 321100 0 x 1/6200500 123 12300300 1320200100 21300300 2311000200 3121002000 3211002000 x 1/6300600900 141

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123 12301015 13201510 21310015 2312500 31210150 3212500 1/6 7040 Crop game: increasing returns ( = 10) 5 distincts marginal contributions vectors v(i) = 0 v(12) = 10 v(13) = 10 v(23) = 0 v(123) = 25 142

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(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 convex envelope of the marginal contribution vectors a convex game 143

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123 1230105 1320510 2131005 2311500 3121050 3211500 1/6 5020 Crop game: decreasing returns ( = 10) v(i) = 0 v(12) = 10 v(13) = 10 v(23) = 0 v(123) = 15 5 distincts marginal contributions vectors 144

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(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) 145

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(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

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123 1230102 1320210 2131002 2311200 3121020 3211200 1/6 4414 Case where the Shapley value does not belong to the core v(i) = 0 v(12) = 10 v(13) = 10 v(23) = 0 v(123) = 12 147

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(12,0,0) (0,12,0) (0,0,12) (2,2,10) (10,0,2) (10,2,0) (0,10,2) (0,2,10) 148

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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 149

150
m F(m)F(m) 0 F(1) F(2) k k+1 F(k)F(k) 1 x F(k) F(1) + F(2) + … + F(m) 150

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m F(m)F(m) 0 L W L > W decreasing returns 151

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m F(m)F(m) 0 L W L = W constant returns 152

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m F(m)F(m) 0 L T L < W increasing returns W 153

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m F(m)F(m) 0 L mixed returns W 154

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Crop game: case of an excess labor supply (k < m) 0 k mx y v(S) = 0if 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)F(k) 155

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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: 156

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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. 157

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The wage of an individual worker is given by: 0 k mx y L W kk 158

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Production game: y = f(x) (one output – one input, with a fixed cost) 1 0 x y 159

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x1x1 x2x2 x3x3 0 15 3 6 (0,9,6) (0,3,12) (6,3,6) imputations x 1 0 x 2 3 x 3 6 x 1 + x 2 + x 3 = 15 160

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(0,15,0)(0,0,15) (15,0,0) x 3 = 9 x 2 = 6 x 1 = 3 (3,6,6) (3,3,9) (0,6,9) 0 x 1 3 3 x 2 6 6 x 3 9 x 3 = 6 x 2 = 3 v(1) = 0 v(2) = 3 v(3) = v(12) = 6 v(13) = 9 v(23) = 12 v(123) = 15 x 1 + x 2 = 6 x 1 + x 2 = 9 x 1 + x 3 = 12 x 1 + x 3 = 9 x 2 + x 3 = 12 core 161

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123 123069 132069 213339 231339 312366 321366 1/6 123048 v(1) = 0 v(2) = 3 v(3) = v(12) = 6 v(13) = 9 v(23) = 12 v(123) = 15 Shapley value The core is the convex hull of the marginal contribution vectors the game is convex. 162

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Modified game: This is a symmetric game and the Shapley value is therefore the egalitarian allocation: (N,u) = (2,2,2) 163

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(6,0,0) (0,6,0) (0,0,6) x 2 + x 3 = 3 x 1 + x 2 = 3 x 1 + x 3 = 3 (3,0,3) (3,3,0) (0,3,3) core of the modified game 164

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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

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1234 13 1010 14 1001 23 0110 24 0101 123 0010 134 1000 234 0100 124 1102 1234 1101 Gloves market: (R,R,L,LL) 166

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123 12 110 13 202 123 201 Gloves market: (RR,L,LL) 167

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3.5 Cartels 168

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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

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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 23 v*(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

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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: 171

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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. 172

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Chapter 4 Cost sharing 173

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4.1 Cost games 174

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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

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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 1Town 2Town 3 Source 182127 Town 1 1512 Town 2 24 S 2 3 18 21 27 15 24 12 1 176

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The minimum cost of the project is c(123) = 45: S 1 2 3 18 21 27 15 24 12 18 + 15 + 12 = 45 177

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Looking at coalitions of towns, the (minimum) costs are given by: c(1) = 18c(12) = 33 c(2) = 21c(13) = 30 c(3) = 27c(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

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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. 179

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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

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A cost game is concave if Concavity implies subadditivity. Equivalently, a cost game is concave if the marginal costs are non-increasing: 181

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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 4.1The surplus game associated to an essential – subadditive – concave cost game is essential – superadditive – convex. 182

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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

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4.2 Imputations and the core 184

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Question: how to share c(N) between the players? A first approach consists in imposing conditions on cost allocations y = (y 1, y 2,..., y n ): individual rationality (incentive compatibility) collective rationality (Pareto efficiency) core (social stability) 185

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Proposition 4.2Core 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. 186

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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: 187

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In terms of surplus sharing, we have successively: 188

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(0,45,0 ) (45,0,0) y 3 = 12 y 2 = 15 (0,0,45) y 3 = 27 y 2 = 21 y 1 = 18 y 1 18 y 2 21 y 3 27 y 1 + y 2 33 y 1 + y 3 30 y 2 + y 3 45 y 1 + y 2 + y 3 = 45 0 y 1 18 15 y 2 21 12 y 3 27 y 1 + y 2 + y 3 = 45 Water distribution (-3,21,27) 189 the yellow is the imputation set

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4.3 The Shapley value 190

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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

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To compute the Shapley value, we construct the vectors of marginal costs that, to each permutation = (i 1,...,i n ) n associates the imputation t( ) defined by: The Shapley value of a cost game (N,c) is defined as the average marginal cost vector: 192

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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): 193

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Water distribution: cost game c(1) = 18c(12) = 33 c(2) = 21c(13) = 30 c(3) = 27c(23) = 45 c(123) = 45 123 123181512 132181512 213122112 23102124 31231527 32101827 1/6 51105114 194

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Water distribution: surplus game v(1) = 0v(12) = 6 v(2) = 0v(13) = 15 v(3) = 0v(23) = 3 v(123) = 21 123 1230615 1320615 2136015 2311803 3121560 3211830 1/6 572148 195

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123 1232015 13220525 213102515 23152520 31215530 32151530 c(1) = 20c(12) = 35 c(2) = 25c(13) = 45 c(3) = 30c(23) = 45 c(123) = 50 5 ≤ y 1 ≤ 20 5 ≤ y 2 ≤ 25 15 ≤ y 3 ≤ 30 x1/6 7590135 12.51522.5 (20,15,15) (10,25,15) (5,25,20) (5,15,30) (15,5,30) (20,5,25) y 3 = 15 y 3 = 30 (50,0,0) (0,50,0) (0,0,50) y 2 = 5 y 2 = 25 y 1 = 20 y 1 = 5 (12.5,15,22.5) 6 distinct vectors 196

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4.4 Airport games 197

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An airport game is defined by individual costs c i 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: 198

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Concavity: let S et T be two coalitions with values: where c S c T (without loss of generality). 199

200
The natural cost allocation is the following: it is the Shapley value ! 200

201
In the case where n = 3: 201

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123 123c1c1 c 2 – c 1 c 3 – c 2 132c1c1 0c 3 – c 1 2130c2c2 c 3 – c 2 2310c2c2 c 3 – c 2 31200c3c3 32100c3c3 x 1/62c12c1 3c 2 – c 1 6c 3 – 3c 2 – c 1 4 distinct vectors 202

203
y 1 c 1 y 2 c 2 y 3 c 3 y 1 + y 2 c 2 y 1 + y 3 c 3 y 2 + y 3 c 3 y 1 + y 2 + y 3 = c 3 (c 3,0,0) (0,c 3,0) (0,0,c 3 ) 0 y 1 c 1 0 y 2 c 2 c 3 – c 2 y 3 c 3 y 1 + y 2 + y 3 = c 3 y 2 = c 2 y 3 = c 3 – c 2 (0,c 2,c 3 -c 2 ) (c1,c2-c1,c3-c2)(c1,c2-c1,c3-c2) (c 1,0,c 3 -c 1 ) y 1 = c 1 203

204
The associated surplus game is given by: Remark The last two players are substitutable, a property that is independent of the number of players. 204

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123 1230c1c1 c2c2 1320c2c2 c1c1 213c1c1 0c2c2 231c1c1 0c2c2 312c1c1 c2c2 0 321c1c1 c2c2 0 x 1/64c14c1 c 1 + 3c 2 4 distinct vectors 205

206
(c 1 + c 2,0,0) x 1 = c 1 x 2 = c 2 x 3 = c 2 x 1 0 x 2 0 x 3 0 x 1 + x 2 c 1 x 1 + x 3 c 1 x 2 + x 3 c 2 x 1 + x 2 + x 3 = c 1 + c 2 0 x 1 c 1 0 x 2 c 2 0 x 3 c 2 y 1 + y 2 + y 3 = c 3 (0,c 1 + c 2,0) (0,0,c 1 + c 2 ) (c 1,c 2,0) (c 1,0,c 2 ) (0,c 2,c 1 )(0,c 1,c 2 ) 206

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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

208
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

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4.5 Data games 209

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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

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Looking at a particular parameter, for a given substance, there are three cases: 1. that data is freely accessible 2. no firm has that data: it has to be produced 3. 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

212
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

213
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

214
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: It turns out to be the Shapley value of some appropriate game. 214

215
A data sharing situation is described by: N = {1,...,n} the set of players M 0 = {1,…,m} the set of available data d h > 0 the cost of reproducing data h M i M 0 the dataset held by player i 215

216
The cost for a coalition S if the cost of completing its dataset: where This defines a cost game called data game. 216

217
Example: N = {1,2,3} and M 0 = {1,2,3,4} M 1 = {1,3}d 1 = 90v 0 = 210 M 2 = {1,2}d 2 = 20 M 3 = {3,4}d 3 = 40 d 4 = 60 c(1) = d 2 + d 4 = 80c(1,2) = d 4 = 60 c(2) = d 3 + d 4 = 100c(1,3) = d 2 = 20 c(3) = d 1 + d 2 = 110c(2,3) = c(1,2,3) = 0 217

218
Proposition 4.4The 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(S T) essential if M i M 0 for some i: i c(i) > c(N) 218

219
Proof essential: subadditive: if S T = non-increasing: S T M S M T and 219

220
Case of a single data The "elementary" data game (N,c h ) associated to data h: where T h denotes the set of players holding data h. 220

221
By working data by data, any data game can be written as a sum of elementary data games 221

222
Because c(N) = 0, we are facing a pure compensation problem. We consider imputations: wherey i > 0 means that i pays y i y i < 0 means that i receives – y i 222

223
Proposition 4.5The 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 223

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the core is nonempty: it always contains the no compensation 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} 224

225
Proof If y C(N,c) we have If instead y satisfies core's definition, we have: Because y(N) = 0, we then have: 225

226
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

227
The core is full dimensional if and its vertices are given by:

228
Core of a 3-player data game 228

229
Looking at the core of the elementary game (N,C h ) 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

230
Core of a 3-player elementary data game where player 1 is alone to hold data h 230

231
The Shapley value of an elementary game c h (S) = 0si S T h c h (S) = d h si S T h = where t h is the number of players in T h. 231

232
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. 232

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Exampled = (90, 20, 40, 60) M 1 = {1,3}, M 2 = {1,2}, M 3 = {3,4} Here y belongs to the core. 233

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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) = (M 1,...,M n,d 1,...,d n ) such that M i = or M 0 : 234

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4.6 Assignment games 235

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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 = [u i (h) | i N, h M] Here u i (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

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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. 237

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Example u1u1 u2u2 u3u3 a399 b1266 c963 c(1) = 12 c(2) = 9 c(3) = 9 c(12) = 21 c(13) = 21 c(23) = 15 c(123) = 27 123 1231296 1321269 2131296 2311296 3121269 3211269 1/6 7245 SV(N,c) = (12, 7.5, 7.5) optimal allocation = (12,6,9): 1 receives house b 2 receives house c 3 receives house a transfers: (12,6,9) – (12,7.5,7.5) = (0, – 1.5, 1.5) players 2 and 3 are substitutes 238

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(0,27,0)(0,0,27) (27,0,0) x 3 = 9 x 2 = 6 x 1 = 12 (12,6,9) (12,9,6) x 1 = 12 6 x 2 9 6 x 3 9 x 3 = 6 x 2 = 9 v(1) = 12 v(2) = 9 v(3) = 9 v(12) = 21 v(13) = 21 v(23) = 15 v(123) = 27 core (12,7.5,7.5) set of imputations (9,9,9) (12,6,9) (12,9,6) optimal allocation before transfers 239

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Chapter 5 Power in decision games 240

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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

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5.1 Decision games 242

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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

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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

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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

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Assumptions D1the grand coalition is always winning: N W D2two disjoint coalitions cannot be simultaneously winning: Consequence: if a coalition is winning, its complement is necessarily loosing: 246

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D3enlarging 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

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5.2 Weighted majority games 248

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Weighted majority games form a particular class of decision games. Decision maker i is characterized by a weight w i 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. 249

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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: 250

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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 FrDeItBeNlLuUKDkIrlGrSpPtSeFiAu 1958 444221 1986 10 552 33585 1994 10 552 33585434 251

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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 w 1,…,w n and quota Q such that: i.e. reproduces the same winning coalitions. 252

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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: w i = 1 for all i and for unanimity: Q = n for simple majority: 253

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-the weighted majority game defined by Q = 1, w 1 = 1 and w 2 = w 3 = 0 is equivalent to the situation where 1 is adictator. -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

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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

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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

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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

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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, w 1 = 3 and w i = 1 (i = 2,...,5) 258

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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

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5.3 Decision games as simple games 260

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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

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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 x i 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

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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

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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

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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: 265

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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. 266

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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

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5.4 Power indices 268

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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

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Shapley et Shubik (1954) … proposed to apply the Shapley value to the associated simple game: Shapley-Shubik index (SSI) 270

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Marginal contributions are equal to 0 or 1: v(S) – v(S\i) = 0if S is loosing v(S) – v(S\i) = 0if S and S\i are winnings v(S) – v(S\i) = 1if 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

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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. 272

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The Banzhaf index and the "normalized" Banzhaf index where 273

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123 1000 1/3 2000 3000 12110 1/6 13101 1/6 23000 1/6 123100 1/3 311 SSI 1 = 2/6 + 1/3 = 2/3 SSI 2 = SSI 3 = 1/6 simple majority with veto BNI 1 = 3/5 BNI 2 = BNI 3 = 1/5 Decision maker 1 is decisive in 3 coalitions while the other two are decisive in only one coalition. 274 BI 1 = 3/4 BI 2 = BI 3 = 1/4

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123 123010 132001 213100 231100 312100 321100 1/6 411 Alternative computation of the Shapley-Shubik index: 275

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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) = 1/5 123,124,125,134,135,145 6 5 (3) = 1/5 1234,1235,1245,1345 4 5 (4) = 1/5 SSI 1 = 3/5 SSI i = 1/10 (i = 2,...,5) 5 = (1/5,1/20,1/30,1/20,1/5) SSI = (60,10,10,10,10) en % 276

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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. NBI 1 = 14/22 and NBI i = 2/22 (i = 2,...,5) NBI = (64, 9, 9, 9, 9) % to be compared to SSI = (60, 10, 10, 10, 10) % 277

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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. 278

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Interpreting Shapley-Shubik index and Banzhaf index as expectations: we observe that in the Banzhaf index, probabilities are independent of coalition size. 1/2 n-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. 279

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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

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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. 281

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The following equation must be satisfied: 5 SSI P + 10 SSI NP = 1. Hence: SSI P 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. 282

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The number of times a decision maker is decisive is: 848 5 + 84 10 = 5080. NBI NP = 84/5080 0.016 0.002 NBI P = 848/5080 0.167 0.196 or, in % NBI NP 1.6 0.2 NBI P 16.7 19.6 98 according to SSI ratio P/NP: 10 according to NBI 283

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Quota games A quota game with n players is a weighted majority game defined by a vector of relative weight 1,…, 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}. 284

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1234 44 240101 1/12 340011 1/12 1231110 1/12 1240101 1/12 1340011 1/12 2340001 1/12 12340000 1/4 4 =(1/4,1/12,1/12,1/4) 285

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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

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Party 1 is decisive in the coalitions: 287

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Party 3 is decisive in the coalitions: 288

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Der Deutsche Bundestag 1994-Today SeatsCDU/CSUSPDFDPGRÜNELINKEfraktionslos 19942942524749300672 19982452984347360669 2002248251475502603 20052232226151532612 Today2391469368760622 NBICDU/CSUSPDFDPGRÜNELINKEfraktionslos 19940.500.17 0.00 100 19980.130.550.13 0.050.00100 20020.33 0.000.330.00 100 20050.30 0.13 0.00100 Today0.500.17 0.000.170.00100 289

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Der Deutsche Bundestag 1994-Today Seats in %CDU/CSUSPDFDPGRÜNELINKEfraktionslos 199444377740100 199837446750100 200241428900100 200536 10890100 Today38231511120100 NBI in %CDU/CSUSPDFDPGRÜNELINKEfraktionslos 19945017 00100 1998135513 50100 200233 0 00100 200530 13 0100 Today5017 0 0100 290

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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

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195819861994 France410 Germany410 Italy410 Belgium255 Netherland255 Luxembourg122 England10 Denmark33 Ireland33 Greece55 Spain88 Portugal55 Sweden4 Finland3 Austria4 Total177687 Quota125462 EU-6EU-12EU-15 23.812.911.2 23.812.911.2 23.812.911.2 14;36.65.8 14.36.65.8 0.01.82.2 12.911.2 4.63.6 4.63.6 6.65.8 10.99.2 6.65.8 4.8 3.6 4.8 EU-6EU-12EU-15 23.313.4211.7 23.313.4211.7 23.313.4211.7 15.06.375.5 15.06.375.5 0.01.22.0 13.4211.7 4.263.5 4.263.5 6.37 5.5 11.129.5 6.375.5 4.5 3.5 4.5 Banzhaf Shapley-Shubik 292

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Chapitre 6 Beyond the core… 293

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The core of a cooperative game with transferable utility (N,v) is the set of imputations x = (x 1,…,x n ) 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. 294

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6.1 The bargaining set 295

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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 = (x 1,…,x n ) is formed of a coalition S and an allocation y such that: 296

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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. T S TSTS T\ST\S 297

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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

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6.2 Stable sets 299

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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: 300

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Proposition 6.1An imputation x belongs to the core if and only if it is never dominated. Proof 301

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302

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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 6.2There may be several stable sets and the core is a subset of all stable sets. Proposition 6.3In the case of a convex game, the core is the unique stable set. 303

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Proposition 6.4A set of imputations K is stable if and only if for all imputations x and y, internal stability external stability 304

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Proof Denote internal stability: external stability: (I is the set of imputations) 305

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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 x i ≤ v(i) = 0 and x i = 0 implies x i = 0. This is in contradiction with y i 0. 306

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(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 6.5In 2-player games, the set of all imputations is the only stable set and it coincides with the core. 307

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Example 3-player simple majority game: The following set is a solution: It is the "symmetric" solution. 308

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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

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310 (1,0,0) (0,1,0) (0,0,1) x 2 = 1/2 x 3 = 1/2 x 1 = 1/2 every imputation not in K is dominated by an imputation in K

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311 (1,0,0) (0,1,0) (0,0,1) x 2 = a 2 x 3 = a 3 x 1 = a 1 A1A1 A3A3 A2A2 a imputations dominated by an interior imputation

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From this we can conclude thatno 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

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313 (1,0,0) (0,1,0) (0,0,1) a b imputations dominated by a or b

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314 (1,0,0) (0,1,0) (0,0,1) a b imputations dominated by imputations on the line segment [a,b] not joining two sides of the triangle

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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

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The next figures show that the only other solutions are line segments of the following type: These are the "discriminatory" solutions. 316

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317 (1,0,0) (0,1,0) (0,0,1) a b x 2 = a 2 x 3 = b 3 x 1 = a 1 = b 1 a 1 = b 1 > 1/2 a 3 = b 2 = 0

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318 (1,0,0) (0,1,0) (0,0,1) a b x 2 = a 2 x 3 = b 3 x 1 = a 1 = b 1 a 1 = b 1 < 1/2 a 3 = b 2 = 0 K1()K1()

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6.3 Least core and nucleolus 319

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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." 320

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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. 321

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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

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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 vector formed 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

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To each x I(N,v) we associate the list of m = 2 n - 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

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Proposition 6.6Given 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 rule that to any game (N,v) associates an imputation NUC(N,v). Proposition 6.7As a rule, the nucleolus satisfies efficiency, symmetry and null player The nucleolus does not satisfy additivity. 325

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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

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Example: auction game (p 1 = 0) v(1) = v(2) = v(3) = v(23) = 0 v(12) = p 2 v(13) = v(123) = p 3 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. 327

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For each p, we order the excesses in a decreasing way: 328

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- p 3 p p3p3 p2p2 0 (p 2 -p 3 )/2 p 2 -p 3 – p p 2 – p p – p 3 p 2 329 least core = core nucleolus

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From this we conclude that the interval [p 2, p 3 ] defines the least core (that coincides with the core). Furthermore, the mid-point of interval [p 2, p 3 ] defines the nucleolus: i.e. (250, 0, 50) in the case where p 3 = 300 and p 2 = 200. 330

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Example: 3-player airport game The associated surplus game is given by: v(1) = v(2) = v(3) = 0 v(12) = v(13) = c 1 v(23) = c 2 v(123) = c 1 + c 2 Players 2 et 3 are substitutable the nucleolus is of the form: 331

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332

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c 1 + c 2 - q q 0 – (c 1 +c 2 ) q – c 1 – c 1 2c 2 > 3c 1 333

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c 1 + c 2 - q q 0 – (c 1 +c 2 ) q – c 1 – c 1 2c 2 = 3c 1 334

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c 1 + c 2 - q q 0 – (c 1 +c 2 ) q – c 1 – c 1 2c 2 < 3c 1 335

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The least-core is a singleton. It is therefore also the nucleolus: nucleolus of the cost game: 336

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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

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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

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339 e w 1 0 w* = 3/7 - w 3(1-w)/4 3/4 w 1/2 1/4 - 3/4 - 1/2 - 1/4 3/7

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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: 340

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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. 341

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The nucleolus of a 3-player data game 342

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Comparing the two allocation rules we observe that they coincide if and only if each data is held by a single player: 343

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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: 344

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Core allocations impose that only player n may be compensated: 0 y i d 1 for all i = 1,…, n – 1 The nucleolus goes further by imposing that the n – 1 other players contribute a same amount: 345

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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 346

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Special case: the case where t h = 1 for all h M 0 is the partition case: M i M j = 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. 347

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In the partition case: where c i = c(i) is the cost of the data that player i is missing. 348

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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: 349

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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. 350

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Exemple N = {1,2,3,4} and M 0 = {1,2,3,4} M 1 = d 1 = 6 M 2 = {1}d 2 = 10 M 3 = {2}d 3 = 4 M 4 = {3,4} d 4 = 12 y 1 = 8 – 0 = 8 y 2 = 8 – 6 = 2 y 3 = 8 – 10 = – 2 y 4 = 8 – 4 – 12 = – 8 351

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In the partition case, data games are concave. It is more easy to check that the associated surplus game is convex: 352

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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

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Core of a 3-player partition data game 354

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(c 1 – v 0, c 2 – v 0, c 3, c 4 – v 0 ) (c 1 – v 0, c 2, c 3 – v 0, c 4 – v 0 ) (c 1, c 2 – v 0, c 3 – v 0, c 4 – v 0 ) (c 1 – v 0, c 2 – v 0, c 3 – v 0, c 4 ) y 3 = c 3 – v 0 y 2 = c 2 – v 0 y 4 = c 4 – v 0 y 1 = c 1 – v 0 y 1 + y 4 = c 1 + c 4 – v 0 y 1 + y 2 = c 1 + c 2 – v 0 y 2 + y 3 = c 2 + c 3 – v 0 y 3 + y 4 = c 3 + c 4 – v 0 y 2 + y 4 = c 2 + c 4 – v 0 355 Core of a 4-player partition data game

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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

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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.il www.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) 357

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