Econ 618: Topic 11 Introduction to Coalitional Games

Econ 618: Topic 11 Introduction to Coalitional Games Sunanda Roy 1 Coalitional games with transferable payoffs, the Core Consider a game with a finite set of players. A coalition is a nonempty subset of the set of players. A coalitional game with transferable payoff consists of (1) a finite set of players, N (2) a function v : S N R. The value of the function for a given S, v(s) is known as the worth of the coalition S. v(s) is the total payoff that is available for division among the members of the coalition S. The coalition can choose to divide v(s) in any way amongst its members. The set of joint actions that the coalition can take consists of all possible divisions of v(s) amongst the members of S. Thus, an action by a coalition is an allocation of v(s) amongst its members. In many situations, the payoff that a coalition can achieve depends on the actions taken by players outside the coalition. We assume this is not the case here. The phrase transferable payoff captures the notion that every action of S generates a distribution of payoffs amongst the members of S that has the same sum or in other words every allocation has the same total value. A coalitional game in which this is not the case, has non-transferable payoff. The following two examples illustrate the difference. Example 1: The coalitional version of the Stag - Hunt game A group of n hunters can catch a stag only if every member (all n of them) remains attentive. Any inattentive member may catch a hare. A coalition may split its spoils in any way amongits members. Each hunter s preferences are given by the payoff function αx + y where x = amount of 1

COALITIONAL GAMES 2 hares and y = amount of stag the hunter obtains and α < 1/n. This situation can be modelled as a coalitional game with transferable payoffs, as follows. Consider a coalition of S hunters, s of which are inattentive (S s are attentive). Thus 0 s S n. Set of actions for the coalition is given by x 1 + x 2 +...x S = s if 0 < s n and S n and by y 1 + y 2...y n = 1ifs = 0 and S = n. The value or worth of the coalition is v(s)=1ifs = 0 and S = n and v(s) =αs if s > 0 and S n. The payoff to each player i in the coalition is αx i if 0 < s n and y i if s = 0. Example 2: The house allocation game or the marriage market game Each member of a group of n people owns an indivisible good - call it a house. Houses differ. Any coalition may reallocate its members houses in any way it wishes - with one house to each person. Each person cares only about the house he or she obtains. People have different preferences across the available houses, that is the values assigned to houses vary among the people. This situation can be modelled as a coalitional game but with non-transferable payoffs. The following explains why: Consider a coalition of only the first two players 1 and 2. Let v i = payoff to player 1 if she gets the house initially owned by player i, where i = 1,2. Let w i = payoff to player 2 if she gets the house initially owned by player i, where i = 1, 2. Then the coalition of the two players can achieve only the distributions (v 1,w 2 ) and (v 2,w 1 ). But v 1 + w 2 v 2 + w 1 in general. A coalitional game with transferable payoff is cohesive if v(n) K v(s k ), for every partition {S 1,S 2...S K } of N k=1 A coalitional game is designed to model situations in which players are better off acting in groups rather than individually. A cohesive game is one in which the worth of the grand coalition of all N players is at least as much as the sum of the worth of all smaller coalitions. Thus it captures the idea that it is possible for players to be better off forming one grand coalition because

COALITIONAL GAMES 3 the latter can achieve outcomes that are at least as desirable for every player as those achieved by any partition. The Stag-hunt game is cohesive because v(n) =1ifs = 0 and for all S n and s > 0, v(s) < 1 because α < 1/n. One can similarly argue that the house allocation game is also cohesive. A coalitional game that is cohesive, allows us to look for equilibrium allocations in which all players coalesce. The analogue of Nash Equilibrium in co-operative game theory is the Core. An allocation is in the Core if no coalition can deviate and obtain an outcome that is better off for all its members. Let < N,v > be a coalitional game with transferable payoff and S any coalition. x(s)= i S x i is a S-feasible allocation if x(s) =v(s). AnN-feasible allocation is simply described as a feasible allocation. The core of a coalitional game with transferable payoff is the set of all feasible allocations {x i } i N such that there is no coalition S and an S-feasible allocation {y i } i S, for which y i > x i for all i S. An equivalent definition that is useful to identify core allocations in specific problems is the following: The core is a set of allocations {x i } i N such that v(s) x(s) for every coalition S. With this definition, the core becomes a set of allocations satisfying a system of weak linear inequalities and hence is closed and convex. The core of a coalitional game may be empty. The core may not be unique. 2 Specific coalitional games and their core 2.1 The Stag-hunt game Suppose {αx 1 + y 1,αx 2 + y 2,...αx n + y n } belongs to the core. If y i > 0 for any i under such an allocation, x i = 0 for all i - since, all n hunters need to be attentive to catch a stag, in which case none can catch a hare. Hence, such a core allocation will be of the form {y 1,y 2...y N } (where y i 0). Since each hunter by himself (one-member coalition) can at least hunt a hare and get α.1 = α, the core allocation must satisfy, y i α, for all i, such that N i=1 y i = 1

COALITIONAL GAMES 4 Can an allocation of the form {αx 1,αx 2...αx N } (where x i 0) be in the core? Note that, under such an allocation, N i=1 αx i = α N i=1 x i αn. Since α < 1/n, N i=1 αx i < 1. Hence it is possible for the coalition to do better than under this allocation. Hence such an allocation cannot be in the core. Thus core allocations will consist of shares of the stag but no hare. 2.2 The three player majority game Three people have access to one unit of output. Any majority - two or three players - can decide how to allocate this output. Each person cares only about his or her share. The core of this game is empty because of the following argument. Suppose it is not and there exists x, such that x 1 +x 2 +x 3 = 1 and no coalition of two players can improve upon it. Under such a core allocation x i > 0 for each i. If not and suppose x 3 = 0, then a coalition of {3, 2} or {3, 1} can allocate amongst themselves {1 x 2 ε,x 2 +ε} or {1 x 1 ε,x 1 +ε},so that each member of the coalition is strictly better off than under the core allocation. If there is a core allocation such that x 1 + x 2 + x 3 = 1 and x i > 0 for all i, any two players can form a coalition and divide up the third player s share amongst themselves and be strictly better off than under x. Hence, x is not in the core. Consider a variant of this game. Suppose player 1 has three votes, players 2 and 3 have one vote each. Then the core is non-empty because player 1 can form a coalition by himself and allocate the output to himself. Thus {x 1 = 1,x 2 = 0,x 3 = 0} is in the core. Further this is the unique core allocation because player 1 can improve upon any other by himself. 2.3 Market games Example 3: A person holds one indivisible unit of a good and each of two potential buyers has an amount of money (strictly greater than one). The owner values money but not the good. Each buyer values both money and values the good as equal to one one unit of money. Each coalition may assign the good (if owned by one of its members) to any of its members and allocates its members money in any way it wishes amongst its members. Thus it is possible in a coalition

COALITIONAL GAMES 5 of three players, for the owner to keep the good and still get a positive sum of money from both players. What is the structure of the core? First we show that for any action in the core, the owner does not keep the good for the following reason. Suppose there is an action x(3) under which the owner keeps the good and each buyer i transfers an amount m i 0 units of money to the owner. Consider an alternative allocation x (3) under which buyer 1 is given the good who transfers m 1 + 2ε units of money to the owner and buyer 2 transfers m 2 ε units of money to the owner, where 0 < ε < 1/2. Under x (3), each of the owner and buyer 2 gets ε > 0 higher payoff than under x(3) and buyer 1 gets 1 2ε > 0 higher payoff than under x(3). Hence the latter is not in the core. Now consider an action under which buyer 1 obtains the good and pays one unit of money to the owner and buyer 2 pays no money to the owner. We claim that such an action is in the core. An alternative action in which buyer 2 pays a positive amount but does not obtain the good is not in the core because buyer 2 can improve on this allocation by himself by not paying anything. By the same reasoning, an alternative action in which buyer 1 pays more than one unit to the owner in exchange of the good is not in the core. An alternative action in which buyer 1 obtains the good but pays m 1 < 1 is not in the core because a coalition of the owner and buyer 2 can improve upon it by allocating the good to buyer 2 and buyer 2 transfers m 1 < (1 + m 1 )/2 < 1 units of money to the owner. Under this allocation the owner has more than m 1, that he had before. Buyer 2 has a payoff of m 2 (1 + m 1 )/2 + 1 > m 2. Hence such an alternative cannot be in the core. Thus the core contains two allocations, in each of which the good is given to one of the two buyers and this buyer transfers one unit of money to the owner, the other buyer transfers 0 to the owner. Example 4: In a variant of the above, assume that there are B buyers and L sellers and that money has no value to the buyers and the good has no value to the seller. Each seller holds one unit of the good and has a reservation price of 0. Each buyer wishes to purchase the good and has a reservation price of 1. What is the structure of the core? N = B L. Note that for any coalition S of buyers and sellers, v(s) =min{ S B, S L }

COALITIONAL GAMES 6 where S B = number of elements in the set S B and so on. If B > L, then the core consists of a type of allocation in which every seller receives a payoff of 1 and every buyer receives a payoff of 0.To see this, suppose x is in the core. Let b denote a buyer whose payoff is minimal and l denote a seller whose payoff is minimal under x. Then x b + x l v({b,l})=1. Note that v(n)= L and v(n)=x(n) B x b + L x l =( B L )x b + L (x b +x l ) ( B L )x b + L. Since v(n)= L and B > L, this last inequality implies that x b = 0 and x l 1. Since l is the worst off seller and v(n)= L, each seller gets exactly 1 and each buyer gets exactly 0. Example 5: (Landowner-worker game) Consider a game with n landless workers and a landowner with a single piece of land. The land produces f (k + 1) units of food when used by k workers, where f is an increasing function (note, land is essential). The landowner and each of the worker cares only about the food that he or she receives and prefers more to less. Characterize (as much as possible) the core allocations. We illustrate with n = 2 workers and one landlord (player 1). The only coalitions that can obtain a positive amount of output are that consisting of the landowner by herself, the landowner and one worker, the landowner and both workers. Thus an allocation (x 1,x 2,x 3 ) is in the core if, x 1 f (1) x 2,x 3 0 x 1 + x 2,x 1 + x 3 f (2) x 1 + x 2 + x 3 = f (3) From the last condition we have x 1 = f (3) x 2 x 3, so that we can rewrite the conditions as, 0 x 2 f (3) f (2) 0 x 3 f (3) f (2) x 2 + x 3 f (3) f (1)

COALITIONAL GAMES 7 x 1 + x 2 + x 3 = f (3) That is in a core allocation, each worker obtains at most the marginal output of a worker and the workers together at most obtain the marginal output produced by the two workers. The observation can be generalized to any number of workers. 2.4 House swap game and the top trading cycle mechanism Each member of a group of n people owns a single, indivisible house. There is no market for a house - that is it cannot be sold in exchange for money. Any subgroup may reallocate its members houses in any way it wishes. One person may be assigned one house only. Each person caresonly about the house she obtains. Assume further that each person has a strict (indifference is not allowed) transitive ranking across the houses. What is a stable assignment of the houses? Because every player has the option of simply keeping his or her house, any player who initially owns her favorite (first ranked) house obtains that house in the core. Now consider three players and three houses. The following possibilities are exhaustive. (1) If at least two people initially own their favorite houses, then the core contains the single assignment in which each person keeps his or her own house. (2) If exactly one person, say player i, owns her favorite house, in any core assignment, i keeps her own house. Drop i from the set of players and i s house from the set of houses. The other two players exchange their houses if each prefers the house of the other to his own. The unique core assignment is that player i keeps her house and the other two swaps their houses. If one of them prefers his own house to the other, then in the unique core assignment, each keeps his own house. (3) Now assume that no person owns her favorite house. There are two cases: (i) There is a two-cycle, that is, there are players i and j such that j owns i s favorite and i own s j s favorite. Then the unique core assignment is that i and j swap houses and the third player keeps his own house. (ii) Suppose there is no two-cycle. Denote by o(i) the person who owns playeri s favorite house. Say o(i) = j. Then o( j) =k, where k is the third player. For, otherwise {i, j} is a two-cycle. And o(k)=i, for otherwise { j,k} is a two-cycle. That is, {i, j,k}is a three-cycle and the assignment

COALITIONAL GAMES 8 that i gets j s house, j gets k s house and k gets i s house is in the core. This is not however the unique core assignment as an example below shows. The construction in 3(i) and 3(ii) can be extended to games with any number of players and is described as the top trading cycle mechanism. Under this mechanism, we first search for cycles (cycles of any length, including length one, which indicates that a person owns her favorite house) at the top or first place of the players rankings and assign to each member of the cycle her favorite house. With three players, there can be at most one such cycle (either a two-cycle or a three-cycle). With more than three players, there can be multiple. Having assigned cycle members their favorite houses, eliminate from consideration the players involved in these cycles and their assigned houses. Next look for cycles at the top of the remains of the players rankings and assign to each member of each of these cycles her favorite house amongst those remaining. Continue the process until all houses have been assigned. The following example demonstrates the method. In the example h i denotes the house owned by player i. Each column provides the player s ranking of the houses in descending order. (Blank spaces denote that the rest of the ranking is irrelevant.) Player1 Player2 Player3 Player4 h 2 h 1 h 1 h 3 h 2 h 2 h 4 h 4 h 3 First, we note that {1,2} is a cycle. So we assign them each other s houses. Then we drop h 1, h 2 and players 1 and 2 from consideration. Then we look at Player 3 and 4 s rankings over the remaining houses. We spot a next level cycle with {3,4}. So we assign them each other s houses. Result: For any strict preferences, the core of the house exchange game contains the assignment induced by the top trading cycle procedure. To see why, note that every player assigned a house in the first round receives her favorite house. Hence, no coalition containing such a player can make all its members better off than they are under this assignment. Now consider a coalition that contains players assigned houses in the

COALITIONAL GAMES 9 second round but no players assigned houses in the first round. Such a coalition does not own any of the houses assigned on the first round. Its members who were assigned in the second round obtain their favorite houses among the houses the coalition owns.thus such a coalition has no action that can make all its members better off than under the assignment. We can continue the argument for later rounds. Result: The top trading cycle assignment is not the unique core assignment, as the following example shows. Player1 Player2 Player3 h 3 h 1 h 2 h 2 h 2 h 3 h 1 h 3 h 1 There is a top trading cycle assignment. Consider however the alternative allocation in which player 1 obtains h 2, player 2 obtains h 1 and player 3 obtains h 3. This is also in the core.