Cooperative games

Pierre DehezCORE, University of Louvain

(Center for Operations Research et Econometrics)

pierre.dehez@uclouvain.be

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

Chapter 1

Characteristic function games

3

1.1 Transferable utility

4

The problem is the following: A group of individuals considers the possibility of cooperating on a common project.

To cooperate means that each member of the group accepts to act in the interest of the group ...

… by adopting strategies aiming at the maximization of the collective welfare

Question: How to share the results of this cooperation ?

5

Example: sharing a euro

Two persons are proposed a euro provided they agree on the way to split it.

Question: How to split it ?

The immediate answer is clearly an equal splitting: 50 cents for each.

This is the fair solution.

6

Example: porters

Imagine three porters in a train station. They are asked to move four (heavy) suitcases for 6 €.

The four suitcases must be moved together and two porters can do it.

Question: How to share the 6 € ?

Here too, the egalitarian solution seems fair: 2 € for each.

7

Example: auction between one seller and two buyers

Each trader has a reservation price.

The object will naturally be sold to the buyer with the largest reservation price.

Questions:

- At which price?

- Should the buyer with the lowest reservation be compensated?

8

Example: sharing a crop

Imagine a landlord and m (identical) workers, and a technology described by a production function y = F(s) where s is the number of workers.

With the landlord's agreement and all workers on the land, the crop is F(m).

Question: How to share the crop between the landlord and the workers?

Workers receiving the same amount is fair but how much should be allocated to the landlord?

9

The four examples are defined in terms of a commodity which is used as a "money": euros or kilos of wheat…

Quantities of these commodities can be added and "monetary" transfers between players can be organized.

These situations are (cooperative) games with transferable utility or games with side payments: TU games.

We shall see that coalitions of player will play a central role in the analysis of TU-games.

10

1.2 Characteristic function

11

Let N = {1,…,n} denote the set of players, n 2.

A coalition is a subset of players S N.

In particular, {i} and N are also considered as coalitions. N is the "grand coalition".

If we add the empty set, there are 2n coalitions and to each coalition S N, is associated a real number, its worth:

with the convention v() = 0.

( )v S

12

This defines a set function, called "characteristic" function:

where v(S) is measures the minimum gain that coalition S can generate if it forms, whatever the other players do.

v(S) is expressed in units of a "money" that is some perfectly divisible commodity (euros, tons de wheat, ...).

The set G of all set functions on a set N is a vector space of dimension 2n-1 that can be identified as

: 2Nv

, ( ) for all ,v w G v w G

2 1n

13

If a coalition S N forms, v(S) can be defined on the basis of the two player noncooperative game where the players are S and N\S: v(S) is then the "security level":

the members of S act to maximize the gain of the coalition assuming that the excluded members act

against S.

This is the MaxiMin associated to prudent strategies.

v(N) is the maximum gain obtained by the "grand" coalition when all players coordinate their actions.

14

Notation:

v(i) = v({i}), v(ij) = v({i,j}), v(ijk) = v({i,j,k}), …

Si = S{i}, S\i = S\{i}

|A| is the cardinal of the set A and n = |N|, s = |S|, t = |T|,…

1for a vector ( ,..., ) and a subset {1,..., }

we write ( )n

ii S

x x x S n

x S x

15

In sharing a euro, N = {1,2} and the characteristic function is given by:

v(1) = v(2) = 0

v(12) = 1

Alone, a player gets nothing. Together they receive 1 €.

16

In the porters' game, N = {1,2,3} and the characteristic function is given by:

v(1) = v(2) = v(3) = 0

v(12) = v(13) = v(23) = 6

v(123) = 6

Alone, a porter cannot move the 4 suitcases, but two or three can do it.

17

In the auction game, if S contains the seller and at least one buyer, v(S) is the sum of the gains of the seller and of the buyer whose reservation price is highest:

v(1) = v(2) = v(3) = 0

v(12) = (p – p1) + (p2 – p) = (p2 – p1)

v(13) = (p – p1) + (p3 – p) = (p3 – p1)

v(23) = 0

v(123) = (p – p1) + (p3 – p) = (p3 – p1)

where p is the price at which the transaction takes place. It does not appear because gains are added!

seller: 1buyers: 2 and 3

p1 ≤ p2 ≤ p3

18

In the crop game, there are n = m + 1 players:

v(S) = 0 if S does not include the landlord

v(S) = F(s – 1) if S includes the landlord (he/she does not work)

In particular, v(i) = 0 for all i and v(N) = F(m).

We suppose that F is increasing with F(0) = 0, not more at this stage.

19

Merging coalitions should be beneficial or, at least, not detrimental:

This is the property of super-additivity.

From now on, a game is a pair (N,v) where N is a set of players and v is a super-additive set function on N.

( ) ( ) ( )S T v S T v S v T

20

Super-additivity implies in particular the following inequalities:

Hence cooperation does not induce a loss.

In case of a strict inequality, there is a potential gain for all and the game is said to be essential:

All the examples we have seen so far are super-additive and essential games.

1

( ) ( ) for all ( ) ( )n

i S i

v i v S S v i v N

1

( ) ( )n

i

v i v N

21

The importance of a player is not only measured by his/her contribution to the "grand coalition" v(N) – v(N\i).

One has to take also into account his/her (marginal) contributions to all those coalitions to which he/she belongs to:

In particular, Cmi(i) = v(i).

By super-additivity we have:

( ) ( ) ( \ )iCm S v S v S i

22

( ) ( \ ) ( ) ( ) ( ) ifiv S v S i v i Cm S v i i S

Two players i et j are substitutable in a game (N,v) if their marginal contributions are identical:

for all coalitions S containing i et j.

Alternatively, i et j are substitutes in (N,v) if:

for all coalitions S containing i et j.

( ) ( \ ) ( ) ( \ )v S v S i v S v S j

( \ ) ( \ )v S i v S j

23

For instance, in the 3-player case, players 2 et 3 are substitutes if and only if:

v(2) = v(3)

v(12) = v(13)

Indeed we have:

v(23) – v(2) = v(23) – v(3)

v(123) – v(13) = v(123) – v(12)

24

Player i is a null player in a game (N,v) if he/she contributes to no coalition:

for all coalition S containing i.

In a 3-player game, player 1 is a null player if:

v(1) = 0

v(12) = v(2)

v(13) = v(3)

v(123) = v(23)

( ) ( \ ) 0v S v S i

25

1.3 Particular classes of games

26

A game is monotonic if enlarging a coalition is never detrimental, in which case the marginal contributions are nonnegative:

Proposition 1.1 A game such that v(S) 0 for all S N is monotonic.

This is an immediate consequence of super-additivity:

( ) ( )S T v S v T

( ) ( / ) ( ) ( ) ( )S T v S v T S v T v S v T

( ) 0iCm S

27

A game is symmetric if the worth of any coalition depends only upon the number of its members, independently of their identity: all players are substitutable.

It means that there exists a function f such that:

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

( ) ( ).v S f s

( ) ( ) ( ) ( )S T s t f s f t v S v T

28

Example: garbage game

Each player has a garbage bag he/she would like to dispose of in a neighbor's garden.

v(S) = – (n – s) for all S N,

v(N) = – n

i.e. the "disutility" is proportional to the numbers of bags.

This game is super-additive and symmetric but it is not monotonic.

29

sn0 1

–n

f(s)

–(n–1)

n–1

–1

v(S) = f(s)

30

A game (N,v) is convex if for all coalitions S et T:

Clearly, convexity implies super-additivity and the set of convex game is a sub-cone of the cone of superadditive games.

Proposition 1.3 A game (N,v) is convex if and only if marginal contributions are non-decreasing:

for all i, S, T such that i S T.

( ) ( ) ( ) ( )v S T v S v T v S T

( ) ( )i iCm S Cm T

31

The crop game is convex if and only if returns to scale are constant or increasing (non-decreasing marginal productivity).

Consider a player i and two distinct coalitions S and T such that i S T:

1if :

( ) ( \ ) ( 1) ( 2) ( 1) ( 2) ( ) ( \ )

i S

v S v S i F s F s F t F t v T v T i

1if \ :

( ) ( \ ) 0

( ) ( \ ) ( 1) ( 2) 0

i T S

v S v S i

v T v T i F t F t

32

1if :

( ) ( \1) ( 1) 0

( ) ( \1) ( 1)

( 1)

i

v S v S F s

v T v T F t F s

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

1if :

( ) ( \ ) ( ) ( \ )

0

i T

v S v S i v T v T i

33

The auction game with p = (0,200,300) is not convex:

v(12) + v(13) – v(1) = 500 > v(123) = 300

Alternatively, in terms of marginal contributions:

v(13) – v(1) = 300 > v(123) – v(12) = 100

The porter's game is not convex:

v(12) + v(13) – v(1) = 12 > v(123) = 6

Alternatively, in terms of marginal contributions:

v(12) – v(1) = 6 > v(123) – v(13) = 0

34

A simple game is defined by a binary characteristic function:

v(S) {0,1} with

The normalized porters' game is a simple (and symmetric) game:

v(S) = 1 if and only if s 2

it is obtained by dividing by 6 all coalitions' worth.

( ) 0 and ( ) 1

( ) 1 ( \ ) 0

v i v N

v S v N S

35

Chapter 2

Imputations and the core

36

Given a TU-game (N,v), the problem is to distribute the result of the cooperation as measured by v(N).

Cooperation means that each player acts in the interest of the collectivity so as to obtain the largest "social output".

The immediate and natural question then concerns the remuneration of the players for their participation to the collective project.

Formally, the problem is to find n numbers x1,...,xn such that:

37

1

( )n

ii

x v N

2.1 Imputations

38

There are two minimal conditions to impose on a distribution of the social output:

(i) collective rationality (Pareto efficiency)

all v(N) should be distributed: no waste

(ii) individual rationality (incentives)

no player should get less that what he/she can get alone

1

( )n

ii

x v N

39

( )ix v i i N

An imputation is an allocation satisfying these two conditions:

Super-additivity ensures that the set of imputations is nonempty:

The set of imputations is a polyhedron of dimension n – 1 if the game is essential. It is actually a regular simplex.

1

( ) ( ) ( , )n

i

v i v N I N v

40

( , ) { | ( ) ( ) and ( ) for all }niI N v x x N v N x v i i N

If the game is inessential the imputation set reduces to the allocation that gives each player his/her worth:

If the game is essential

and there are imputations (x1,...,xn) which are better for all members of S:

( )ix v i i N

1

( ) ( )n

i

v i v N

41

( , ) { (1),..., ( )}I N v v v n

In the porters' game, an imputation is defined by three nonnegative numbers x1, x2, x3 summing to 6:

x1, x2, x3 0

x1 + x2 + x3 = 6

In the auction game, an imputation is defined by three nonnegative numbers x1, x2, x3 summing to 300:

x1, x2, x3 0

x1 + x2 + x3 = 300

42

x2

x1

imputations line for n = 2:xi v(i) i = 1,2x1 + x2 = v(N)

43

( ) (2), (2)v N v v

(1), ( ) (1)v v N v

0 v(N)

v(N)

v(1)

v(2)

v(N) – v(2) – v(3)x1

x2 x3

(v(1),v(2),v(3))

imputations' triangle for n = 3:xi v(i) for i = 1,2,3x1 + x2 + x3 = v(N)

44

v(N) – v(1) – v(3) v(N) – v(1) – v(2)

x2 = v(2)x 3

= v

(3)

x1 = v(1)

45

( ) (2) (3), (2), (3)v N v v v v

(1), ( ) (1) (3), (3)v v N v v v (1), (2), ( ) (1) (2)v v v N v v

imputations' triangle for n = 3:xi v(i) for i = 1,2,3x1 + x2 + x3 = v(N)

x1

x2 x3

0x

2 = 0x 3 =

0

x1 = 0

v(N)

v(N) v(N)

46

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

47

( ),0,0v N

0, ( ),0v N 0,0, ( )v N

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

2.2 The core

48

Extending individual rationality to coalitions leads to the following condition:

no coalition should be allocated less than what it could get by itself

Indeed if coalition S would be in a position to object against the allocation x = (x1,…,xn)…

( ) ( ) pour toutx S v S S N

( ) ( )x S v S

( ) ii S

x S x

49

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

( ) ( ) there exists ( | ) such that:

( ) ( ) and for all

i

i i

x S v S y i S

y S v S y x i S

50

( , ) ( , )C N v I N v

The set of allocations against which no coalition can formulate an objection defines the core (Gillies, 1953):

i.e. core allocations are "socially" stable.

Core allocations are imputations:(just take S = {i} and S = N)

Geometrically, the core is an intersection of half-spaces. Hence it is a convex polyhedron whose dimension does not exceed n – 1.

( , ) { | ( ) ( ) and ( ) ( ) for all }nC N v x x N v N x S v S S N

51

In the porters' game, the core is defined by:

x1, x2, x3 0x1 + x2 + x3 = 6x1 + x2 6x1 + x3 6x2 + x3 6

No allocation does satisfy all seven conditions: they are not compatible.

the core of the porter's game is empty: C(N,v) =

x3 = 0 ... x2 = 0 ... x1 = 0

52

Whatever is the allocation proposed, there will be a coalition to object.

In particular, 2-player coalitions will refuse the egalitarian allocation (2,2,2): they can get 4€ instead of 6€.

Coalition {1,2} can indeed get 6€ and propose the allocation (3,3,0) against which coalition {1,3} will object by proposing

(4,0,2), an allocation which will be rejected by coalition {2,3}, …

Endless… 53

Proposition 2.1 Core allocations x C(N,v) satisfy the following inequalities:

Indeed, if x C(N,v), we have

Hence, in particular, core allocations x satisfy

( ) ( ) ( \ ) for alliv i x v N v N i i N

( ) ( ) ( ) ( \ ) for allv S x S v N v N S S N

54

( ) ( ) ( ) ( \ ) ( ) ( \ )v N x N x S x N S x S v N S

The core of the auction game is defined by:

x1, x2, x3 0x1 + x2 + x3 = p3 – p1 x1 + x2 p2 – p1

x1 + x3 p3 – p1

x2 + x3 0 (redundant)

First observation: (p3 – p1, 0, 0) belongs to the core.

Second observation: x2 = 0 in all core allocations.

x2 = 0

55

i.e. - buyers cannot object against the fact that the seller gets all!

- any allocation giving a positive amount to the firstbuyer is not stable.

The allocation (p2 – p1, 0, p3 – p2) is also in the core. It is the allocation which is most favorable to the second buyer.

56

To summarize, the core of the auction game is defined by:

We observe that the maximum the second buyer can expect is equal to the price difference p3 – p2. The minimum the seller can expect is the lowest price p2.

if the buyers' prices would be equal, say p3, the core then reduces to the allocation (p3 – p1, 0, 0) !

1 3 2 3( , ) {( ,0, ) | }C N v p p p p p p p

57

Indeed, the core conditions are then given by:

x1, x2, x3 0

x1 + x2 + x3 = p3 – p1

x1 + x2 p3 – p1

x1 + x3 p3 – p1

x2 = x3 = 0 x1 = p3 – p1

3 1( , ) {( ,0,0)}C N v p p

58

Core of the crop game with 2 workers and constant return to scale: m = 2 and F(s) = s

x1, x2, x3 0

x1 + x2 + x3 = 2x1 + x2 x1 + x3

x2 + x3 0 (redundant)

First observation: (2, 0, 0) and (0, , ) belong to the core

Second observation: the maximum a worker can expect is

landlord: 1travailleurs: 2 et 3

v(i) = 0v(12) = v(13) = v(23) = 0v(123) = 2

59

We first observe that the extreme allocation (F(m), 0, …,0) always belongs to the core.

Furthermore, the maximum a worker can expect within the core is determined by the marginal productivity of the "last" worker. For all i 1, we have:

This is a consequence of Proposition 2.1. Indeed we have:

( ) ( \ ) ( ) ( 1)ix v N v N i F m F m

( ) ( 1)ix F m F m

60

Example: 4-player gloves market (perfect complements)

Consider four players and the following initial distribution of gloves :

(R, R, L, LL)

The worth of a coalition is equal to the number of pairs it can form.

With two left gloves, player 4 seems to be in an advantageous position...

61

There are 15 coalitions and the characteristic function is defined by:

v(i) = 0 for all i N

v(12) = v(34) = 0

v(13) = v(14) = v(23) = v(24) = 1

v(123) = v(134) = v(234) = 1

v(124) = v(N) = 2

The core consists of a unique allocation: (1, 1, 0, 0) !

(R, R, L, LL)

62

(1, 1, 0, 0) is indeed the unique allocation satisfying the core conditions :

x1, x2, x3, x4 0

x1 + x2 + x3 + x4 = 2

x1 + x3 1

x1 + x4 1

x2 + x3 1

x2 + x4 1

x1 + x2 + x4 2

(ignoring redundant inequalities)

x3 = 0

x1, x2 1 x4 = 0

(R, R, L, LL)

63

Example: 3-player gloves market

Initial distribution of gloves:

(RR, L, LL)

The characteristic function is defined by:

v(1) = v(2) = v(3) = v(23) = 0

v(12) = 1

v(13) = v(N) = 2

64

The core conditions are given by:

x1, x2, x3 0,

x1 + x2 + x3 = 2

x1 + x2 1

x1 + x3 2

In the absence of transfers between players, only two allocations actually satisfy these conditions:

(1, 0, 1) and (2, 0, 0)

(RR, L, LL)

( , ) {(1 ,0,1 ) | 0 1}C N v z z z

65

Game where some v(i) are nonzero can be modified by a translation in such a way that individual values are zero:

starting from a game (N,v), we define the game (N,u) by:

( ) ( ) ( )i S

u S v S v i S N

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

66

If (y1,…,yn) belongs to the core of the game (N,u), the allocation (x1,…,xn) defined by

belongs to the core of the game (N,v). Indeed:

( )i ix y v i

( ) ( ) ( ) ( ) ( ) ( )i S i S

x S y S v i u S v i v S S N

( ) ( )y S u S S N

( ) ( )x S v S S N

67

translation invariance

x1

x2 x3

0c

x1 + x3 = v(N) – c

fixing the coordinate of one player,we get a line segment in the triangle which is the locus of allocations distributing what is left between theother two players

x2 = c, 0 c v(N)

v(N)

v(N) v(N)

68

(v(N),0,0)

(0,v(N),0) (0,0,v(N))x

1 + x

3 = v(N

) – c

x1 +

x3 =

v(N)x

2 = c

x2 =

0

69

(v(N),0,0)

(0,v(N),0) (0,0,v(N))

the segment divides the triangle in two parts

x2 cx1 + x3 v(N) - c

(v(N) – c, c, 0)

(0, c, v(N) – c)

x2 cx1 + x3 v(N) - c

x2 = cx1 + x3 = v(N) - c the imputations (v(N) – c, c, 0)

and (0, c, v(N) – c) are the twoextreme points of the line segment

70

(v(N),0,0)

(0,v(N),0) (0,0,v(N))

the center of gravity of the triangle is the egalitarian allocation x defined byxi = v(N)/3 (i = 1,2,3)

71

(v(N),0,0)

(0,v(N),0)

x1 + x2 = v(1,2)

x2 + x3 = v(2,3)

(0,0,v(N))

x1 + x3 = v(1,3)

x1 +

x3 =

v(N)

x2 =

0

x 1 +

x 2 =

v(N

)

x 3 =

0

x2 + x3 = v(N) x1 = 0

nonempty core

72

(v(N),0,0)

(0,v(N),0)

x1 + x2 = v(1,2)

x2 + x3 = v(2,3)

(0,0,v(N))

x1 + x3 = v(1,3)x

1 + x

3 = v(N

)

x2 =

0

x 1 +

x 2 =

v(N

)

x 3 =

0

x2 + x3 = v(N) x1 = 0

core reduced to a single point

73

(v(N),0,0)

(0,v(N),0)

x1 + x2 = v(1,2)

x2 + x3 = v(2,3)

(0,0,v(N))

x1 + x3 = v(1,3)x

1 + x

3 = v(N

)

x2 =

0

x 1 +

x 2 =

v(N

)

x 3 =

0

x2 + x3 = v(N) x1 = 0

empty core

74

(15,0,0)

(0,15,0)

x1 + x2 = 10

(0,0,15)

x1 + x3 = 10

Crop game with decreasing returns

(5,5,5)

(10,0,5)(10,5,0)

(0,10,5) (0,5,10)75

v(i) = 0v(12) = v(13) = 10v(23) = 0v(123) = 15

(300,0,0)

(0,300,0)

x1 + x2 = 200

(0,0,300)

x1 + x3 = 300

(200,0,100)

Auction game:v(i) = 0v(12) = 200v(13) = 300v(23) = 0v(123) = 300

76

(2,0,0)

(0,2,0)

x1 + x2 = 1

(0,0,2)

x1 + x3 = 2

(1,0,1)

Gloves market: (RR, L, LL)

77

2.3 Nonemptiness of the core

78

Super-additivity is not sufficient (nor necessary) to ensure non-emptiness of the core. There is a general theorem (due to Shapley and to Bondareva) that gives necessary and sufficient conditions under which the core is nonempty.

For n = 3 and assuming super-additivity and v(i) = 0 for all i, these conditions reduce to a single inequality:

v(12) + v(13) + v(23) ≤ 2 v(N)

Indeed, adding the three inequalities xi + xj v(ij) for all i j, we get:

2(x1 + x2 + x3) v(12) + v(13) + v(23)

79

Proposition 2.2 A game (N,v) whose core is nonempty satisfies the following inequalities:

Indeed, if x C(N,v) we have for all S N:

( ) ( \ ) ( ) pour toutv S v N S v N S N

( ) ( )and ( ) ( \ ) ( ) ( \ )( \ ) ( \ )

( ) ( \ ) ( ) ( )

v S x Sv S v N S x S x N S

v N S x N S

v S v N S x N v N

80

There are classes of games whose cores are empty. The core of a inessential game is nonempty: it reduces to the allocation which gives to each player his/her value: xi = v(i) for all i.

The largest class of games whose cores are nonempty are the convex games.

Proposition 2.3 The core of a convex game is nonempty.

This is a result due to Shapley (1971).

81

Proposition 2.4 The core of a simple game is nonempty if and only if there are veto players who then share the worth of the game.

Proof Consider a decision game (N,v). Let V N denote the set of players with a veto right. We proceed in three steps.

(i) V = C(N,v) =

If V = , then v(N\i) = 1 for all i N. Let x be a core allocation. Combining x(N) = 1 with x(N\i) v(N\i) = 1, we get x = 0, a contradiction.

82

(ii) V C(N,v)

Consider the allocation x defined by

where the xi's for i V are choosen in such a way that x(N) = 1, belongs to the core.

Indeed v(S) = 1 if and only if V S. Hence x(S) ≤ v(S) for all S N.

83

0 for all

0 for alli

i

x i V

x i V

(iii) x C(N,v) xi = 0 for all i V

Consider a core allocation x such that xi > 0 for some i V. Then, using the fact that xi 0 for all i N, we get:

a contradiction: the coalition of veto players can improve upon the allocation x.

84

1 ( )ii V

x v V

Chapter 3

The Shapley value

85

3.1 Looking for a fair compromise

86

The core is a set which may be empty or contain many allocations: they are "stable". The notion of core does not include any value judgement on allocations.

A game with an empty core means that the situation it describes is such that it is impossible to distribute the "social output" without facing objections from some coalitions. A compromise based on accepted principles is then necessary.

A compromise is also necessary when there are many stable allocations or when core allocations are not fair.

The Shapley value is such a compromise.

87

What can a player expect from playing a cooperative game ?

In the porters' game, a possible neutral mechanism consists in ordering the players at random and let them take the suitcases sequentially:

being the first is too early: two suitcases left

being second means 6 € in the pocket

being last is too late: no more suitcases

Each porter has a probability 1/3 of being 2nd

each porter may expect to receive 2 €

88

(0,6,0)

1(6 6 0) 2

6ix

1each branch has the same probability:

6

(3,1,2)

(3,2,1)

(6,0,0)

(1,2,3) (0,6,0)

(1,3,2) (0,0,6)

(2,1,3) (6,0,0)

(2,3,1) (0,0,6)

89

In this case, the egalitarian solution is the natural allocation:

the three porters are in a symmetric position:no one is more "important" than an other

In situations where players are identical ("substitutable"), fairness requires an identical treatment:

each player must have the same expectation

Let's apply this random mechanism to the auction game.

90

1

2

3

1 1100(200 300 300 300) 183

6 61 200

200 336 61 500

(300 100 100) 836 6

x

x

x

(3,1,2)

(3,2,1)

(1,2,3) (0, p2 – p1, p3 – p2) = (0, 200, 100)

(1,3,2)

(2,1,3)

(2,3,1)

(0,0, p3 – p1) = (0, 0, 300)

(p2 – p1, 0, p3 – p2) = (200, 0, 100)

(p3 – p1 ,0, 0,) = (300, 0, 0)

(p3 – p1, 0, 0) = (300, 0, 0)

(p3 – p1, 0, 0) = (300, 0, 0)p = (0, 200, 300)

91

1 2 3

123 0 200 100

132 0 0 300

213 200 0 100

231 300 0 0

312 300 0 0

321 300 0 0

1/6 1100 200 500

550 100 250( , , ) (183,33,83)

3 3 3x

v(i) = 0v(12) = 200v(13) = 300v(23) = 0v(123) = 300

each row corresponds to a permutation

each column corresponds to a player

92

The object is sold at a price

p = 183

which means a gain equal to 116 for the second buyer…

... who gives 33 to the first and keeps 83 for him/herself.

What happens if the two buyers have the same reservation price (say 300) ?

93

1 2 3

123 0 300 0

132 0 0 300

213 300 0 0

231 300 0 0

312 300 0 0

321 300 0 0

1/6 1200 300 300

(200,50,50)x

v(i) = 0v(12) = 300v(13) = 300v(23) = 0v(123) = 300

94

The object is sold at a price

p = 200

and the two buyers have get the same, 50.

... we just don't know which one gets the object !

They are anyway indifferent.

It could be allocated by flipping a coin in which case the buyer who gets the object compensates the other buyer.

95

3.2 The Shapley value

96

The procedure is the following...

- players are ordered at random,

- they enter in a room following the given order,

- when a player gets in, he/she receives his/her marginal contribution to the set of players already present.

Each player expects to receive his/her average "marginal contribution", all orders having the same probability.

We denote by n the set of n! orderings (permutations) of the n players.

97

In this way, v(N) is entirely allocated.

Indeed, if the players are ordered according to the permutation = (i1,…,in) n we have successively:

v(i1) – v()

+ [v(i1,i2) – v(i1)]

+ [v(i1,i2,i3) – v(i1,i2)]

+ …

+ [v(i1,i2,…,in-1) – v(i1,i2,…,in-2)]

+ [v(i1,i2,…,in) – v(i1,i2,…,in-1)] = v(N)

98

To each permutation = (i1,…,in) we associate the vector of marginal contributions () defined by:

1 1 1

1 1 1

( ) ( ) ( ) ( )

( ) ( ,..., ) ( ,..., ) ( 2,..., 1)

( ) ( ) ( \ )

k

n

i

i k k

i n

v i v v i

v i i v i i k n

v N v N i

There are n! vectors, not necessarily distinct.

Proposition 3.1 A game is convex if and only if its core is a (Shapley, 1971) convex polyhedron whose vertices are the

marginal contribution vectors.

99

Crop game: constant returns to scale

1 2 3

123 0

132 0

213 0

231 2 0 0

312 0

321 2 0 0

1/6 6 3 3 ( , , )2 2

x

the crop is shared equally between the landlord and the workers who receive 1/2 of the marginal product

v(i) = 0v(12) = v(13) = v(23) = 0v(123) = 2

100

the share of the landlord is greater than the share of the workers who receive 2/3 of the marginal product

1 2 3

123 0 /2

132 0 /2

213 0 /2

231 3/2 0 0

312 /2 0

321 3/2 0 0

1/6 5 2 25

( , , )6 3 3

x

v(i) = 0v(12) = v(13) = v(23) = 0v(123) = 3/2

Crop game: decreasing returns to scale

101

1 2 3

123 0 3/2

132 0 3/2

213 0 3/2

231 5/2 0 0

312 3/2 0

321 5/2 0 0

1/6 7 4 4

v(i) = 0v(12) = v(13) = v(23) = 0v(123) = 5/2

7 2 2( , , )

6 3 3x

Crop game: increasing returns to scale

the share of the landlord is smaller than the share of the workers who receive 4/9 of the marginal product

102

1 2 3

123 0 6 0

132 0 0 6

213 6 0 0

231 0 0 6

312 6 0 0

321 0 6 0

1 2 3

123 0 10 10

132 0 10 10

213 10 0 10

231 20 0 0

312 10 10 0

321 20 0 0

1 2 3

123 0 10 5

132 0 5 10

213 10 0 5

231 15 0 0

312 10 5 0

321 15 0 0

6 0 0

0 6 0

0 0 6

0 10 10

10 0 10

10 10 0

20 0 0

0 10 5

0 5 10

10 0 5

10 5 0

15 0 0

porter game crop game with decreasing returns crop game with decreasing returns

103

This procedures allocates to player i his/her average marginal contribution:

1( )

!n

i ixn

It defines a rule SV which, to any game (N,v) associates an allocation, called "value":

SV : (N,v) (x1,...,xn) = SV(N,v)

It is precisely the value introduced by Loyd Shapley in 1953. He has proposed a set of criteria (axioms) that a rule should meet and that actually characterize uniquely the above rule.

104

3.3 Axiomatization of the value

105

Efficiency (Eff): the rule is such that

for all and all ( ) : ( , ) ( )ii N

N v G N N v v N

This is a requirement of collective rationality:

only rules which distribute the entirevalue of the game are considered

106

Symmetry (SYM): the rule is such that

if players i and j are substitutable in a game (N,v), they should be allocated the same amount:

This is a fairness requirement:

"equal treatment of equals"

( , ) ( , )i jN v N v

107

In a symmetric game, all players are substitutable. As aconsequence, any rule satisfying efficiency and symmetry allocates a same amount to each player:

( )( , ) ( , )i j

v NN v N v

n

The porters' game is symmetric:

6( , ) 2 for 1,2,3

3i N v i

108

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

( )( , ) 1,...,i

v NED N v i n

n

Is this egalitarian solution always fair ?

The way players contribute to a game may be different.

In particular, should a player who never contributes to a coalition be compensated ?

109

Null player (NULL): the rule is such that

if player i is null in a game (N,v), he/she should receive nothing within that game:

With this axiom, null players can be neglected:

if L is the set of null players in a game (N,v), a rule satisfying A3 can be restricted to the reduced game

(N\L,v') where v'(S) = v(S\L).

( , ) 0i N v

110

Let's modify the porters' game by assuming that player 1 can carry only one suitcase. The characteristic function is then given by:

v(1) = v(2) = v(3) = 0

v(12) = v(13) = 0

v(23) = v(123) = 6

Applying any rule satisfying the three axioms we get:

1 3 26

2( , ) 0 et ( , ) ( , ) 3N v N v N v

player 1 is null

111

This example shows that equal division does not satisfy the null player axiom.

An other rule consists in taking only into account marginal contributions to the grand coalition, and to share the resulting surplus (or deficit) equally among the players:

1

1( , ) [ ( ) ( \ )] ( ) [ ( ) ( \ )]

surplus or déficit

n

ij

AC N v v N v N i v N v N v N jn

112

This rule satisfies efficiency by construction.

It also satisfies symmetry: two substitutes players have identical marginal contributions to the grand coalition.

However it does not satisfy the null player axiom.

Applied to the modified porters' game, this rule results in an absurd outcome:

1

2 3

( , ) 0 2 2

( , ) ( , ) 6 2 4

N v

N v N v

113

There are actually many rules satisfying the above three axioms.

One may for instance modify the rules ED and AC by proceeding in two steps: first allocate zero to the null players and then apply the rule to the remaining player set.

Applied to the modified porters' game the two modified rules lead to the same outcome: (0,3,3).

114

We need additional axioms to reduce the possible rules. Actually, one additional axiom will reduce the set of possible rules to a single rule.

We have seen that games defined on a given set of players can be added to define a new game:

the sum of the games (N,v1) and (N,v2) is the game (N,v) whose characteristic function v is defined by:

1 2( ) ( ) ( )v S v S v S

115

Consider the games (N,v1) and (N,v2) defined by:

v1(i) = 0 v2(i) = 0

v1(12) = v1(13) = 0 v2(12) = v2(23) = 0

v1(23) = 6 v2(13) = 6

v1(123) = 6 v2(123) = 6

v(i) = 0

v(12) = 0

v(13) = v(23) = 6

v (123) = 12

v = v1 + v2

116

Additivity (ADD): the rule is such that

the value of the sum of two games is the sum of the values:

This is a independence axiom:

facing differents games, players evaluate them independently

1 2 1 2 1( , ) ( , ) ( , ) ,...,i i iN v v N v N v i n

117

The modified rules ED and AC do not satisfy additivity. This can be seen by applying them to the games v1, v2 and v = v1 + v2. The modified ED rule gives:

1

2

1 2

1 2

( , ) (0,3,3)

( , ) (3,0,3)

( , ) ( , ) (3,3,6)

( , ) (4, 4,4) (3,3,6)

N v

N v

N v N v

N v v

118

Shapley (1953) has proved the following remarquable proposition:

Proposition 3.2 There is one and only one rule that satisfies (Shapley, 1953) simultaneously EFF, SYM, NULL and ADD.

That unique rule is the Shapley value. Indeed, it is easily verified that the rule defined by the average of the marginal contribution vectors satisfies all four axioms.

There are actually other axiomatizations of the Shapley value.

The most remarquable is due to Young (1985).119

A rule is a marginalist if what it allocates to a player in a game depends only upon his/her marginal contributions:

Proposition 3.3 The Shapley value is the unique marginalist (Young, 1985) rule that satisfies EFF and SYM.

1 1 2 2

1 2

( ) ( \ ) ( ) ( \ )

implies

( , ) ( , )i i

v S v S i v S v S i S N

N v N v

120

Alternative definition

Proposition 3.4 The Shapley value allocates to each player a weighted sum of his/her marginal

contributions:

where the weights n(s) depend only on the size of the coalitions:

( , ) ( ) [ ( ) ( \ )]i nS N

SV N v s v S v S i

( 1)!( )!( )

!n

s n ss

n

121

Proof Let denote the coalition formed by player i and the players who preecede him in the permutation . In particular, if i is last and if i is first.

For a given coalition S containing i, there are (s – 1)!(n – s)! permutations such that

1( , ) [ ( ) ( \ )]

!

1( 1)!( )![ ( ) ( \ )]

!

n

i i i

S N

SV N v v S v S in

s n s v S v S in

iS

{ }iS i iS N

.iS S

122

The coefficients 's are at first sight surprising: they give more weight to small or large coalitions and less weight to coalitions of intermediary size.

But there are less coalitions of small or large size than coalitions of intermediary size …

The number de coalitions of a given size containing a given player is the number de combinations of n – 1 players in groups of size s – 1:

11

1

1

( )!( )

( )!( )!s

n n

ns C

n s s

!

!( )!kn

nC

k n k

123

Hence the sum of the n(s) is equal to 1 if we limit ourself to coalitions containing a given player:

the Shapley value is a weighted sum that is uniform with respect to the size of coalitions

1( )nS NS i

s

1Indeed we have: ( ) ( ) pour tout , 0 .n ns s s s n

n

124

2 2

3 3

4 4

5 5

6 6

1 12 ( , ) (1,1)

2 2

1 1 13 ( , , ) (1,2,1)

3 6 3

1 1 1 14 ( , , , ) (1,3,3,1)

4 12 12 4

1 1 1 1 15 ( , , , , ) (1,4,6,4,1)

5 20 30 20 5

1 1 1 1 1 16 ( , , , , , ) (1,5,10,10,5,1)

6 30 60 60 30 6

n

n

n

n

n

125

Auction game

1 2 3 (s)

1 0 0 0 1/3

2 0 0 0 1/3

3 0 0 0 1/3

12 200 200 0 1/6

13 300 0 300 1/6

23 0 0 0 1/6

123 300 0 100 1/3

1

2

3

1 1 1 1100200 300 300 183

6 6 3 61

200 3361 1 500

300 100 836 3 6

x

x

x

126

v(i) = 0v(12) = 200v(13) = 300v(23) = 0v(123) = 300

Crop game: constant returns

1 2 3 (s)

1 0 0 0 1/3

2 0 0 0 1/3

3 0 0 0 1/3

12 a a 0 1/6

13 a 0 a 1/6

23 0 0 0 1/6

123 2a a a 1/3

1

2

3

2 12

6 31 1

6 3 21 1

6 3 2

x a a a

ax a a

ax a a

v(i) = 0v(12) = av(13) = av(23) = 0v(123) = 2a

127

3.4 Properties of the value

128

By construction, the Shapley value satisfies the four axioms.

Individual rationality is not included in the axioms and the Shapley value may not be individually rational when applied to a game which is not superadditive.

The following proposition is an immediate consequence of super-additivity:

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

129

Proof By superadditivity, marginal contributions vectors are individually rational allocations and Milnor-reasonable and the Shapley value is an average of these vectors.

The Shapley value does not necessarily define a core allocation:

in the auction game, the Shapley value compensates the buyer with the smallest reservation price.

However for some classes of games, the Shapley value defines a core allocation. This is the case of convex games.

130

We know that the core of a convex game is a polyhedron whose vertices are the marginal contribution vectors.

The Shapley value is a convex combination of the marginal contribution vectors. The following proposition then follows:

Proposition 3.6 The Shapley value of a convex game defines a core allocation.

Remark: The n! vectors all appear in the computation of the value: if two vectors are identical, they are taken into account twice.

131

Example: bankruptcy game

n players (creditors)

E > 0 value to be distributed

d1,...,dn 0 the claims 1

n

ii

d E

( ) 0, ( \ )v S Max E d N S

( )v N E

v(S) is the maximum S can expect to receive if all other creditors have been compensated possibly partly

( ) [0, ( ) ]

where ( ) 0

v S Max d S K

K d N E

or

132

Case where n = 3, with E = 600 and d = (150, 300, 450):

v(1) = v(2) = 0

v(3) = v(12) = 150

v(13) = 300

v(23) = 450

v(123) = 600

The bankruptcy game is convex: its core is nonempty andcontains the Shapley value.

( ) 0, ( \ )v S Max E d N S

133

Convexity:

( ) ( ) ( ) ( )

( ( ) ( ) 2 ) ( ( ) ) ( )

( ( ) ) ( ) 0

si ( ) et ( )

(0 ( ) ) ( ( ) ) 0 ( \ ) 0

si ( ) et ( )

0 0 (0, ( ) ) 0 0

si ( ) et ( )

v S v T v S T v S T

d S d T K d S T K v S T

d S T K v S T

d S K d T K

d T K d S T K d S T

d S K d T K

Max d S T K

d S K d T K

( ) ( ) ( ) ( )d S d T d S T d S T

( ) [0, ( ) ]v S Max d S K

134

1 2 3

123 0 150 450

132 0 300 300

213 150 0 450

231 150 0 450

312 150 300 150

321 150 300 150

x 1/6 600 1050 1950

1

2

3

600100

61050

1756

1950325

6

x

x

x

losses = (50, 125, 125)

(150)

(300)

(450)

v(1) = v(2) = 0v(3) = v(12) = 150v(13) = 300 v(23) = 450v(123) = 600

Shapley value

135

1 2 3 (s)

1 0 0 0 1/3

2 0 0 0 1/3

3 0 0 150 1/3

12 150 150 0 1/6

13 150 0 300 1/6

23 0 300 450 1/6

123 150 300 450 1/3

1

2

3

300 150100

6 3450 300

1756 3

750 600325

6 3

x

x

x

v(1) = v(2) = 0v(3) = v(12) = 150v(13) = 300 v(23) = 450v(123) = 600

136

(600,0,0)

(0,600,0)

x3 = 150

x1 = 150

(0,0,600)

x2 = 300

(150,0,450)(150,300,150)

(0,300,300) (0,150,450)

x3 = 4500 x1 150

0 x2 300150 x3 450

v(1) = v(2) = 0v(3) = v(12) = 150v(13) = 300 v(23) = 450v(123) = 600

imputations' triangle

(100,175,325)

imputations and core

137

Observations:

- there are 4 distinct marginal contributions vectors:

(150,300,150), (150,0,450), (0,300,300), (0,150,450)

- they are the vertices of the core: it is a characteristic of convex games

- the Shapley value is somewhere in the center of the core, an other characteristic of convex games.

138

The Talmud

A man dies and his three wives have each a claim on his estate, following past promises. The value of the estate falls short of the total of the claims. Here is what a Mishnah suggests:

The solution is the nucleolus in all 3 cases.

d1=100 d2=200 d3=300

E=100 33.3 33.3 33.3

E=200 50 75 75

E=300 50 100 150

Shapley valueNucleolus(equal division)

Shapley value

Nucleolus

139

The Shapley value for the three Talmudic cases

In the case where E = 100, the game is symmetric: v(S) = 0 for all S N and the allocation is equal division, as suggested by the Talmud.

In the two other cases, we have the following outcomes:

E = 200 (33,3, 83,3, 83,3)

E = 300 (50, 100, 150)

140

They result from the following tables:

1 2 3

123 0 0 200

132 0 200 0

213 0 0 200

231 100 0 100

312 0 200 0

321 100 100 0

x 1/6 200 500 500

1 2 3

123 0 0 300

132 0 200 100

213 0 0 300

231 100 0 200

312 100 200 0

321 100 200 0

x 1/6 300 600 900

141

1 2 3

123 0 10 15

132 0 15 10

213 10 0 15

231 25 0 0

312 10 15 0

321 25 0 0

1/6 70 40 40

35 20 20( , , )

3 3 3x

Crop game: increasing returns ( = 10)

5 distincts marginal contributions vectors

v(i) = 0v(12) = 10v(13) = 10v(23) = 0v(123) = 25

142

(25,0,0)

(0,25,0) (0,0,25)

Core of the crop game with increasing returns

(10,0,15)(10,15,0)

(0,15,10) (0,10,15)

the core is the convexenvelope of the marginalcontribution vectors

35 20 20( , , )

3 3 3a convex game

143

1 2 3

123 0 10 5

132 0 5 10

213 10 0 5

231 15 0 0

312 10 5 0

321 15 0 0

1/6 50 20 2025 10 10

( , , )3 3 3

x

Crop game: decreasing returns ( = 10)

v(i) = 0v(12) = 10v(13) = 10v(23) = 0v(123) = 15

5 distincts marginal contributions vectors

144

(15,0,0)

(0,15,0) (0,0,15)

Core of the crop game with decreasing returns

(5,5,5)

(10,0,5)(10,5,0)

(0,10,5) (0,5,10)

25 10 10( , , )

3 3 3

2 320

3x x

145

(15,0,0)

(0,15,0) (0,0,15)

(10,0,5)(10,5,0)

(0,10,5) (0,5,10)

The core is a subset of the convex envelope of the marginal contribution vectors called the "Weber set"

146

1 2 3

123 0 10 2

132 0 2 10

213 10 0 2

231 12 0 0

312 10 2 0

321 12 0 0

1/6 44 14 1422 7 7

( , , ) noyau3 3 3

x

Case where the Shapley value does not belong to the core

v(i) = 0v(12) = 10v(13) = 10v(23) = 0v(123) = 12

147

(12,0,0)

(0,12,0) (0,0,12)

(2,2,10)

(10,0,2)(10,2,0)

(0,10,2) (0,2,10)

22 7 7( , , )

3 3 3

2 320

3x x

148

Crop game: general case

Workers are substitutes: they get the same wage and we need only to compute what the value allocates to the landlord. In a given permutation, only the position of the landlord counts.

if the landlord is in position k, he gets F(k-1)

and there are m + 1 positions possibles

1

11 1

1 11

1 1( , ) ( ) ( )

m m

k k

SV N v F k F km m

149

m

F(m)

0

F(1)

F(2)

k k+1

F(k) 1 x F(k)F(1) + F(2) + … + F(m)

150

m

F(m)

0

L

W

L > W

decreasingreturns

0

( )m

k

F k

151

m

F(m)

0

L

W

L = W

constantreturns

152

m

F(m)

0

L

T

L < W

increasingreturns

W

153

m

F(m)

0

L

mixedreturns

W

154

Crop game: case of an excess labor supply (k < m)

0 k m x

y( ) if 0

( ) if

y F x x k

F k x k

v(S) = 0 if 1 S

v(S) = F(s – 1) if 1 S and s k + 1

v(S) = F(k) if 1 S and s > k + 1

F(k)

155

The core reduces to the allocation which gives all to the landlord: C(N,v) = {(F(k), 0,…, 0)}. Indeed we have seen that the maximum a worker can hope to get is equal to the marginal product that is zero here!

Alternatively:

( ) ( )

( \ ) ( \ ) ( )

( ) ( ) 0 0j j

x N F k

x N j v N j F k

x F k F k x

156

The Shapley value allocates a rent

to the landlord. What is left is uniformly distributed to the workers. If F(k) = k, we have:

There is equality between the rent r and the wage bill mw iff k = m r = mw = k/2.

1

1( ) ( ) ( )

1

k

s

r F s m k F km

1

1 ( 1) 2 1( ) ( )

1 1 2 1 2

k

s

k k k m kr s m k k m k k

m m m

157

The wage of an individual worker is given by:

2 1 1 1

1 2 2 1

k m k k kw k

m m m

0 k m x

y

L

W

k

158

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

1

1 1

( ) 0 if 0

3( ) if

f x x

x x

10 x

y

1 2 3input endowments: 1, 2, 3 ( 3)n

(1) (1) 0 (12) (3) 6

(2) (2) 3 (13) (4) 9

(3) (3) 6 (23) (5) 12 (123) (6) 15

v f v f

v f v f

v f v f v f

159

x1

x2 x3

0

15

15 15

36

(0,9,6) (0,3,12)

(6,3,6)

imputations

x1 0x2 3x3 6x1 + x2 + x3 = 15

160

(0,15,0) (0,0,15)

(15,0,0)

x3 = 9x2 = 6

x1 = 3 (3,6,6) (3,3,9)

(0,6,9)

0 x1 3

3 x2 66 x3 9

x3 = 6x2 = 3v(1) = 0

v(2) = 3v(3) = v(12) = 6v(13) = 9 v(23) = 12v(123) = 15

x1 + x2 = 6

x1 + x2 = 9

x1 + x3 = 12

x1 + x3 = 9

x2 + x3 = 12

core

161

1 2 3

123 0 6 9

132 0 6 9

213 3 3 9

231 3 3 9

312 3 6 6

321 3 6 6

1/6 12 30 48 (2,5,8)x

v(1) = 0v(2) = 3v(3) = v(12) = 6v(13) = 9 v(23) = 12v(123) = 15

Shapley value

The core is the convex hull of the marginal contribution vectors the game is convex.

162

Modified game:

This is a symmetric game and the Shapley value is therefore the egalitarian allocation:

(N,u) = (2,2,2)

(1) 0 (12) 3

(2) 0 (13) 3

(3) 0 (23) 3 (123) 6

u u

u u

u u u

( ) ( ) ( )i S

u S v S v i

163

(6,0,0)

(0,6,0) (0,0,6)

x2 + x3 = 3

x1 + x2 = 3x1 + x3 = 3

(3,0,3)(3,3,0)

(0,3,3)

core of the modified game

164

1

1

If ( ,..., ) is an imputation of the game ( , ), the imputation ( ,..., ) corresponding to the game ( , )is given by:

( )

n

n

i i

y y N ux x N v

x y v i

We obtain in this way the Shapley value of the original game:

(N,v) = (2,2,2) + (0,3,6) = (2,5,8)

which is at the center of the core.

165

1 2 3 4

13 1 0 1 014 1 0 0 123 0 1 1 024 0 1 0 1

123 0 0 1 0134 1 0 0 0234 0 1 0 0124 1 1 0 2

1234 1 1 0 1

Gloves market: (R,R,L,LL)

4

1 1 1 1( , , , )4 12 12 4

7 7 3 7( , ) ( , , , )

12 12 12 12N v

( , ) ( , )N v C N v

( , ) {(1,1,0,0)}C N v

166

1 2 3

12 1 1 013 2 0 2

123 2 0 1

Gloves market: (RR,L,LL)

3

1 1 1( , , )3 6 3

7 4( , ) ( , , )

6 6

1

6N v

( , ) ( , )N v C N v

0( , ) {(1 , ,1 ) | 0 1}C N v z z z

167

3.5 Cartels

168

The formation of cartels or syndicates, viewed as agreements between players who decide to act as a single player can be analyzed using the Shapley value .

In the porters' game, we immediately see that any two players have an interest to form a cartel cartel, but this does not tell you which cartel will form !

If players 1 et 2 get together, the game reduces to a 2-player game where player 3 is null:

v*(12) = 6 et v*(3) = 0

v*(12,3) = 6

169

In the auction game, the buyers have an interest to form a cartel. The game becomes a 2-player game:

v*(i) = 0 pour i = 1 et 23v*(1,23) = 300

which calls naturally for an equal division between the seller and the cartel of buyers.

In the case where the buyers have the same reservation price, we have:

the buyers get 150 when in a cartel while the Shapley value allocates them 100

170

In the crop game, if workers form a syndicate, the game becomes a 2-player symmetric game whose solution is:

to be compared to the allocation derived from the Shapley value:

1 23(2)

2

Fx x

1

2 3

(1) (2)

32 (2) (1)

3

F Fx

F Fx x

171

The difference is given by:

It is positive if and only if returns to scale are decreasing:

F(2) < 2F(1).

workers have an incentive to form a syndicate if returns to scale are decreasing.

(2) 2 (2) (1) 1 (2)(1)

2 3 3 2

F F F FF

172

Chapter 4

Cost sharing

173

4.1 Cost games

174

A set N = {1,…,n} of players have a common project that costs c(N).

The (minimum) cost c(S) of realizing this project to the benefit of the members of any coalition S N is also known.

This defines a cost function c: S N c(S) with c() = 0.

It is a characteristic function and the couple (N,c) defines a cooperative game with transferable utility, here interpreted as a cost sharing game.

175

Example: water distribution

Three towns consider building of a water distribution system linked to a source. The costs of connecting each town to the source and to the other towns are given by:

Town 1 Town 2 Town 3

Source 18 21 27Town 1 15 12Town 2 24

S 2

3

18

21

27

15

24

12

1

176

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

Looking at coalitions of towns, the (minimum) costs are given by:

c(1) = 18 c(12) = 33

c(2) = 21 c(13) = 30

c(3) = 27 c(23) = 45

Together with c(123) = 45, this defines the cost game (N,c).

S

1

2

3

18

21

27

15

24

12

178

Subadditivity:

Cost functions defining cost games are assumed to be subadditive :

Subadditivity is typically verified by cost function derived from problems involving networks: the water distribution example is subadditive.

( ) ( ) ( )S T c S T c S c T

179

Subadditivity implies that there is no loss in cooperating:

A cost game is essential if

The water distribution example is essential.

The cost function of the water distribution example is actually concave.

180

( ) ( )i N

c i c N

( ) ( )i S

c i c N

A cost game is concave if

Concavity implies subadditivity.

Equivalently, a cost game is concave if the marginal costs are non-increasing:

( ) ( ) ( ) ( ) for all ,c S T c S c T c S T S T N

for all , , such that :

( ) ( \ ) ( ) ( \ )

i S T i S T

c S c S i c T c T i

181

In a given cost sharing situation (N,c), the surplus generated by cooperation is measured by the difference

The natural surplus sharing game (N,v) associated to a cost game (N,c), is then defined by:

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

( ) ( )i N

c i c N

182

( ) ( ) ( )i S

v S c i c S

Subadditivity of the cost function implies superadditivity of the surplus function:

The surplus game (N,v) associated to an essential cost game (N,c) is essential. Indeed v(i) = 0 for all i and

Concavity of the cost function implies convexity of the surplus function:

183

( ) ( ) ( ) ( ) ( ) ( )S T v S T v S v T c S c T c S T

( ) ( ) ( ) 0i N

c N c i v N

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )v S T v S v T v S T c S c T c S T c S T

4.2 Imputations and the core

184

Question: how to share c(N) between the players?

A first approach consists in imposing conditions on cost allocations y = (y1, y2,..., yn):

individual rationality(incentive compatibility)

collective rationality(Pareto efficiency)

core (social stability)

( ) ( )y N c N

( ) for alliy c i i N

( ) ( ) for ally S c S S N

185

Proposition 4.2 Core conditions y satisfy the following inequalities:

Proof The left hand inequalities is part of core's definition. sing core's inequalities, we get:

Hence no coalition pays less than its "additionnal cost": there are no cross-subsidies.

( \ ) ( \ )

( ) ( )

y N S c N S

y N c N

( ) ( ) ( \ )y S c N c N S

186

( ) ( \ ) ( ) ( ) for allc N c N S y S c S S N

In particular, core allocations y satisfy the inequalities:

i.e. players pay at least their marginal cost to the grand coalition.

Surplus allocations x and cost allocations y are related by the following equations:

( ) ( \ ) ( )ic N c N i y c i i N

187

( )i ix y c i ( )i iy c i x

( )i ix c i y

In terms of surplus sharing, we have successively:

( ) ( ) ( ) ( ) ( ) ( )ii N i N

y N c N c i x c N x N v N

( ) ( ) ( ) 0 ( )i i iy c i c i x c i x v i

( ) ( ) ( ) ( ) ( ) ( )ii S i S

y S c S c i x c S x S v S

( )i ix y c i

188

(0,45,0)

(45,0,0)

y3 = 12

y2 = 15

(0,0,45)

y3 = 27y2 = 21

y1 = 18

y1 18 y2 21 y3 27y1 + y2 33 y1 + y3 30y2 + y3 45y1 + y2 + y3 = 45

0 y1 18 15 y2 2112 y3 27y1 + y2 + y3 = 45

Water distribution

(-3,21,27) 189

the yellow is the imputation set

4.3 The Shapley value

190

We need to define a sharing rule that associates to each cost game an allocation y =  (N,c)

There exist various such rules. The most "popular" are the Shapley value and the nucleolus. They can be obtained equivalently from the cost game or indirectly from the surplus game.

The nucleolus defines a core allocation, if the core is nonempty, while the cost allocation derived from the Shapley value may not be stable.

191

To compute the Shapley value, we construct the vectors of marginal costs that, to each permutation  = (i1,...,in)  n associates the imputation t() defined by:

The Shapley value of a cost game (N,c) is defined as the average marginal cost vector:

1 1 1

1 1 1 2 1)

( ) ( ) ( ) ( )

( ) ( ,..., ) ( ,..., ) ( ,...,

( ) ( ) ( \ )k

n

i

i k k

i n

t c i c c i

t c i i c i i k n

t c N c N i

11

( , ) ( ) ,...,!

n

i iSV N c t i nn

192

Given the surplus game (N,v) associated to the cost game (N,c), the marginal contribution vector () and the marginal cost vector t() associated to permutation   n are related by the equations

Hence, the value of the cost game (N,c) can be obtained from the value of the surplus game (N,v):

1( , ) ( ) ( , ) ,...,i iSV N c c i SV N v i n

193

1( ) ( ) ( ) ,...,i it c i i n

Water distribution: cost game

c(1) = 18 c(12) = 33c(2) = 21 c(13) = 30c(3) = 27 c(23) = 45c(123) = 45

1 2 3

123 18 15 12

132 18 15 12

213 12 21 12

231 0 21 24

312 3 15 27

321 0 18 27

1/6 51 105 114

( , ) (8.5,17.5,19)SV N c

194

Water distribution: surplus game

v(1) = 0 v(12) = 6v(2) = 0 v(13) = 15v(3) = 0 v(23) = 3v(123) = 21

1 2 3

123 0 6 15

132 0 6 15

213 6 0 15

231 18 0 3

312 15 6 0

321 18 3 0

1/6 57 21 48

( , ) (9.5, 3.5, 8)SV N v

195

( , ) ( , ) (9.5, 3.5, 8) (8.5,17.5,19) (18, 21, 27)SV N v SV N C

1 2 3

123 20 15 15

132 20 5 25

213 10 25 15

231 5 25 20

312 15 5 30

321 5 15 30

c(1) = 20 c(12) = 35c(2) = 25 c(13) = 45c(3) = 30 c(23) = 45c(123) = 50

5 ≤ y1 ≤ 20 5 ≤ y2 ≤ 2515 ≤ y3 ≤ 30

x1/6 75 90 135

12.5 15 22.5

(20,15,15)

(10,25,15)

(5,25,20) (5,15,30)

(15,5,30)

(20,5,25)

y3 = 15

y3 = 30

(50,0,0)

(0,50,0) (0,0,50)

y2 = 5

y2 = 25

y1 = 20

y1 = 5

(12.5,15,22.5)

6 distinct vectors

196

4.4 Airport games

197

An airport game is defined by individual costs ci 0 and the associated cost function is given by:

Airport games are monotone increasing and concave (hence subadditive). They are essential if costs are not all equal.

Without loss of generality, players are ordered in terms of their costs:

( ) i S ic S Max c

198

1 20 ... ( )n nc c c c N c

Concavity: let S et T be two coalitions with values:

where cS cT (without loss of generality).

( ) et ( )S Tc S c c T c

( ) ( ) ( ) ( ) ( ) 0Tc S c T c S T c S T c c S T

( ) Max ( , )

( ) Min ( , )T S S

T S T

c S T c c c

c S T c c c

199

The natural cost allocation is the following:

11

1 2 12

11 2 1

1 21 2 11

1

1 1

1 2

...

...

...

...

k kk

n nn n n

cy

nc c c

yn n

c cc c cy

n n n k

c cc c cy c c

n n

it is the Shapley value !

200

In the case where n = 3:

1

2

3

(1)

(2) (12)

(3) (13) (23) (123)

c c

c c c

c c c c c

11

1 2 1 2 12

1 2 1 2 13 3 2 3

3

3 2 2 6

3 2 2 6

cy

c c c c cy

c c c c cy c c c

201

1 2 3

123 c1 c2 – c1 c3 – c2

132 c1 0 c3 – c1

213 0 c2 c3 – c2

231 0 c2 c3 – c2

312 0 0 c3

321 0 0 c3

x 1/6 2c1 3c2 – c1 6c3 – 3c2 – c1

1 2 1 2 13( , ) ( , , )

3 2 6 2 6

c c c c cSV N c c

4 distinct vectors

202

y1 c1 y2 c2 y3 c3

y1 + y2 c2 y1 + y3 c3

y2 + y3 c3

y1 + y2 + y3 = c3

(c3,0,0)

(0,c3,0) (0,0,c3)

0 y1 c1 0 y2 c2 c3 – c2 y3 c3

y1 + y2 + y3 = c3

y2 = c2y3 = c3 – c2

(0,c2,c3-c2)

(c1,c2-c1,c3-c2) (c1,0,c3-c1)y1 = c1

203

The associated surplus game is given by:

( ) Maxi i S ii S

v S c c

1

2

1 2

(1) (2) (3) 0

(12) (13)

(23)

(123)

v v v

v v c

v c

v c c

Remark The last two players are substitutable, a property that is independent of the number of players.

204

1 2 3

123 0 c1 c2

132 0 c2 c1

213 c1 0 c2

231 c1 0 c2

312 c1 c2 0

321 c1 c2 0

x 1/6 4c1 c1 + 3c2 c1 + 3c2

1 1 2 1 22( , ) ( , , )

3 6 2 6 2

c c c c cSV N v 4 distinct vectors

205

1 2 3( , ) ( , ) ( , , )SV N v SV N C c c c

(c1 + c2,0,0)

x1 = c1

x2 = c2x3 = c2

x1 0 x2 0x3 0x1 + x2 c1 x1 + x3 c1

x2 + x3 c2

x1 + x2 + x3 = c1 + c2

0 x1 c1 0 x2 c2 0 x3 c2

y1 + y2 + y3 = c3

(0,c1 + c2,0) (0,0,c1 + c2)

(c1,c2,0) (c1,0,c2)

(0,c2,c1) (0,c1,c2)

206

The Shapley value of the airport game (N,c) can be written compactly as:

SV(N,c) = B.c

where B is the n x n triangular matrix defined by:

207

1if

( 1)( )

1for all

1

ij

ii

b j in j n j

b in i

The matrix B has a simple recursive structure. For n = 4, it is given by:

The matrices are overlapping, starting from the lower right element 1. For instance, if n = 5 the first column starts with 1/5, followed by –1/20...

208

1/ 4 0 0 0

1/12 1/ 3 0 0

1/12 1/ 6 1/ 2 0

1/12 1/ 6 1/ 2 1

4.5 Data games

209

Initial motivation: REACH program (Registration, Evaluation, Authorisation and Restriction of Chemicals) imposed on EU Chemical Industry:

about 30.000 substances and

100 parameters per substance

Submission process: started in 2008, it will extend until 2018.

Cooperation between firms is encouraged both in terms of data acquisition and data sharing: there is an exchange forum for each substance (SIEF) whose role is to facilitate the exchange of data among firms.

210

Looking at a particular parameter, for a given substance, there are three cases:

1. that data is freely accessible2. no firm has that data: it has to be produced3. that data is held by some but not all firms

In case 1, the data is freely accessible and there is no property rights: no need for compensations

In case 2, no firm has the data: it must be acquired jointly at a cost that has to be shared among the firms

equal splitting is the natural rule in this case.

211

We shall consider only case 3 that opens the possibility of compensations among firms.

Consider first the simplest case involving 2 firms where only one holds the data (say firm 1).

The data has a value that corresponds to the cost of duplicating it, say d > 0.

What would be a fair compensation: how much should firm 2 be asked to pay to firm 1 within a joint submission ?

212

If that data was held by no player, they should each pay d/2.

So firm 2 could be asked to pay d/2 to firm 1. Equivalently, both firm pay d/2 and firm 1 gets d back.

With 3 firms, only firm 1 holding the data, the same argument suggests that the other firms should each pay d/3 to firm 1.

In the situation where only firm 3 does not hold the data, firm 3 should still pay d/3, an amount divided equally between firm 1 and 2: each receives d/6.

213

In case of n firms, the data being held by t firms, 0 < t ≤ n, the n – t firms without the data each pay d/n each.

The resulting allocation is then given by:

0 if owns the data

if does not

i

i

d dy i

n t

dy i

n

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

214

A data sharing situation is described by:

N = {1,...,n} the set of players

M0 = {1,…,m} the set of available data

dh > 0 the cost of reproducing data h

Mi  M0 the dataset held by player i

01with

n

iiM M

215

The cost for a coalition S if the cost of completing its dataset:

where

This defines a cost game called data game.

0

0\

( )S S

h hh M M h M

C S d v d

0

0

is the data set held by coalition

is the value of the complete dataset

S ii S

hh M

M M S

v d

( ) 0C N

216

Example: N = {1,2,3} and M0 = {1,2,3,4}

M1 = {1,3} d1 = 90 v0 = 210

M2 = {1,2} d2 = 20

M3 = {3,4} d3 = 40

d4 = 60

c(1) = d2 + d4 = 80 c(1,2) = d4 = 60

c(2) = d3 + d4 = 100 c(1,3) = d2 = 20

c(3) = d1 + d2 = 110 c(2,3) = c(1,2,3) = 0

217

Proposition 4.4 The cost function c defining a data game (N,c) is:

decreasing: S T c(T) c(S)

subadditive: S T = c(S) + c(T) c(ST)

essential if Mi M0 for some i: i c(i) > c(N)

218

Proof

essential:

subadditive: if S  T = 

non-increasing: S  T MS  MT  and

0 01 1

for some ( ) 0 ( )i

n n

i hi i h M

M M i c i nv d c N

0

0

( ) ( ) 2

( ) ( )S T

S T

h hh M h M

hh M M

c S c T v d d

c S T v d c S T

219

( ) ( ) 0S T

h hh M h Mc T c S d d

Case of a single data

The "elementary" data game (N,ch) associated to data h:

where Th denotes the set of players holding data h.

( ) if

0 if

h h h S

h S

c S d S T h M

S T h M

220

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

\

( ) ( )S

h hh M h M M

c S d c S

( ) if

0 if

h h S

S

c S d h M

h M

221

Because c(N) = 0, we are facing a pure compensation problem.

We consider imputations:

where yi > 0 means that i pays yi

yi < 0 means that i receives – yi

1

( , ) | 0 and ( ) for alln

ni i

i

I N c y y y c i i N

222

Proposition 4.5 The core of a data game is then by:

where

is the value of the data player i is alone to hold and is the set of data held by single players

the core of a data game depends only on the data held by single players

and | for somei

i h i i jh M

v d M h M h M j i

( , ) { | 0 and for all }ni i i

i N

C N c y y y v i

( \ )iv c N i

223

iM

the core is nonempty: it always contains the nocompensation allocation 0 = (0,0,…0)

the maximum amount a player can expect to receive is the value of the data he/she is alone to hold

no player can expect to be compensated for data he/she is not alone to hold

if all data are each owned by at least 2 players, the core reduces to {0}

( , ) { | 0 and for all }ni i i

i N

C N C y y y v i N

224

Proof If y C(N,c) we have

If instead y satisfies core's definition, we have:

Because y(N) = 0, we then have:

( \ ) ( \ ) i i iy N i c N i v y v

225

\

( \ ) ( ) for allii N S

y N S v C S S

0\ \

( ) ( ) for allS

i hi N S h M M

y S v d C S S

226

The core of a data game

is a regular simplex:

adding the vector to core allocation and dividing by result in the standard unit simplex

1( ,..., )nv v

0 iv v

( , ) { | ( ) 0 and for all }ni iC N C y y N y v i N

{ | 1 and 0 for all }nn i i

i N

y y y i N

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

10 1 2

21 0 2

1 2 0

( , ,..., )

( , ,..., )

...

( , ,..., )

n

n

nn

v v v v

v v v v

v v v v

for someiM i

0 1 2 3( , , )v v v v

1 0 2 3( , , )v v v v 1 2 0 3( , , )v v v v

3 3y v2 2y v

1 1y v

Core of a 3-player data game

2y

1y

3y

228

Looking at the core of the elementary game (N,Ch) associated to data h:

either it reduces to {0} because data h is held by more than one player

or it is a full dimensional regular simplex because only one player holds data h

229

(0,0,0)

( , ,0)h hd d ( , 0, )h hd d

3 0y 2 0y

1 hy d

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

1

2 3 0

hv d

v v

230

The Shapley value of an elementary game

ch (S) = 0 si S Th

ch (S) = dh si S Th =

where th is the number of players in Th.

1( , ) ( )[ ( ) ( )] hh i h n h h

di T N c c i c

n

1( , ) h h h

h i h hh h

n t d di T N c d

t n n t

231

By additivity

- the cost of the complete dataset is uniformly allocated among all players

- the cost of each data is uniformly redistributed to the players holding it.

0

( , ) ( , )i i hh M

N c N c

0 1( , ) ,...,i

hi

h M h

v dN c i n

n t

0

0 hh M

v d

232

Example d = (90, 20, 40, 60)

M1 = {1,3}, M2 = {1,2}, M3 = {3,4}

Here y belongs to the core.

0210

703

v

n

0( , )i

hi

h M h

v dN c

n t

1

2

3

90 4070 5

2 290

70 20 52

4070 60 10

2

y

y

y

233

Axiomatization of the value on the set of data games

The Shapley value is uniquely determined by 4 axioms: efficiency, symmetry, null player and additivity.

There are no null players in data games.

Keeping efficiency, symmetry and additivity, one possible additional axiom could be:

for all data sharing situations (M,d) = (M1,...,Mn,d1,...,dn) such that Mi = or M0:

234

0( , )i i

vM M d

n

4.6 Assignment games

235

A set N = {1,…,n} of agents and a set M = {1,…,m} of indivisible objects (say houses) to be allocated, one for each agent (m n).

Data: a "utility" matrix U = [ui(h) | i N, h M]

Here ui(h) is the reservation price of agent i for house h (i.e. the maximum price i is willing to pay for house h).

It is expressed in monetary terms. It is the value that agent i attach to house h.

236

Side payments being allowed, the associated TU-game is given by:

where F is the set of all functions f: N M that associate a house to each player.

c(S) is the cost of the houses that are optimally allocated to its members. (N,c) is the assignment game studied by Shapley and Shubik (1972).

It is a concave (hence subadditive) cost game.

( ) ( ( ))f F ii Sc S Max u f i

237

Example

u1 u2 u3

a 3 9 9

b 12 6 6

c 9 6 3

c(1) = 12c(2) = 9c(3) = 9c(12) = 21c(13) = 21c(23) = 15c(123) = 27

1 2 3

123 12 9 6

132 12 6 9

213 12 9 6

231 12 9 6

312 12 6 9

321 12 6 9

1/6

72 45 45

SV(N,c) = (12, 7.5, 7.5)optimal allocation = (12,6,9):1 receives house b2 receives house c3 receives house a

transfers: (12,6,9) – (12,7.5,7.5) = (0, –1.5, 1.5)

players 2 and 3 are substitutes

238

(0,27,0) (0,0,27)

(27,0,0)

x3 = 9

x2 = 6

x1 = 12

(12,6,9)(12,9,6)

x1 = 12

6 x2 96 x3 9 x3 = 6

x2 = 9v(1) = 12v(2) = 9v(3) = 9v(12) = 21v(13) = 21 v(23) = 15v(123) = 27

core

(12,7.5,7.5)

set ofimputations

(9,9,9)

(12,6,9)(12,9,6)

optimal allocation before transfers

239

Chapter 5

Power in decision games

240

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

5.1 Decision games

242

A decision games is defined by:

players = decision makers i N = {1,...,n}

rules = a collection W of winning coalitions

A paire (N,W) defines a decision game.

There are no restrictions; all coalitions are a priori possible.

A coalition minimal winning if removing any of its members makes it loosing:

243

if and only if and \S M S W S i W

Simple 3-player decision games:

unanimity: W = { {1,2,3} }

simple majority: W = { {1,2}, {1,3}, {2,3}, {1,2,3} }

simple majority + veto: W = { {1,2}, {1,3}, {1,2,3} }

dictatorship: W = { {1}, {1,2}, {1,3}, {1,2,3} }

244

Minimals winning coalitions:

unanimity: M = {1,2,3}

simple majority: M = {1,2}, {1,3}, {2,3}

simple majority + veto: M = {1,2}, {1,3}

dictatorship: M = {1}

245

Assumptions

D1 the grand coalition is always winning: N W

D2 two disjoint coalitions cannot be simultaneously winning:

Consequence: if a coalition is winning, its complement is necessarily loosing:

246

andS W S T T W

\S W N S W

D3 enlarging a winning coalition keeps it winning :

the set of all winning coalitions can be obtained from the set of minimal winning coalitions by adding players

247

andS W S T T W

5.2 Weighted majority games

248

Weighted majority games form a particular class of decision games.

Decision maker i is characterized by a weight wi 0 and a coalition is winning if and only if its weight is not below some given quota Q:

where the quota and the weights satisfy the inequalities:

In this way, assumptions D1 et D2 are verified.

iff ( ) : ii S

S W w S w Q

1( ) ( )

2w N Q w N

249

Assumption D3 is automatically satisfied: adding a player to a winning coalition does not decrease its weight.

Apparent power of players are given by their relative weights:

By construction:

1

where ( )( )

ni

i jj

ww N w

w N

1

0 1 for all

1

i

n

ii

i

250

EU Council of Ministers

Distribution of votes (EU-6 à EU-15)

Quota = minimum number of votes (environ 70%):

12/17 in 1958 54/76 in 1986 62/87 in 1994

Fr De It Be Nl Lu UK Dk Irl Gr Sp Pt Se Fi Au

1958 4 4 4 2 2 1

1986 10 10 10 5 5 2 10 3 3 5 8 5

1994 10 10 10 5 5 2 10 3 3 5 8 5 4 3 4

251

A voting game is said to be weighted if it is equivalent to a weighted majority game coalitions winnings:

starting from a voting game (N,W), there exist weights w1,…,wn and quota Q such that:

i.e. reproduces the same winning coalitions.

iff ( )S W w S Q

252

In situations involving three decision makers:

- unanimity and simple majority (qualified majority too) are by definition of weighted majority games where each decision makers have the same weight: wi = 1 for all i and

for unanimity: Q = n

for simple majority: 1 if is even2

1if is odd

2

nQ n

nQ n

253

- the weighted majority game defined by Q = 1, w1 = 1 and w2 = w3 = 0 is equivalent to the situation where 1 is a dictator.

- the situation where decision maker 1 has a veto right is also equivalent to a weighted majority game:

- assign weight 1 to decision makers 2 et 3

- assign weight x to decision maker 1

- choose a quota Q

such that {1,2}, {1,3} et {1,2,3} are winning.

254

The largest loosing coalition is {2,3} with a weight equal to 2.

The smallest winning coalition est {1,2} with a weight equal to x + 1.

The following inequalities must be verified:

2 < Q x + 1

It works for x = 2 and Q = 3.

255

The UN Security Council has 15 members: 5 permanent with veto right and 10 non-permanent The quota is 9. It is weighted:

- assign weight 1 to non-permanent members

- assign weight x to permanent members

- choose a quota Q

so as to reproduce the same winning coalitions.

The largest loosing coalition is of the form {4p, 10np} with a weight equal to 4x + 10.

256

The smallest winning coalition is of the form {5p, 4np} with a weight equal to 5x + 4.

The following inequalities must be verified:

4x + 10 < Q 5x + 4

It works for x = 7 and Q = 39.

257

Example: Apex game (partial veto)

This is a 5-player decision game defined by;

S W if and only if either 1 S and |S| 2

or |S| 4

This is a weighted decision game:

Q = 4, w1 = 3 and wi = 1 (i = 2,...,5)

258

Assign weight 1 to the players 2 to 5 and weight x au player 1.

the largest loosing coalition is of the type {i,j,k} where i,j,k {2,3,4,5} and its weight is equal to 3

the smallest winning coalition is of the form {1,i } where i {2,3,4,5} and its weight is equal to x + 1

The following inequalities must be verified:

3 < Q x + 1

It works for x = 3 and Q = 4.

259

5.3 Decision games as simple games

260

A decision game can be written as a simple game (N,v) with:

Assumptions D1, D2 et D3 imply that this simple game is super-additive and monotone.

It is essential if and only if there is no dictator. Indeed, v(i) = 0 for all i in the absence of a dictator and v(N) = 1.

261

( ) 1 v S if and only if S W

Concepts introduced to solve TU-games can therefore be applied to these games, in particular the Shapley value.

Remember that the core of a simple game is nonempty if and only if there are veto players (Proposition 2.5).

Transposing the question of allocating v(N) between players, the question is to measure how decision power is distributed:

where xi is interpreted as a (relative) measure of power of decision maker i or as his/her share in the"cake" resulting from the decision taken.

262

11

( ,..., ) such that ( ) 1 ( 100%)n

n ii

x x x v N

We look for a measure of the power of each decision maker that results from the decision rules, assuming that all coalitions can form and without taking into account the nature of the proposition put to vote nor the preferences of the decision makers.

On can measure the power of a political party in a parliamentary system by identifying parties to single decision makers (assuming party discipline!):

each party has a weight equal to its number of seats.

263

In simple or qualified majority (that includes unanimity), decision makers are equal and they have therefore the same power.

Within a coalition, the power of a decision maker is linked to his/her capacity to make loosing coalitions winning by joining them.

A decision maker is decisive (or key) in a winning coalition if that coalition is loosing without him/her:

264

and \S W i S S i W

A coalition is minimal winning if all its members are decisive.

A decision maker has a veto right if he/she is decisive in all winning coalitions.

A dictator is decisive in all coalitions.

Two decision makers are substitutable if they are decisive in the same coalitions:

for all such that , : \ \S W i j S S i W S j W

265

Alternatively:

In unanimity and in simple/qualified majority games, decision makers are all substitutable.

A decision maker who is never decisive is null:

A null decision maker has no power. It was the case of Luxembourg in EU-6.

or[ \ and \ ] [ \ and \ ]S i W S j W S i W S j W

\ for all S i W S W

266

In a weighted majority game, two decision makers with identical weights are substitutable...

though two decision makers may be substitutable while having different weights !

In the game defined by w = (10, 20, 30, 40) and Q = 51 (simple majority), we have:

W = {24, 34, 123, 124, 134, 234, 1234}

2 and 3 are substitutable: they are decisive in coalition {123} and they are not decisive in coalitions {234} and {1234}.

267

5.4 Power indices

268

We need a method – a rule – to compute power in any given voting game: a power index.

Banzhaf (1965)

... proposed to simply compute the number of coalitions in which decision makers are decisive

Banzhaf power index (BI)

Banzhaf (normalized) power index (NBI)

269

Shapley et Shubik (1954)

… proposed to apply the Shapley value to the associated simple game:

Shapley-Shubik index (SSI)

( ) 1 if is winning

( ) 0if is loosing

v S S

v S S

270

Marginal contributions are equal to 0 or 1:

v(S) – v(S\i) = 0 if S is loosing

v(S) – v(S\i) = 0 if S and S\i are winnings

v(S) – v(S\i) = 1 if S is winning and S\i is loosing

a decision maker is decisive in a coalition if and only if

his/her marginal contribution to that coalition is equal to 1.

271

The number of coalitions in which decision maker i is decisive is given by the sum of his/her marginal contributions:

This is the "raw" Banzhaf index from which two indices can be defined.

( , ) [ ( ) ( \ )]iS N

N v v S v S i

272

The Banzhaf index

and the "normalized" Banzhaf index

where

1 1

1 1( , ) [ ( ) ( \ )] ( , )

2 2i in nS N

BI N v v S v S i N v

1

( , ) ( , )n

ii

N v N v

1 ( , )( , ) [ ( ) ( \ )]

( , ) ( , )i

iS N

N vNBI N v v S v S i

N v N v

273

1 2 3

1 0 0 0 1/3

2 0 0 0 1/3

3 0 0 0 1/3

12 1 1 0 1/6

13 1 0 1 1/6

23 0 0 0 1/6

123 1 0 0 1/3

3 1 1

SSI1 = 2/6 + 1/3 = 2/3SSI2 = SSI3 = 1/6

simple majority with veto

BNI1 = 3/5BNI2 = BNI3 = 1/5

Decision maker 1 is decisive in 3 coalitions while the other two are decisive in only one coalition.

274

BI1 = 3/4BI2 = BI3 = 1/4

1 2 3

123 0 1 0

132 0 0 1

213 1 0 0

231 1 0 0

312 1 0 0

321 1 0 0

1/6 4 1 12 1 1

( , , )3 6 6

SSI

Alternative computation of the Shapley-Shubik index:

275

Apex game (n = 5)

1 S and |S| 2 S W iff or

|S| 4

Coalitions in which 1 is decisive:

12,13,14,15 4 5(2) = 1/5123,124,125,134,135,145 6 5(3) = 1/51234,1235,1245,1345 4 5(4) = 1/5

SSI1 = 3/5 SSIi = 1/10 (i = 2,...,5)

5 = (1/5,1/20,1/30,1/20,1/5)

SSI = (60,10,10,10,10) en %

276

Decision maker 1 is decisive in 14 coalitions and the other decision makers (substitutes) are decisives in 2 coalitions.

For example, decision maker 2 is decisive in 2 coalitions: 12 et 2345.

NBI1 = 14/22 and NBIi = 2/22 (i = 2,...,5)

NBI = (64, 9, 9, 9, 9) %

to be compared to SSI = (60, 10, 10, 10, 10) %

277

Comparing the two normalized indices:

we observe that they differ in the way they weight marginal contributions: in Banzhaf, the weights do not depend upon coalition size.

1[ ( ) ( \ )]

( , )iS N

NBI v S v S iN v

( ) [ ( ) ( \ )]i nS N

SSI s v S v S i

278

Interpreting Shapley-Shubik index and Banzhaf index as expectations:

we observe that in the Banzhaf index, probabilities are independent of coalition size.

1/2n-1 is the probability that a coalition containing a given player forms while n(s) is the probability that a coalition of size s containing a given player forms.

1

1[ ( ) ( \ )]

2i nS N

BI v S v S i

( ) [ ( ) ( \ )]i nS N

SSI s v S v S i

279

NBI and SSI satisfy the first three axioms of Shapley:

Efficiency:

they sum up to 1 (= the worth of the game v(N))

Symmetry:

substitutable decision makers have a same power

Null player:

null decision makers has no power

280

Security Council

A permanent member i is decisive in the coalitions

{4 P, k NP, i} where k {4, 5, 6, …, 9, 10}

A non-permanent member i is decisive in the coalitions

{5 P, 3 NP, i}

The number of these coalitions is given by the number of combinaisons of 3 elements among 9.

39 15

9! 8!6!(9) 0.002

3!6! 15!NPSSI C

!

! ( )!km

mC

k m k

1( )!( )!

( )!n

s n ss

n

281

The following equation must be satisfied:

5 SSIP  + 10 SSINP  = 1.

Hence:

SSIP 0.196

To compute the normalized Banzhaf index, we must compute the number of coalitions in which a permanent membre is decisive. They are of the type {4 P, k NP, i} where k = 4,…,10.

4 5 6 7 8 9 1010 10 10 10 10 10 10 848C C C C C C C

282

The number of times a decision maker is decisive is:

8485 + 8410 = 5080.

NBINP = 84/5080 0.016 0.002

NBIP = 848/5080 0.167 0.196

or, in %

NBINP 1.6 0.2

NBIP 16.7 19.6

98 according to SSIratio P/NP:

10 according to NBI

283

Quota games

A quota game with n players is a weighted majority game defined by a vector of relative weight1,…,n such that a coalition S is winning iff

For = (10, 20, 30, 40) in % we have:

W = {24, 34, 123, 124, 134, 234, 1234}

Decision makers 2 et 3 are substitutable: they are decisive in only one coalition of which they are members:{123}.

1

2ii S

284

1 2 3 4 4

24 0 1 0 1 1/12

34 0 0 1 1 1/12

123 1 1 1 0 1/12

124 0 1 0 1 1/12

134 0 0 1 1 1/12

234 0 0 0 1 1/12

1234 0 0 0 0 1/4

4=(1/4,1/12,1/12,1/4)

1 3 3 5, , ,

12 12 12 12SSI NBI

285

On large party and small parties:

(40, 20, 20, 20) % SSI = (50, 17, 17, 17) %

Two large parties and small parties:

(40, 40, 20) % SSI = NBI = (33, 33, 33) %

(35, 35, 20, 10) % SSI = NBI = (33, 33, 33, 0) %

(30, 30, 10, 10, 10, 10) %

SSI = (30, 30, 10, 10, 10, 10) !

NBI = (28.5, 28.5, 11, 11, 11, 11) %286

Party 1 is decisive in the coalitions:

1 2

6 1 2

12 1

12 {3,4,5,6} 4

12 , {3,4,5,6} 6 ( , ) ( , ) 16

1 , , {3,4,5,6} 4

13456 1

1 1 1 1 1 1 1 4 6 4 1 3( ) ( , , , , , )

6 30 60 60 30 6 30 60 60 60 30 10

3 3 1 1 1 1( , , , , , )10 10 10 10 10 10

i i

ij i j N v N v

ijk i j k

s SSI SSI

SSI

287

Party 3 is decisive in the coalitions:

13 , {4,5,6} 3

23 , {4,5,6} 3

( , ) 6 {3,4,5,6}

( , ) 2 16 4 6 56

8 8 3 3 3 3( , , , , , )28 28 28 28 28 28

i

ij i j

ij i j

N v i

N v

NBI

288

Der Deutsche Bundestag 1994-Today

Seats CDU/CSU SPD FDP GRÜNE LINKE fraktionslos

1994 294 252 47 49 30 0 672

1998 245 298 43 47 36 0 669

2002 248 251 47 55 0 2 603

2005 223 222 61 51 53 2 612

Today 239 146 93 68 76 0 622

NBI CDU/CSU SPD FDP GRÜNE LINKE fraktionslos

1994 0.50 0.17 0.17 0.17 0.00 0.00 100

1998 0.13 0.55 0.13 0.13 0.05 0.00 100

2002 0.33 0.33 0.00 0.33 0.00 0.00 100

2005 0.30 0.30 0.13 0.13 0.13 0.00 100

Today 0.50 0.17 0.17 0.00 0.17 0.00 100

289

Der Deutsche Bundestag 1994-Today

Seats in % CDU/CSU SPD FDP GRÜNE LINKE fraktionslos

1994 44 37 7 7 4 0 100

1998 37 44 6 7 5 0 100

2002 41 42 8 9 0 0 100

2005 36 36 10 8 9 0 100

Today 38 23 15 11 12 0 100

NBI in % CDU/CSU SPD FDP GRÜNE LINKE fraktionslos

1994 50 17 17 17 0 0 100

1998 13 55 13 13 5 0 100

2002 33 33 0 33 0 0 100

2005 30 30 13 13 13 0 100

Today 50 17 17 0 17 0 100

290

The actual coalitions are underlined.

We observe that the last elections result in a situation similar to 1994 in terms of coalition (CDU-FDP) and power distribution. This time the Grüne have no power. This party is now a null player while SPD, FDP and Linke are substitutes. Hence any power index which gives the same power to substitutable voters (symmetry) and no power to null voters must end up with the same power distribution. In particular, Banzhaf (normalized) and Shapley-Shubik indices coincide: BNI = SSI.

291

1958 1986 1994

France 4 10 10

Germany 4 10 10

Italy 4 10 10

Belgium 2 5 5

Netherland 2 5 5

Luxembourg 1 2 2

England 10 10

Denmark 3 3

Ireland 3 3

Greece 5 5

Spain 8 8

Portugal 5 5

Sweden 4

Finland 3

Austria 4

Total 17 76 87

Quota 12 54 62

EU-6 EU-12 EU-15

23.8 12.9 11.2

23.8 12.9 11.2

23.8 12.9 11.2

14;3 6.6 5.8

14.3 6.6 5.8

0.0 1.8 2.2

12.9 11.2

4.6 3.6

4.6 3.6

6.6 5.8

10.9 9.2

6.6 5.8

4.8

3.6

4.8

EU-6 EU-12 EU-15

23.3 13.42 11.7

23.3 13.42 11.7

23.3 13.42 11.7

15.0 6.37 5.5

15.0 6.37 5.5

0.0 1.2 2.0

13.42 11.7

4.26 3.5

4.26 3.5

6.37 5.5

11.12 9.5

6.37 5.5

4.5

3.5

4.5

Banzhaf Shapley-Shubik

292

Chapitre 6

Beyond the core…

293

The core of a cooperative game with transferable utility (N,v) is the set of imputations x = (x1,…,xn) against which no coalition can object:

This defines a set that may be empty or contain a large number of imputations.

Several concepts have been proposed in relation to the core: the bargaining set, the stable sets, the least-core and the nucleolus .

( ) ( ) pour toutx S v S S N

294

6.1 The bargaining set

295

The idea behind the notion of bargaining set is to limit the possibilities of objection, by only considering "credible" objections.

We follow here the definition proposed by Mas Colell (198x).

An objection to an allocation x = (x1,…,xn) is formed of a coalition S and an allocation y such that:

( ) ( )

pour touti i

y S v S

y x i S

296

An objection (S,y) to an allocation x faces a counter-objection (T,z) if:

An objection is credible if it has no counter-objection.

The bargaining set is the set of imputations against which there are no credible objection.

( ) ( )

for all

for all \

i i

i i

z T v T

z y i T S

z x i T S

T

S

TS

T\S

297

In the porter game, there are no credible objections against the egalitarian allocation x = (2, 2, 2) :

- a single player or the grand coalition cannot object.

- all objection (S,y) where S = {1,2} and y = (a, 6 – a, 0)face an objection (T,z) where T = {1,3}, z = (b, 0, 6 – b) for 2 < a < 4 and a < b < 4.

For instance the objection (S,y) where S = {1,2} and y = (3, 3, 0) may face the conter-objection (T,z) where T = {1,3} and z = (3.5, 0, 2.5).

298

6.2 Stable sets

299

Stable sets have been introduced by von Neumann and Morgenstern (1944). It used to be called "solution".

Given a TU-game (N,v) and two allocations x and y:

dominates via coalition if

for all

( ) ( )

i i

y x S

y x i S

y S v S

Sy x

dominates if for some Sy x y x S N y x

300

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

Proof

If is in the core and is dominated by , we have:

( ) ( ) ( )

contredicting the inequality ( ) ( ).

x y

v S x S y S

y S v S

301

If is not in the core:

( ) ( ) for some

Then the allocation defined by:

1( ) ( ) if

0 otherwise

is such that and ( ) ( )

i i

i i S

x

x S v S S N

y

y x v S x S i Ss

y x y S v S y x

302

The idea of the von Neumann et Morgenstern solution is to exclude allocations that are dominated by allocations that are themself dominated.

A set K of imputations est stable if K is a set of all imputations that are non-dominated by imputations in K.

Proposition 6.2 There may be several stable sets and the core is a subset of all stable sets.

Proposition 6.3 In the case of a convex game, the core is the unique stable set.

303

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

internal stability

external stability

(i) , Sx y K y x

(ii) if , there exists such thaty K x K x y

304

Proof Denote

internal stability:

external stability:

(I is the set of imputations)

( ) set of imputations dominated by

( ) set of imputations dominated by some

Dom x x

Dom K x K

( ) ( )x K

Dom K Dom x

\ ( )K I Dom K

\ ( )I Dom K K \ ( )I Dom K K

305

Two observations:

(i) no imputation dominates another imputation via a single player

Consider a 0-normalized game (N,v) and two imputations x and y such that for some i N.

Then xi ≤ v(i) = 0 and xi = 0 implies xi = 0. This is in contradiction with yi 0.

ix y

306

(ii) no imputation dominates another imputation via the grand coalition

Consider two imputations x and y such that

Then x(N) > y(N). This is in contradiction with the equality x(N) = y(N) = v(N).

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

.Nx y

307

Example 3-player simple majority game:

The following set is a solution:

It is the "symmetric" solution.

(1) (2) (3) 0

(12) (13) (23) (123) 1

v v v

v v v v

1 1 1 1 1 1, ,0 , ,0, , 0, ,

2 2 2 2 2 2K

308

Indeed, domination can only occur through 2-player coalitions. As a consequence, internal stability holds.

External stability holds as well. Consider any imputation x K such that for instance

It is dominated by (1/2, 0, 1/2) .

If instead it is dominated either by (1/2, 0, 1/2) or by (1/2, 1/2, 0).

309

1 2 1

11 and

2x x x

1 2 1x x

310

(1,0,0)

(0,1,0) (0,0,1)

x2 = 1/2 x3 = 1/2

x1 = 1/2

every imputation notin K is dominated by an imputation in K

1 1 1 1 1 1, ,0 , ,0, , 0, ,

2 2 2 2 2 2K

311

(1,0,0)

(0,1,0) (0,0,1)

x2 = a2

x3 = a3

x1 = a1

31 2 3{ | 1, 0 for all }j i iA x x x x x a i j

A1

A3A2

a\i N iy A a y

imputations dominated by an interior imputation

From this we can conclude that no singleton can be a solution.

Furthermore, if K is a stable set, the line segment joining any two points in K must be parallel to one side of the imputation triangle. Otherwise one would dominate the other.

The following figures show:

- if a line segment parallel to one side of the imputation triangle is a solution, it must join two of its sides

- the interior vertices of some triangle cannot be a solution

312

313

(1,0,0)

(0,1,0) (0,0,1)

a b

imputations dominated by a or b

314

(1,0,0)

(0,1,0) (0,0,1)

a b

imputations dominated by imputations on the line segment [a,b] not joiningtwo sides of the triangle

315

(1,0,0)

(0,1,0) (0,0,1)

a

(1,0,0)

(0,1,0) (0,0,1)

b

(1,0,0)

(0,1,0) (0,0,1)

c

the interior vertices of some triangle cannot be a solution

The next figures show that the only other solutions are line segments of the following type:

These are the "discriminatory" solutions.

1

2

3

( ) {( , ,1 ) | 0 1 }

( ) {( , ,1 ) | 0 1 }

( ) {( ,1 , ) | 0 1 }

1with 0

2

K x x x

K x x x

K x x x

316

317

(1,0,0)

(0,1,0) (0,0,1)

a b

x2 = a2x3 = b3

x1 = a1 = b1

a1 = b1 > 1/2a3 = b2 = 0

318

(1,0,0)

(0,1,0) (0,0,1)

a b

x2 = a2x3 = b3

x1 = a1 = b1

a1 = b1 < 1/2a3 = b2 = 0

K1()

6.3 Least core and nucleolus

319

We define the excess associated to an allocation x and a coalition S by:

In the words of Maschler, Peleg and Shapley (1979) who have introduced the notion of least core:

"It represents the gain (or loss if negative) to the coalition S if its members depart from an agreement that yields x in order to form their own coalition."

( , ) ( ) ( )e x S v S x S

320

The core is then equivalently defined as the set of imputations for which no excess is positive.

The –core is defined for some > 0 by the set of imputations x such that no excess is larger than :

The inequalities defining the –core can be written as:

The core corresponds to = 0.

( ) ( )x S v S

321

( , ) ( , ) | ( , ) for all ,C N v x I N v e x S S N S N

The least-core is the intersection of all nonempty –cores.

Equivalently it is defined by the smallest for which the –core is nonempty:

The –cores have all dimension n–1 or less (if nonempty), except for the least-core which has dimension n–2 or less.

322

*

( , ),

( , ) ( , )

where * ( , ).x I N v S NS N

LC N v C N v

Min Max e S x

The idea of the least core is to minimize the largest excess. This defines a set of imputations, a subset of the core if nonempty.

Schmeidler has proposed a procedure that goes further to eventually retain a unique imputation.

To each imputation, we associate the vectorformed by the excesses placed in a decreasing order.

Imputations are then compared lexicographically in terms of the ordered vectors to which they are associated.

323

To each x I(N,v) we associate the list of m = 2n - 2 proper coalitions (all coalitions except and N) ordered in terms of excesses and the corresponding vector of excesses:

with

We then retain the imputations x* I(N,v) such that:

324

1,..., ( ) mmx S S x

( ) ( , ) and ( ) ( ) for all .i i i jx e S x x x i j

( *) ( ) for all ( , )Lx x x I N v

Proposition 6.6 Given any game (N,v), this procedure leads (Schmeidler, 1969) to one and only one imputation.

The resulting imputation is called the nucleolus. It defines a rulethat to any game (N,v) associates an imputation NUC(N,v).

Proposition 6.7 As a rule, the nucleolus satisfies efficiency, symmetry and null player

The nucleolus does not satisfy additivity.

325

The nucleolus is included in any nonempty –core. It is therefore also an element of the least core which can be alternatively defined by:

In the case where the least core reduces to a single imputation, that imputation defines the nucleolus.

326

*( , ) ( , )LC N v C N v

( , ) 1where * ( ).x I N vMin x

Example: auction game (p1 = 0)

v(1) = v(2) = v(3) = v(23) = 0

v(12) = p2

v(13) = v(123) = p3

The core is the set of allocations of the form

The nucleolus being contained in the core, it has this form and the parameter p suffices to identify the imputations.

3 2 3( ,0, ) wherep p p p p p

327

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

3

2

( , ) if {1}

0 if {2} and {13}

if {3} and {23}

if {12}

e p S p S

S S

p p S S

p p S

2 3 3 2

3 3 2 3

2 3

(0, 0, , , , ) if [ , ]

(0, 0, , , , ) if [ , ]

where2

p p p p p p p p p p

p p p p p p p p p p

p pp

328

- p3

pp3p20

(p2-p3)/2

p2-p3

– p

p2 – p

p – p3

p2

2 3

2

p pp

329

least core = core

nucleolus

From this we conclude that the interval [p2, p3] defines the least core (that coincides with the core).

Furthermore, the mid-point of interval [p2, p3] defines the nucleolus:

i.e. (250, 0, 50) in the case where p3 = 300 and p2 = 200.

3 2 3 2,0,2 2

p p p pNUC

330

Example: 3-player airport game

The associated surplus game is given by:

v(1) = v(2) = v(3) = 0v(12) = v(13) = c1

v(23) = c2

v(123) = c1 + c2

Players 2 et 3 are substitutable the nucleolus is of the form:

1 21 2 3 1 2, where 0

2

c c qx q x x q c c

331

1 2

1 2

1

( , ) if {1}

( )if {2} et {3}

2

if {12} et {13}2

if {23}

e q S q S

q c cS S

c c qS S

q c S

1 21 1Solution: ,

2 3

c cx q Max c

332

c1 + c2

- q

q0

– (c1+c2)

q – c1

– c1

2c2 > 3c1

1 2

2c c

1

2c

1 2

2c c

1 2( )2

q c c

1 22

c c q

1

2c

123

cc

333

1c

c1 + c2

- q

q0

– (c1+c2)

q – c1

– c1

2c2 = 3c1

11

22 3c

cc

1 2

2c c

1 2

2c c

1 22

c c q

1 2( )2

q c c

1

2c

334

c1 + c2

- q

q0

– (c1+c2)

q – c1

– c1 2c2 < 3c1

1 2

2c c

1 2

2c c

1 22

c c q

1 2( )2

q c c 1

23

cc

1

2c

1

2c

335

The least-core is a singleton. It is therefore also the nucleolus:

nucleolus of the cost game:

2 2 21 1 2

1 2 1 2 11 2

2 2( , ) , , if 3 2

3 3 3

, , if 3 22 2 4 2 4

c c cNUC N v c c c

c c c c cc c

2 2 23 1 2

1 2 1 2 13 1 2

2( , ) , , if 3 2

3 3 3

, , if 3 22 2 4 2 4

c c cNUC N c c c c

c c c c cc c c

( , ) ( , )i i iNUC N c NUC N v c

336

Example: apex games

The core of the apex game is empty: no player has a veto right.

The nucleolus being symmetric, the problem can be reduced to a single variable, say w, the share of player 1.

For each w  [0,1] and each coalition S, S  and S  N, we first compute the excess of S at w

 

337

Depending upon the coalition, we have:

The locus of maximum excesses can easily be identified graphically: only the first and the third actually matter and the nucleolus is defined by their intersections.

338

5( , ) (1 ) if 1 and 2 4

4

( , ) (1 ) if 1 and 1 34

( , ) if {2,3,4,5}

( , ) if {1}

se w S w S S

se w S w S S

e w S w S

e w S w S

339

e

w10 w* = 3/7

-1

- w

3(1-w)/4

3/4 w

1/2

1/4

- 3/4

- 1/2

- 1/4

3/7

3 1 1 1 1( , , , , )7 7 7 7 7

NUC

The nucleolus coincides with the least core which is the -core for  = 3/7. Indeed, v(S) – x(S)  3/7 for all S  N with equality for S = {2,3,4,5}.

Comparison with the SSI and NBI:

340

3 1 1 1 1( , , , , ) 0.43,0.14,0.14,0.14,0.147 7 7 7 7

NUC

3 1 1 1 1( , , , , ) (0.60,0.10,0.10,0.10,0.10)5 10 10 10 10

SSI

7 1 1 1 1( , , , , ) (0.64,0.09,0.09,0.09,0.09)11 11 11 11 11

NBI

Example: data games

We have seen that the core of a data game is is a regular simplex given by:

As a consequence, the core's center of gravity is simply the average of its vertices:

It defines the least core and, thereby, the nucleolus.

( , ) { | ( ) 0 and for all }ni iC N C y y N y v i N

341

0( , )i i

vNUC N C v

n

0 1 2 3( , , )v v v v

1 0 2 3( , , )v v v v 1 2 0 3( , , )v v v v

3 3y v 2 2y v

1 1y v

( , )N C

The nucleolus of a 3-player data game

0( , )3i i

vNUC N C v

342

Comparing the two allocation rules

we observe that they coincide if and only if each data is held by a single player:

0( , )i

hi

h M h

v dN c

n tSV

0( , )i

i hh M

vNUC N c d

n

01 for all and for allh i it h M M M i N

343

Outside the partition case, the Shapley value may not be in the core… but the nucleolus may be unfair as the following example illustrates.

Consider a n-player situation, n 3, where only the last two players hold data and their datasets differ by a single data:

1

2

{1,..., }

{2,..., }

1,...,

n

n

i

M m

M m

M i n

1

0

( ) 0 if

( ) if and 1

( ) if and 1

C S n S

C S d n S n S

C S v n S n S

344

Core allocations impose that only player n may be compensated:

0 yi d1 for all i = 1,…, n – 1

The nucleolus goes further by imposing that the n – 1 other players contribute a same amount:

1

1

( 1,... 1)

( 1)

i

n

dy i n

n

dy n

n

345

The Shapley value instead is given by:

the first n – 2 players contribute more

player n – 1 contributes less and may be compensated

the last player gets a higher compensation

0

0 1 11 0

2 2

0 11 0 1 1

2

( 1,... 2)

2 2

2 2 2 2

20

2 2 2

i

m mh

n hh h

mh

n nh

vy i n

n

v d n d d ny v d

n n n n

v d n dy d v y d

n n

346

Special case: the case where th = 1 for all h M0 is the partition case:

Mi  Mj =  for all i  j.

It fits patent and copyright pooling aiming for instance at developing new products or standard technologies.

We shall see that:

- the resulting cost game is concave

- the Shapley value and the nucleolus coincide.

347

In the partition case:

where ci = c(i) is the cost of the data that player i is missing.

0( ) (1 ) ii S

c S s d c

0 0\

0 0\

:

( ) (1 )

i i i

S i

i h h h ih M M h M h M

h ih M M i S

hh M i S

c d d d d d c

c S dd d s d c

348

The associated surplus game

is given by:

It is a symmetric game (all players are substitutes) and the Shapley value and the nucleolus coincide:

01( ) ( )v S s d

( ) ( ) ( )i S

v S c i c S

0

( ) ( 1)( , ) ( , )i i

v N nSV N v NUC N v d

n n

349

The compensations derived from the Shapley value or the nucleolus are then given by:

A player is compensated if and only if the value of the data he/she owns exceeds the per capita value of the complete database.

00 0

0

( , ) ( ) ( , )

( 1)( )

( , )i

i i

i i

h ih M

SV N c c i SV N v

n vc d v c

n n

vd NUC N c

n

350

Exemple N = {1,2,3,4} and M0 = {1,2,3,4}

M1 = d1 = 6M2 = {1} d2 = 10M3 = {2} d3 = 4M4 = {3,4} d4 = 12

y1 = 8 – 0 = 8

y2 = 8 – 6 = 2

y3 = 8 – 10 = – 2

y4 = 8 – 4 – 12 = – 8

032

84

v

n

0( , ) ( , )i

i i hh M

vSV N c NUC N c d

n

351

In the partition case, data games are concave.

It is more easy to check that the associated surplus game is convex:

0

0 0

( ) ( ) ( ) ( )

[( 1) ( 1) ( | | 1) (| | 1)] 0

if

[( 1) ( 1) ( 1)] 0

if

v S v T v S T v S T

s t s t S T S T v

S T

s t s t v v

S T

352

By concavity, the core of a partition data game is the polyhedron whose vertices are the marginal cost vectors. Using Proposition 4.5, they are given by:

353

10 1 2 1 2 0 0

21 0 2 1 0 2 0

1 2 0 1 0 2 0

( , , ... , ) ( , , ... , )

( , , ... , ) ( , , ... , )

...

( , , ... , ) ( , , ... , )

n n

n n

nn n

v v v v c c v c v

v v v v c v c c v

v v v v c v c v c

1 2 0 3 0( , , )c c v c v

1 0 2 3 0( , , )c v c c v

3 3y v 2 2y v

1 1y v

Core of a 3-player partition data game

354

1 0 2 0 3( , , )c v c v c

(c1 – v0, c2 – v0, c3, c4 – v0)

(c1 – v0, c2, c3 – v0, c4 – v0)

(c1, c2 – v0, c3 – v0, c4 – v0)

(c1 – v0, c2 – v0, c3 – v0, c4)

y3 = c3 – v0

y2 = c2 – v0y4 = c4 – v0

y1 = c1 – v0

y1 + y

4 = c1 + c

4 – v0

y 1 + y

2 = c

1 + c

2 – v

0

y2 + y

3 = c2 + c

3 – v0 y 3 +

y4 =

c3 +

c4 –

v 0

y 2 + y 4 = c 2 + c 4 – v 0

355

Core of a 4-player partition data game

Some references

Luce R.D. and H. Raiffa Games and decisions, Wiley 1957.

Meyerson R.B. Game theory. Analysis of conflict, Harvard University Press, 1991.

Moulin H. Cooperative microeconomics, Princeton University Press, 1995.

Osborne M.J. An introduction to game theory, Oxford University Press, 2004.

Osborne M.J. and A Rubinstein, A course in game theory, MIT Press, 1994.

Peeters H. Game theory: a multi-leveled approach, Springer Verlag, 2008.

Peleg B. and P. Sudhölter, Introduction to the theory of cooperative games, Kluwer, 2003.

Shubik M. Game theory in the social sciences, MIT Press, 1982.

Shubik M. A game-theoretic approach to political economy, MIT Press, 1984.

von Neumann J. and O. Morgenstern, Theory of games and economic behavior, Princeton University Press, 1944.

Young, H.P. Cost allocation: methods, principles, applications, North-Holland, 1985.

Young, H.P. (ed.) Fair allocation, American Mathematical Society, 1985. 356

Some interesting web sites:

theory of games (cooperative and non-cooperative):

www.citg.unige.it/siti_internet_web.html (in italien)www.econ.canterbury.ac.nz/personal_pages/paul_walker (historical)arielrubinstein.tau.ac.ilwww.economics.utoronto.ca/osborne/igt (web site of his book)

cooperative games:

www.econ.usu.edu/acaplan/tugames.htm (a nice software for 3-person games)

webs.uvigo.es/matematicas/campus_vigo/profesores/mmiras/TUGlabWeb/TUGlab.html(a matlab toolbox for 2 to 4 player TU-games)

power indices:

powerslave.val.utu.fi www.warwick.ac.uk/~ecaae (online computation)

357