EC941 - Game Theory

EC941 - Game Theory Lecture 8 Prof. Francesco Squintani Email: f.squintani@warwick.ac.uk

Structure of the Lecture • Coalitional Games and the Core • Ownership and the Distribution of Wealth • Horse Trading and House Exchanges • Voting and Matching

Coalitional Games A coalitional game is a model of interaction by groups of players. A set of actions is associated with every group of players, not only with individual players. An outcome of a coalitional game consists of a partition of the set of players into groups, together with an action for each group in the partition. In general, players care about all actions, but we focus on the case in which players in the same group care only about their common action.

Definition A coalitional game consists of a set of players N for each coalition S subset of N, a set of actions for each player, preferences over the set of all actions of all coalitions of which she is a member. Definition A coalitional game is cohesive if, for every partition {S1, . . . , SK} of the set of all players and every combination (a(S1), . . . , a(SK) ) of actions, the grand coalition N has an action that is at least as desirable for every player i as the action a(Sj) of the member Sj of the partition to she player i belongs.

Transferable Utility In many applications, each coalition controls some quantity of a good, distributed among its members. Each action of a coalition S is a distribution among the members of S of the good that S controls, and it is called allocation. When utility can be transferred in one-to-onefashion among players (as it is the case for money), the game is called game with transferable utility: There is a unique action v(S) for each coalition S: the worth of the whole coalition S.

Examples: Games with Transferable Utility 1. Distribution of resources. A landowner’s estate, when used by k workers, produces the output f(k) of food, where f is a increasing function for which f(0) = 0. The total number of workers is n. The landowner and each worker care only about the amount of output received, and prefer more to less.

The following coalitional game models this situation. Players: The landowner and the m workers. Worth: The worth v(S) of coalition S is zero if S does not include the landowner. Otherwise, v(S)=f(k), where k is the number of workers in S. Preferences Each player’s preferences are represented by the amount of worth she obtains. This is a cohesive game.

2. A voting game. Three people have access to one unit of output. Any majority may control the allocation of this output. Each person cares only about the amount of output she obtains. Players: The three people. Worth: The worth is v(i)=0 for i=1,2,3; and v(S)=1 for all other coalitions. Preferences: Each player’s payoff equals herworth.

Examples: Games without Transferable Utility 3. House allocation Each one ofn people has a single house. Any subgroup may reallocate its members’ houses in any way it wishes (one house to each person). The values assigned to houses vary among the people; each person cares only about the house she obtains.

3. House allocation coalitional game. Players The n people. Actions The set of actions of a coalition S is the set of all assignments to members of S of the houses originally owned by members of S. Preferences Each player prefers one outcome to another according to the house she is assigned. The game is cohesive.

4. Matching market. A group X of men and Y of women may be matched in pairs. Each person cares only about her partner. A matching of the members of a coalition S is a partition of S into male-female pairs and singles. Players: The set X of the men and Y of the women. Actions: The set of actions of a coalition S is the set of all matchings of the members of S. Preferences: Each player prefers one outcome to another according to the partner she is assigned. The game is cohesive.

The Core The solution concept in cohesive games is the core. An action of the grand coalition is “stable” if no coalition can break away and choose an action that all its members prefer. The set of all stable actions of the grand coalition is called the core. DefinitionThe core of a coalitional game is the set of actions aN of the grand coalition N such that no coalition has an action that all its members prefer to aN.

Distribution of Resources Reconsider Example 1 on the distribution of resources. Assume that f(k) – f(k-1) decreases in k. The core is the set of allocations in which each worker i receives xi at most f(n) – f(n-1). In fact, because f(k) – f(k-1) decreases in k, it follows that f(n) – f(k) > (n-k)[f(n)-f(n-1)]. Thus, there is no coalition of k workers plus the landowner which can give to all its workers more than xi and more than f(n) – Si xi to the landowner.

Suppose now that the land is owned collectively and the distribution of output is determined by majority voting (including the output of the minority) Actions The set of actions of a coalition S consisting of more than n/2 players is the set of all S-allocations of the output f(n) between the members of S. The set of actions of a coalition S consisting of at most n/2 players is the single S-allocation in which no player in S receives any output.

The core of this coalitional game is empty. For every action of the grand coalition, at least one player obtains a positive amount of output. But if player i obtains a positive amount of output then the coalition of the remaining players, which is a majority, may improve upon the action, distributing the output f (n) among its members (so that player i gets nothing).

Homogeneous Good Markets Homogeneous good markets may be modelled as coalitional games. The actions of each coalition S are the set of S-allocations of the good initially owned by the members of S. The core of such a game is the set of allocations of the good, robust to the trading opportunities of coalitions. If aN is in the core, then no coalition can trade among themselves, so that all members prefer the trade to aN.

Definition: A horse-trading game is a coalitional game defined as follows: Players. n traders among horse owners and nonowners, each with different valuation vi for owning a horse. Actions. For each coalition S, a S-allocation of the horses and the total amount of money owned by S such that each player has at most one horse. Preferences.Each player i’s preferences are given by r, if i obtains no horse, and by vi+r if i obtains a horse, where r is the net amount of money obtained by i in the trade.

Core of Horse-Trading Games Number owners in ascending order and nonownersin descending order according to their horse valuations. Letowner i’s valuation be siand nonowneri’s valuation be bi. Denote by k* the largest iwith bi>si. Proposition.In every action aNin the core of a horse trading game, every nonowner pays the same price p for a horse, p lies between max{sk*,bk*+1} and min{bk*,sk*+1}, every owner i with vi< p trades her horse, every nonowneri with vi>pobtains a horse.

To prove this, denote by L* the set of sellers, and by B* the set of buyers. Denote by rithe money received by owner i and by pj the sum paid by nonownerj in aN. Ifirst claim that pj= 0 for every nonownerj not in B*. In fact, if pj> 0 for some nonownerj not in B*, then j’s payoff is negative, and she can unilaterally improve upon aNby retaining her original money. If pj< 0 for some nonowner j not in B*,then the coalition of all players other than j, has pj less money than initially owned, and the same number of horses. This coalition can improve by assigning horses as in aNand distributing pj/(n-1) more money to its members.

By a similar argument, ri = 0 for every owner not in L*. I now argue that in aN, all trades occur at the same price p*, i.e., ri= pj=p* for every seller iand buyer j. Suppose that ri< pjfor seller i and buyer j. Then, the coalition {i, j} can improve upon aN: i can sell her horse to j at a price between riand pj. Under aN, seller i’s payoff is riand buyer j’s payoff is bj-pj. If i sells her horse to j at the price ½(ri+pj ) then her payoff is ½(ri+pj ) > ri and j’s payoff is bj - ½(ri+pj ) > bj-pj, so that both iand jare better off than they are in aN. Thus ri>pjfor every seller i and every buyer j.

Now, the sum of all the amounts ri received by sellers is equal to the sum of all the amounts pjpaid by buyers, and L* and B* have the same number of members. Thus we have ri = pjfor every seller i and buyer j. To pin down the range of p*, first note that every owner iwith vi < p* must sell her horse. In fact, if she did not, then the coalition consisting of herself and any buyer j would improve by trading i’s horse at a price between vi and p*. Second, note that no owner i with vi > p* trades, because her payoff from doing so is negative.

Similarly, every nonowneri with vi > p* must buy a horse, and no nonowneri with vi < p* does so. • As a result, the trading price p* cannot be smaller than sk*or than bk*+1, nor larger than bk*or than sk*+1. • Finally, I show that any player i with vi= p* trades. • Suppose nonowner i’s valuation equals p*. Then owner ihas vi< p* and owner i+1 has vi > p*, so that exactly iowners trade. Thus exactly inonowners must trade, implying that nonowneritrades. • Symmetrically, a owner i with vi= p* must trade. 22

Exchanging Houses Reconsider Example 3 on House Allocation. TheoremEvery allocation in the core can be found with the top trading cycle procedure. Definition A trading cycle is a cycle of players in which each player prefers the house of the next player more than her own.

Definition The top trading cycles procedure is defined as follows. First we look for cycles among the houses at the top of the players’ rankings, and assign to each member of each cycle her favorite house. Then we eliminate from consideration the players involved in these cycles and the houses they are allocated, and start again, until exhausting all houses.

To illustrate the procedure, consider the game with four players whose preferences as follows: p1: h2, -, -, -; p2: h1, -, -, -; p3: h1, h2, h4, h3; p4: h3, -, -, - We see that 12 is a 2-cycle, so at the first step players 1 and 2 are assigned their favourite houses (h2 and h1 respectively). After eliminating these players and their houses, 34 becomes a 2-cycle, so that player 3 is assigned h4 and player 4 is assigned h3. If player 3’s ranking of h3 and h4 were reversed then at the second stage 3 would be a one-cycle, so that player 3 would be assigned h3, and then at the third stage player 4 would be assigned h4.

Voting Reconsider example 2 on voting by majority rule, but allow for any odd number of voters and for any policy. We model the interaction as a variant of a coalitional game, in which all players care about the actions of any majority coalition. Say that i prefers outcome x to outcome y if and only if: either there are majority coalitions in the partitions associated with both x and y and she prefers the majority policy associate to x to the one associated to y, or there is a majority coalition in x by not in y.

The set of actions available to any coalition containing a majority of the players is the set of all policies; every other coalition has a single action. The definition of the core of this variant of a coalitional game is as follows: the set of actions aN of the grand coalition N such that no majority coalition has an action that all its members prefer to aN.

The Core of Voting Games TheoremIf players’ preferences are strict, then an outcome is in the core of a voting game if and only if it is a Condorcet winner. Suppose that the policy x is in the core of this game. Then no policy is preferred to x by a coalition consisting of a majority of the players. If we assume that every player’s preferences are strict, then for every policy y, the set of players who prefer x to y is a majority. That is, x is a Condorcet winner.

TheoremIf the policy space is uni-dimensional and all players’ preferences have a single peak, then the unique Condorcet winner is m, the median of the players’ preferred positions. • To prove this, note that m defeats in a majority all policies to its left (right), as it is strictly preferred by all voters whose bliss point is to the right (left) of m. • This immediately implies that m is a Condorcet winner (as it defeatsall other policies), and that no other policy is a Condorcet winner (as it is defeated by m).

The existence of Condorcet winners is in general problematic. Euclidean preferences are described by a bliss point (a vector in Rn), and rank policies in Rn with utility decreasing in the distance from the bliss point. It is easy to find examples of preference profiles in multi-dimensional spaces such that there does not exist any Condorcet winner.

Each policy in the intersection between 1 and 2’s contour sets defeats all policies not in the union of 1 and 2’s sets. • By repeating the argument for all sets, we see that there is no Condorcet winner. 1 2 3

Matching Reconsider example 4, on matching between male and female, or workers and firms. PropositionA matching m is in the core if and only if each player prefers her partner to being single for no pair (i, j) with i in X and j in Y, it is the case that i prefers j to μ(i) and j prefers i to μ(j). TheoremAll matchings in the core can be found with the deferred acceptance procedure.

Deferred Acceptance Procedure Definition The deferred acceptance procedure with proposals by X’s is defined as follows. Initially, each player in X proposes to her favourite player in Y, and each player in Y either rejects all the proposals she receives (if she prefers to be alone), or rejects all but the best proposal. Each proposal that is not rejected results in a tentative match between a player in X and one in Y.

If every offer is accepted, the process ends, and the tentative matches become definite. If there is a second stage, each player in X whose proposal was rejected in the first stage proposes to the player in Y she ranks second. Each player in Y chooses among the set of players in X who proposed to her and the one with whom she was tentatively matched in the first stage, rejecting all but her favourite among these players in X.

Again, if every offer is accepted, the process ends, and the tentative matches become definite, whereas if some offer is rejected, there is another round of proposals. • The procedure stops when the proposal of no player X is rejected or when every player in X whose offer was rejected has run out of acceptable players in Y.

Consider for example the following preferences. X Y x1 x2 x3 y1 y2 y3 y2 y1 y1 x1 x2 x1 y1 y2 y2 x3 x1 x3 y3 x2 x3 x2 First x1 proposes to y2 and both x2 and x3 propose to y1; y1 rejects x2’s proposal. Then x2 proposes to y2, so that y2 may choose between x2 and x1 (with whom she was tentatively matched at the first stage).

Player y2 chooses x2, and rejects x1, who then proposes to y1. Player y1 now chooses between x1 and x3 (with whom she was tentatively matched at the first stage), and rejects x3. Finally, x3 proposes to y2, who rejects her offer, and keeps the tentative match with x2. The final matching is thus (x1, y1), (x2, y2), x3 (alone), and y3 (alone).

Summary of the Lecture • Coalitional Games and the Core • Ownership and the Distribution of Wealth • Horse Trading and House Exchanges • Voting and Matching

Preview of the Next Lecture • Ultimatum Game and Hold Up Problem • Rubinstein Alternating Offer Bargaining • Nash Axiomatic Bargaining

EC941 - Game Theory