What is the sum of the following infinite series 1+x+x2+x3+…xn… where 0<x<1? Infinity 1+2x 1+3x 1/x 1/(1-x)

Cooperative Game Theory

Coalitional Games Focus on what groups can accomplish if they work together. Contrast to Nash equilibrium which focuses on what individuals can do acting alone. (sometimes known as non-cooperative game theory)

Coalitional Game with transferable payoffs A set of players N. A coalition S is a subset of N. Grand coalition is N itself. Coalitional game with transferable payoffs assigns a value v(S) to every subset of S. An action for the coalition S is a distribution of Its total value to its members. Think of v(S) as an amount of “money” that the coalition can earn on its own and can divide this money in any way that adds to v(S).

Two Player unanimity game: An Almost Trivial Example Players 1 and 2. Non-empty subsets are {1},{2},{1,2}. Let v({1})=v({2})=0 and v({12})=1. Set of actions available to coalition {1,2}, are distributions of payoffs (x1,x2) such that x1≥0, x2≥0, and x1+x2=1.

Three player majority game: A (slightly) more interesting example Players 1, 2, and 3. Non-empty subsets of N={1,2,3} are N, {12}, {13}, {23},{1},{2},{3}. A cake whose total value is 1 is to be divided. Any coalition that is a majority can choose how to divide it. Then v({12})=v({23})=v({13})=v({123})=1 and v({1})=v({2})=v({3})=0. Actions available to any coalition are possible divisions of the cake. For example, coalition{12} can choose any division such that x1≥0, x2≥0, and x1+x2=1.

Landowner-workers example Player 1 owns all of the land. There are k landless workers. Total food produced is given by a production function f(x) where x is the number of people who work on the land. Landless workers can produce nothing on their own. Landowner can only produce f(1) with his own labor.

Coalition values for landowner worker model Player 1 is the landowner. By himself he gets v({1})=f(1). Workers get nothing without land, so any coalition S to which the landowner does not belong has v(S)=0. A coalition S to which the landowner does belong gets V(S)=f(k+1) where k is the number of workers in the coalition.

The Core The core of a coalitional game is the set of outcomes x (actions by the grand coalition) such that no coalition has an action that all of its members prefer to x.

Examples: Two person unanimity game. Core consists of all possible divisions. Why? Three person majority game. Core is the empty set. Why?

Five people must decide how to divide a pie of fixed size Five people must decide how to divide a pie of fixed size. Any 4 of them can agree to overturn the status quo agreement and redistribute the pie among themselves. One outcome in the core of this game is that in which each player gets 1/5 of the pie. One outcome in the core of this game is that in which 4 players each get ¼ of the pie and one player gets nothing. The core of this game is empty.

Landowner-worker game, 2 workers Let x1,x2,x3 be an allocation of the output f(3) from 3 people working on landowner’s land. If this in the core, then it must be that x1≥f(1)—otherwise coalition {1} could do better by self. x1+x2 ≥f(2)—otherwise {12} could do better by themselves x1+x3 ≥f(2)—otherwise {13} could do better by themselves. x1+x2 +x3=f(3).

Conclusions x1+x3 ≥f(2) and x1+x2 +x3=f(3) imply that x2≤f(3)-f(2) Similarly x1+x2 ≥f(2) and x1+x2 +x3=f(3) imply that x3≤f(3)-f(2) Thus no worker gets more than the “marginal product” of a third worker.

Example Suppose f(1)=3, f(2)=5, f(3)=6. Then we must have x1≥3, x2≤1, and x3≤1. Any allocation where 2 and 3 each gets a non-negative amount and 1 gets more than 3 units will be in the core.

One owner two possible buyers Owner (person 1) has an object that is worthless to him, worth $1 to either of two possible buyers (persons 2 and 3). Persons 2 and 3 each start out with more than $1. Trade is possible. Two outcomes are in the core. Person 1 sells object to 2 for $1. Person 1 sells object to 3 for $1. Why is nothing else in the core?

Previous example except that Person 2 values object at 1. Person 3 values it at $v<1. What is in core? Person 2 gets the object and pays person 1 a price p that is between v and 1.