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COOPERATIVE GAME THEORY

In non-cooperative game theory, we focus on the individual players’ strategies and

their influence on payoffs, and try to predict what strategies players will choose

(equilibrium concept).

In cooperative game theory, we abstract from individual players’ strategies and instead

focus on the coalition players may form. We assume each coalition may attain some

payoffs, and then we try to predict which coalitions will form (and hence the payoffs

agents obtain).

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non-cooperative or strategic game theory, vis-à-vis cooperative or coalitional game theory from an interview with Robert Aumann (2005 Nobel prize) Strategic game theory [he means non-cooperative game theory] is concerned with strategic equilibrium–individual utility maximization given the actions of other people, Nash equilibrium and its variants, correlated equilibrium, that kind of thing. It asks how people should act, or do act. Coalitional game theory [namely, cooperative game theory], on the other hand, concentrates on division of the payoff, and not so much on what people do in order to achieve those payoffs. Practically speaking, strategic game theory deals with various equilibrium concepts and is based on a precise description of the game in question. Coalitional game theory deals with concepts like the core, Shapley value, von Neumann-Morgenstern solution, bargaining set, nucleolus. Strategic game theory is best suited to contexts and applications where the rules of the game are precisely described, like elections, auctions, internet transactions. Coalitional game theory is better suited to situations like coalition formation or the formation of a government in a parliamentary democracy or even the formation of coalitions in international relations; or, what happens in a market, where it is not clear who makes offers to whom and how transactions are

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consummated. Negotiations in general, bargaining, these are more suited for the coalitional, cooperative theory. On the one hand, negotiations can be analyzed from a strategic viewpoint, if one knows exactly how they are conducted. On the other hand, they can be analyzed from a viewpoint of where they lead, which will be a cooperative solution. There is the “Nash program“ –basing cooperative solutions on non-cooperative implementations. For example, the alternating offers bargaining, which is a very natural strategic setup, and leads very neatly to the axiomatic solution of Nash –as shown by Rubinstein and Binmore. The big advantage of the cooperative theory is that it does not need a precisely defined structure for the actual game. It is enough to say what each coalition can achieve; you need not say how. For example, in a market context you say that each coalition can exchange among its own members whatever it wants. You don‘t have to say how they make their offers or counteroffers. In a political context, it is enough to say that any majority of parliament can form a government. You don‘t have to say how they negotiate in order to form a government. That already defines the game, and then one can apply the ideas of the coalitional theory to make some kind of analysis, some kind of prediction.

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Solution concepts:

1. the Nash-bargaining solution (basically when there are 2 players),

2. the core and

3. the Shapley value.

References:

Mas-Colell, Whinston and Green, Appendix A in chapter 18.

Brandenburger, course notes

Osborne and Rubinstein, A course in game theory, chapters 13-15

Osborne and Rubinstein, Bargaining and markets, for the Nash-bargaining solution and

the Nash program.

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Notation of a cooperative game:

1. There is a set of players, { }nN ,...,1= ,

2. A characteristic function that indicates what each coalition S of players can get (it is

the sum of utilities of players in the coalition). Formally, the characteristic function is a

function

+ℜ→Nv 2:

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i. We will assume transferable utility, that is, players within a coalition may

compensate one another through payments.

ii. The characteristic function could be superadditive,

If ∅=∩ 21 SS , then ( ) ( ) ( )2121 SSvSvSv ∪≤+

iii. Finally, there are no externalities: for remaining players, it is indifferent whether

members of coalitions 1S and 2S group into a coalition 21 SS ∪ or not. Or look

at the payoff of coalition 1S : it is ( )1Sv , independently of which other coalitions

form. [For cooperative games with externalities see Maskin (2004)]

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The Nash-bargaining solution

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The Nash program (see Osborne and Rubinstein, chapters…)

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The Core of a game: ( )⎭⎬⎫

⎩⎨⎧

≥∀≤≡ ∑ ∑= ∈

n

i Siiin SvxSNvxxxvcore

11 )(),( / ,...,)(

In words: the core is an allocation ( ) ,...,1 nxx , where xi is the payoff for player i, of

total surplus )(Nv , that satisfies:

(1) It is feasible, ∑=

≤n

ii Nvx

1)(

(2) A set of players S obtain at least what they would obtain forming a coalition S,

∑∈

≥∀Si

i SvxS )( , (otherwise they would not accept the allocation and would blockade

its formation).

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Intuition about what the allocation in the core shows:

Consider the following situation (from Brandenburger’s notes): a seller A has one unit

of a product at cost c=4; there are 2 buyers B and C, with valuations UB=9 and UC=11.

What do you expect to happen?

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A plausible cooperative game representation would be

v(A,B,C)=v(A,C)=7 v(A,B)=5 v(A)=v(B)=v(C)=0

The core must satisfy: 7=++ CBA xxx , 5≥+ BA xx , 7≥+ CA xx , 0≥ix

We obtain that the core is the set ( ) ( ){ }xxxxx CBA −= 7,0,,, with x at least 5.

Player B obtains nothing (since he does not contribute to create value in equilibrium);

the core does not tell us how the additional surplus created when A trades with C over

the value created when A trades with B is shared between A and C.

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Sometimes the core does not give a unique prediction; sometimes the core is empty!

The core is never empty when the game is convex: the marginal contribution of a

player i is larger the larger the coalition:

If TS ⊂ , then )(}){()(}){( TviTvSviSv −∪≤−∪

More properly the core of a cooperative game is not empty if and only if a game is

balanced (see the definition of balanced game in Osborne and Rubinstein, A course in

game theory)

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For a game with an empty core, look at the following example:

v(∅) = 0, v(1) = π1, v(2) = π2, v(1,2) = Π

If Πππ <+ 21 , the core is the set ( ) ( ){ }2121 ,, πΠπΠ −≤≤−= xxxBB .

If Πππ >+ 21 , the core is empty (in this very particular example, you may think,

well, it makes sense not no create a grand coalition; with more than 2 players, maybe

some opportunities are lost when the grand coalition do not form).

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Now look at this example:

v(1,2,3) = 4,

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

v(∅) = v(1) = v(2) = v(3) = 0

In this game the core is empty (but notice that the game is not convex)

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The Shapley value of a game

Let’s see an example of a game with 3 players:

We may set 3! = 6 different orderings of the players:

{1, 2, 3}, {1, 3, 2}, {2, 1, 3}, {2, 3, 1}, {3, 1, 2}, {3, 2, 1}.

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1. There are 2 possibilities that 1 is the first member to arrive, {1, 2, 3} and {1, 3, 2}.

2. There is one possibility that 1 is the second member to arrive when 2 has already

arrived, {2, 1, 3}.

3. There is one possibility that 1 is the second member to arrive when 3 has already

arrived, {3, 1, 2}.

4. There are 2 possibilities that 1 is the last member to arrive, {2, 3, 1} and {3, 2, 1}.

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We evaluate which is the marginal contribution of player 1 to an already formed

coalition (forgetting about additional members to come),

1. v(1) – v(∅)

2. v(1, 2) – v(2)

3. v(1, 3) – v(3)

4. v(1, 2, 3) – v(2, 3)

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Then the Shapley Value of player 1 is the sum of its average marginal

contributions:

[ ] [ ] [ ] [ ]( ))3,2()3,2,1(2)3()3,1()2()2,1()()1(261

1 vvvvvvvvB −+−+−+∅−=

And similarly for players 2 and 3:

[ ] [ ] [ ] [ ]( ))3,1()3,2,1(2)3()3,2()1()2,1()()2(261

2 vvvvvvvvB −+−+−+∅−=

[ ] [ ] [ ] [ ]( ))2,1()3,2,1(2)2()3,2()1()3,1()()3(261

3 vvvvvvvvB −+−+−+∅−=

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Notice that )3,2,1()()3,2,1(321 vvvBBB =∅−=++ if 0)( =∅v .

In general, we define the Shapley value of a game with n players as

( ))(}){(!

)!1(!SviSv

nSnS

BNSi

i −∪−−

= ∑⊆∉

(n is |N|, i.e. there are n players, and |S| is the number of player is set S).

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The idea behind the Shapley Value: to distribute the gains from trade equally.

With 2 players: v(1,2) > v(1) + v(2), where v(1), v(2) ≥ 0. Then we could consider fair

that

{ })2()1()2,1()( 21 vvvivBi −−+=

The Shapley value generalizes this idea for more than 2 players (see Mas-Colell,

Whinston and Green, Appendix A in Chapter 18)

The Nash-bargaining solution does indeed the same for 2 players.

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Properties of the Shapley Value (it is the only solution concept that satisfies these

properties and another one at the same time, see Mas-Colell et al.):

1. )(NvBNi

i =∑∈

. The total gain is distributed, no utility is wasted, the Shapley value

is “efficient”.

2. If for a player i, })({)(}){( ivSviSv +=∪ for all coalitions }{iNS −⊆ , then

})({ivBi = . If a player never adds anything to other players, only receives her

own value standing alone.

3. If for players I and j, }){(}){( jSviSv ∪=∪ for all coalitions }{}{ jiNS −−⊆ ,

then ji BB = . Players with the same impact on surplus are treated similarly.

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An example: There are 1 seller (player 1) and 2 buyers (players 2 and 3), each with a

willingness-to-pay of $1 for one unit. The cost parameters are = c = 0.

If the seller has one unit to sell, the characteristic function may be:

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

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

==

===∅===

vv

vvvvvv

If the seller has 2 units to sell, the characteristic function may be:

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

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

==

===∅===

vv

vvvvvv

If the seller has one unit to sell, the core of the resulting game is (1,0,0).

If the seller has 2 units to sell, the core of the resulting game is (2 − x − y, x, y), for 0 ≤

x ≤ 1 and 0 ≤ y ≤ 1.

If the seller has one unit to sell, the Shapley Value of the resulting game is

(x1, x2, x3) = (2/3, 1/6, 1/6).

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If the seller has 2 units to sell, the Shapley Value of the resulting game is

(x1, x2, x3) = (1, 1/2, 1/2).

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We can embed a cooperative game into a non-cooperative game.

This is a useful way to simplify some stages of a game in order to concentrate the

analysis in a particular feature. This makes sense also (maybe) when we do not want

that our results depend on a particular bargaining procedure.

Consider as an example the following two-stage game (borrowed from Brandenburger

and Stuart, 2005):

In the first stage, a seller chooses capacity s. The costs of building capacity are k.

The seller faces 2 buyers. In the second stage, once the seller has chosen s, players

bargain over the product. I use our former example as the second stage of the game.

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If we use the Shapley Value as the solution for the second stage of the game, in the

first stage the seller chooses capacity to maximize what he gets (net of costs) under

the Shapley Value in the resulting game.

If k=0, the seller will always choose to serve the whole market. Under the Shapley

Value, a seller only chooses to undersupply (k=1 instead of k=2) to avoid the cost

of capacity if 31>k .

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The fact that we do not consider externalities between coalitions does not allow us to

model with a cooperative game some natural interactions. Consider for instance a

Cournot game with 3 firms, and firms may merge.

Maskin sets another example in the same spirit (example 2 in pages 5-7) and his paper

discusses the generalization of the Shapley Value for games with externalities.

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Further references

Brandenburger, Adam and Stuart, Harborne, A Note on Biform Analysis of Monopoly

Maskin, Eric, Bargaining, Coalitions and Externalities,