Cooperative game theory
In game theory, a cooperative game (or coalitional game) is a game with competition between groups of players ("coalitions") due to the possibility of external enforcement of cooperative behavior (e.g. through contract law). Those are opposed to non-cooperative games in which there is either no possibility to forge alliances or all agreements need to be self-enforcing (e.g. through credible threats).
Cooperative games are often analysed through the framework of cooperative game theory, which focuses on predicting which coalitions will form, the joint actions that groups take and the resulting collective payoffs. It is opposed to the traditional non-cooperative game theory which focuses on predicting individual players' actions and payoffs and analyzing Nash equilibria.
Cooperative game theory provides a high-level approach as it only describes the structure, strategies and payoffs of coalitions, whereas non-cooperative game theory also looks at how bargaining procedures will affect the distribution of payoffs within each coalition. As non-cooperative game theory is more general, cooperative games can be analyzed through the approach of non-cooperative game theory (the converse does not hold) provided that sufficient assumptions are made to encompass all the possible strategies available to players due to the possibility of external enforcement of cooperation. While it would thus be possible to have all games expressed under a non-cooperative framework, in many instances insufficient information is available to accurately model the formal procedures available to the players during the strategic bargaining process, or the resulting model would be of too high complexity to offer a practical tool in the real world. In such cases, cooperative game theory provides a simplified approach that allows the analysis of the game at large without having to make any assumption about bargaining powers.
A cooperative game is given by specifying a value for every coalition. Formally, the coalitional game consists of a finite set of players , called the grand coalition, and a characteristic function  from the set of all possible coalitions of players to a set of payments that satisfies . The function describes how much collective payoff a set of players can gain by forming a coalition, and the game is sometimes called a value game or a profit game.
Conversely, a cooperative game can also be defined with a characteristic cost function satisfying . In this setting, players must accomplish some task, and the characteristic function represents the cost of a set of players accomplishing the task together. A game of this kind is known as a cost game. Although most cooperative game theory deals with profit games, all concepts can easily be translated to the cost setting.
The Harsanyi dividend (named after John Harsanyi, who used it to generalize the Shapley value in 1963) identifies the surplus that is created by a coalition of players in a cooperative game. To specify this surplus, the worth of this coalition is corrected by the surplus that is already created by subcoalitions. To this end, the dividend of coalition in game is recursively determined by
Harsanyi dividends are useful for analyzing both games and solution concepts, e.g. the Shapley value is obtained by distributing the dividend of each coalition among its members, i.e., the Shapley value of player in game is given by summing up a player's share of the dividends of all coalitions that she belongs to, .
Let be a profit game. The dual game of is the cost game defined as
Intuitively, the dual game represents the opportunity cost for a coalition of not joining the grand coalition . A dual profit game can be defined identically for a cost game . A cooperative game and its dual are in some sense equivalent, and they share many properties. For example, the core of a game and its dual are equal. For more details on cooperative game duality, see for instance (Bilbao 2000).
Let be a non-empty coalition of players. The subgame on is naturally defined as
In other words, we simply restrict our attention to coalitions contained in . Subgames are useful because they allow us to apply solution concepts defined for the grand coalition on smaller coalitions.
Properties for characterizationEdit
Characteristic functions are often assumed to be superadditive (Owen 1995, p. 213). This means that the value of a union of disjoint coalitions is no less than the sum of the coalitions' separate values:
whenever satisfy .
Larger coalitions gain more:
This follows from superadditivity. i.e. if payoffs are normalized so singleton coalitions have zero value.
Properties for simple gamesEdit
A coalitional game v is considered simple if payoffs are either 1 or 0, i.e. coalitions are either "winning" or "losing".
Equivalently, a simple game can be defined as a collection W of coalitions, where the members of W are called winning coalitions, and the others losing coalitions. It is sometimes assumed that a simple game is nonempty or that it does not contain an empty set. However, in other areas of mathematics, simple games are also called hypergraphs or Boolean functions (logic functions).
- A simple game W is monotonic if any coalition containing a winning coalition is also winning, that is, if and imply .
- A simple game W is proper if the complement (opposition) of any winning coalition is losing, that is, if implies .
- A simple game W is strong if the complement of any losing coalition is winning, that is, if implies .
- If a simple game W is proper and strong, then a coalition is winning if and only if its complement is losing, that is, iff . (If v is a coalitional simple game that is proper and strong, for any S.)
- A veto player (vetoer) in a simple game is a player that belongs to all winning coalitions. Supposing there is a veto player, any coalition not containing a veto player is losing. A simple game W is weak (collegial) if it has a veto player, that is, if the intersection of all winning coalitions is nonempty.
- A dictator in a simple game is a veto player such that any coalition containing this player is winning. The dictator does not belong to any losing coalition. (Dictator games in experimental economics are unrelated to this.)
- A carrier of a simple game W is a set such that for any coalition S, we have iff . When a simple game has a carrier, any player not belonging to it is ignored. A simple game is sometimes called finite if it has a finite carrier (even if N is infinite).
- The Nakamura number of a simple game is the minimal number of winning coalitions with empty intersection. According to Nakamura's theorem, the number measures the degree of rationality; it is an indicator of the extent to which an aggregation rule can yield well-defined choices.
A few relations among the above axioms have widely been recognized, such as the following (e.g., Peleg, 2002, Section 2.1):
- If a simple game is weak, it is proper.
- A simple game is dictatorial if and only if it is strong and weak.
More generally, a complete investigation of the relation among the four conventional axioms (monotonicity, properness, strongness, and non-weakness), finiteness, and algorithmic computability has been made (Kumabe and Mihara, 2011), whose results are summarized in the Table "Existence of Simple Games" below.
|Type||Finite Non-comp||Finite Computable||Infinite Non-comp||Infinite Computable|
The restrictions that various axioms for simple games impose on their Nakamura number were also studied extensively. In particular, a computable simple game without a veto player has a Nakamura number greater than 3 only if it is a proper and non-strong game.
Relation with non-cooperative theoryEdit
Let G be a strategic (non-cooperative) game. Then, assuming that coalitions have the ability to enforce coordinated behaviour, there are several cooperative games associated with G. These games are often referred to as representations of G. The two standard representations are:
- The α-effective game associates with each coalition the sum of gains its members can 'guarantee' by joining forces. By 'guaranteeing', it is meant that the value is the max-min, e.g. the maximal value of the minimum taken over the opposition's strategies.
- The β-effective game associates with each coalition the sum of gains its members can 'strategically guarantee' by joining forces. By 'strategically guaranteeing', it is meant that the value is the min-max, e.g. the minimal value of the maximum taken over the opposition's strategies.
The main assumption in cooperative game theory is that the grand coalition will form. The challenge is then to allocate the payoff among the players in some fair way. (This assumption is not restrictive, because even if players split off and form smaller coalitions, we can apply solution concepts to the subgames defined by whatever coalitions actually form.) A solution concept is a vector (or a set of vectors) that represents the allocation to each player. Researchers have proposed different solution concepts based on different notions of fairness. Some properties to look for in a solution concept include:
- Efficiency: The payoff vector exactly splits the total value: .
- Individual rationality: No player receives less than what he could get on his own: .
- Existence: The solution concept exists for any game .
- Uniqueness: The solution concept is unique for any game .
- Marginality: The payoff of a player depends only on the marginal contribution of this player, i.e., if these marginal contributions are the same in two different games, then the payoff is the same: implies that is the same in and in .
- Monotonicity: The payoff of a player increases if the marginal contribution of this player increase: implies that is weakly greater in than in .
- Computational ease: The solution concept can be calculated efficiently (i.e. in polynomial time with respect to the number of players .)
- Symmetry: The solution concept allocates equal payments to symmetric players , . Two players , are symmetric if ; that is, we can exchange one player for the other in any coalition that contains only one of the players and not change the payoff.
- Additivity: The allocation to a player in a sum of two games is the sum of the allocations to the player in each individual game. Mathematically, if and are games, the game simply assigns to any coalition the sum of the payoffs the coalition would get in the two individual games. An additive solution concept assigns to every player in the sum of what he would receive in and .
- Zero Allocation to Null Players: The allocation to a null player is zero. A null player satisfies . In economic terms, a null player's marginal value to any coalition that does not contain him is zero.
An efficient payoff vector is called a pre-imputation, and an individually rational pre-imputation is called an imputation. Most solution concepts are imputations.
The stable setEdit
The stable set of a game (also known as the von Neumann-Morgenstern solution (von Neumann & Morgenstern 1944)) was the first solution proposed for games with more than 2 players. Let be a game and let , be two imputations of . Then dominates if some coalition satisfies and . In other words, players in prefer the payoffs from to those from , and they can threaten to leave the grand coalition if is used because the payoff they obtain on their own is at least as large as the allocation they receive under .
A stable set is a set of imputations that satisfies two properties:
- Internal stability: No payoff vector in the stable set is dominated by another vector in the set.
- External stability: All payoff vectors outside the set are dominated by at least one vector in the set.
Von Neumann and Morgenstern saw the stable set as the collection of acceptable behaviours in a society: None is clearly preferred to any other, but for each unacceptable behaviour there is a preferred alternative. The definition is very general allowing the concept to be used in a wide variety of game formats.
- A stable set may or may not exist (Lucas 1969), and if it exists it is typically not unique (Lucas 1992). Stable sets are usually difficult to find. This and other difficulties have led to the development of many other solution concepts.
- A positive fraction of cooperative games have unique stable sets consisting of the core (Owen 1995, p. 240).
- A positive fraction of cooperative games have stable sets which discriminate players. In such sets at least of the discriminated players are excluded (Owen 1995, p. 240).
Let be a game. The core of is the set of payoff vectors
In words, the core is the set of imputations under which no coalition has a value greater than the sum of its members' payoffs. Therefore, no coalition has incentive to leave the grand coalition and receive a larger payoff.
- The core of a game may be empty (see the Bondareva–Shapley theorem). Games with non-empty cores are called balanced.
- If it is non-empty, the core does not necessarily contain a unique vector.
- The core is contained in any stable set, and if the core is stable it is the unique stable set; see (Driessen 1988) for a proof.
The core of a simple game with respect to preferencesEdit
For simple games, there is another notion of the core, when each player is assumed to have preferences on a set of alternatives. A profile is a list of individual preferences on . Here means that individual prefers alternative to at profile . Given a simple game and a profile , a dominance relation is defined on by if and only if there is a winning coalition (i.e., ) satisfying for all . The core of the simple game with respect to the profile of preferences is the set of alternatives undominated by (the set of maximal elements of with respect to ):
- if and only if there is no such that .
The Nakamura number of a simple game is the minimal number of winning coalitions with empty intersection. Nakamura's theorem states that the core is nonempty for all profiles of acyclic (alternatively, transitive) preferences if and only if is finite and the cardinal number (the number of elements) of is less than the Nakamura number of . A variant by Kumabe and Mihara states that the core is nonempty for all profiles of preferences that have a maximal element if and only if the cardinal number of is less than the Nakamura number of . (See Nakamura number for details.)
The strong epsilon-coreEdit
In economic terms, the strong -core is the set of pre-imputations where no coalition can improve its payoff by leaving the grand coalition, if it must pay a penalty of for leaving. Note that may be negative, in which case it represents a bonus for leaving the grand coalition. Clearly, regardless of whether the core is empty, the strong -core will be non-empty for a large enough value of and empty for a small enough (possibly negative) value of . Following this line of reasoning, the least-core, introduced in (Maschler, Peleg & Shapley 1979), is the intersection of all non-empty strong -cores. It can also be viewed as the strong -core for the smallest value of that makes the set non-empty (Bilbao 2000).
The Shapley valueEdit
The Shapley value is the unique payoff vector that is efficient, symmetric, and satisfies monotonicity. It was introduced by Lloyd Shapley (Shapley 1953) who showed that it is the unique payoff vector that is efficient, symmetric, additive, and assigns zero payoffs to dummy players. The Shapley value of a superadditive game is individually rational, but this is not true in general. (Driessen 1988)
Let be a game, and let be an efficient payoff vector. The maximum surplus of player i over player j with respect to x is
the maximal amount player i can gain without the cooperation of player j by withdrawing from the grand coalition N under payoff vector x, assuming that the other players in i's withdrawing coalition are satisfied with their payoffs under x. The maximum surplus is a way to measure one player's bargaining power over another. The kernel of is the set of imputations x that satisfy
- , and
for every pair of players i and j. Intuitively, player i has more bargaining power than player j with respect to imputation x if , but player j is immune to player i's threats if , because he can obtain this payoff on his own. The kernel contains all imputations where no player has this bargaining power over another. This solution concept was first introduced in (Davis & Maschler 1965).
Let be a game, and let be a payoff vector. The excess of for a coalition is the quantity ; that is, the gain that players in coalition can obtain if they withdraw from the grand coalition under payoff and instead take the payoff .
Now let be the vector of excesses of , arranged in non-increasing order. In other words, . Notice that is in the core of if and only if it is a pre-imputation and . To define the nucleolus, we consider the lexicographic ordering of vectors in : For two payoff vectors , we say is lexicographically smaller than if for some index , we have and . (The ordering is called lexicographic because it mimics alphabetical ordering used to arrange words in a dictionary.) The nucleolus of is the lexicographically minimal imputation, based on this ordering. This solution concept was first introduced in (Schmeidler 1969).
Although the definition of the nucleolus seems abstract, (Maschler, Peleg & Shapley 1979) gave a more intuitive description: Starting with the least-core, record the coalitions for which the right-hand side of the inequality in the definition of cannot be further reduced without making the set empty. Continue decreasing the right-hand side for the remaining coalitions, until it cannot be reduced without making the set empty. Record the new set of coalitions for which the inequalities hold at equality; continue decreasing the right-hand side of remaining coalitions and repeat this process as many times as necessary until all coalitions have been recorded. The resulting payoff vector is the nucleolus.
- Although the definition does not explicitly state it, the nucleolus is always unique. (See Section II.7 of (Driessen 1988) for a proof.)
- If the core is non-empty, the nucleolus is in the core.
- The nucleolus is always in the kernel, and since the kernel is contained in the bargaining set, it is always in the bargaining set (see (Driessen 1988) for details.)
Convex cooperative gamesEdit
Introduced by Shapley in (Shapley 1971), convex cooperative games capture the intuitive property some games have of "snowballing". Specifically, a game is convex if its characteristic function is supermodular:
that is, "the incentives for joining a coalition increase as the coalition grows" (Shapley 1971), leading to the aforementioned snowball effect. For cost games, the inequalities are reversed, so that we say the cost game is convex if the characteristic function is submodular.
Convex cooperative games have many nice properties:
- Supermodularity trivially implies superadditivity.
- Convex games are totally balanced: The core of a convex game is non-empty, and since any subgame of a convex game is convex, the core of any subgame is also non-empty.
- A convex game has a unique stable set that coincides with its core.
- The Shapley value of a convex game is the center of gravity of its core.
- An extreme point (vertex) of the core can be found in polynomial time using the greedy algorithm: Let be a permutation of the players, and let be the set of players ordered through in , for any , with . Then the payoff defined by is a vertex of the core of . Any vertex of the core can be constructed in this way by choosing an appropriate permutation .
Similarities and differences with combinatorial optimizationEdit
Submodular and supermodular set functions are also studied in combinatorial optimization. Many of the results in (Shapley 1971) have analogues in (Edmonds 1970), where submodular functions were first presented as generalizations of matroids. In this context, the core of a convex cost game is called the base polyhedron, because its elements generalize base properties of matroids.
However, the optimization community generally considers submodular functions to be the discrete analogues of convex functions (Lovász 1983), because the minimization of both types of functions is computationally tractable. Unfortunately, this conflicts directly with Shapley's original definition of supermodular functions as "convex".
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