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).^{ [1] }

- Mathematical definition
- Harsanyi dividend
- Duality
- Subgames
- Properties for characterization
- Superadditivity
- Monotonicity
- Properties for simple games
- Relation with non-cooperative theory
- Solution concepts
- The stable set
- The core
- The core of a simple game with respect to preferences
- The strong epsilon-core
- The Shapley value
- The kernel
- The nucleolus
- Convex cooperative games
- Properties 4
- Similarities and differences with combinatorial optimization
- See also
- References
- Further reading
- External links

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.^{ [2] }^{ [3] }

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*^{ [4] } 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^{ [5] }) 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

An explicit formula for the dividend is given by . The function is also known as the Möbius inverse of .^{ [6] } Indeed, we can recover from by help of the formula .

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.

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.

A coalitional game v is considered **simple** if payoffs are either 1 or 0, i.e. coalitions are either "winning" or "losing".^{ [7] }

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
- 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^{ [8] }):

- 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^{ [9] } has been made (Kumabe and Mihara, 2011^{ [10] }), whose results are summarized in the Table "Existence of Simple Games" below.

Type | Finite Non-comp | Finite Computable | Infinite Non-comp | Infinite Computable |
---|---|---|---|---|

1111 | No | Yes | Yes | Yes |

1110 | No | Yes | No | No |

1101 | No | Yes | Yes | Yes |

1100 | No | Yes | Yes | Yes |

1011 | No | Yes | Yes | Yes |

1010 | No | No | No | No |

1001 | No | Yes | Yes | Yes |

1000 | No | No | No | No |

0111 | No | Yes | Yes | Yes |

0110 | No | No | No | No |

0101 | No | Yes | Yes | Yes |

0100 | No | Yes | Yes | Yes |

0011 | No | Yes | Yes | Yes |

0010 | No | No | No | No |

0001 | No | Yes | Yes | Yes |

0000 | No | No | No | No |

The restrictions that various axioms for simple games impose on their Nakamura number were also studied extensively.^{ [12] } 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.

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:^{ [13] }

- 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.^{ [14] } 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 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 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.

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.)

Because the core may be empty, a generalization was introduced in ( Shapley & Shubik 1966 ). The *strong -core* for some number is the set of payoff vectors

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 value* is the unique payoff vector that is efficient, symmetric, and satisfies monotonicity.^{ [15] } 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.)

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:

It can be shown (see, e.g., Section V.1 of ( Driessen 1988 )) that the supermodularity of is equivalent to

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 .

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".

In mathematics, in particular abstract algebra, a **graded ring** is a ring such that the underlying additive group is a direct sum of abelian groups such that . The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid. The direct sum decomposition is usually referred to as **gradation** or **grading**.

In the mathematical discipline of set theory, **forcing** is a technique for proving consistency and independence results. It was first used by Paul Cohen in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory.

The **Shapley value** is a solution concept in cooperative game theory. It was named in honor of Lloyd Shapley, who introduced it in 1951 and won the Nobel Prize in Economics for it in 2012. To each cooperative game it assigns a unique distribution of a total surplus generated by the coalition of all players. The Shapley value is characterized by a collection of desirable properties. Hart (1989) provides a survey of the subject.

**Vapnik–Chervonenkis theory** was developed during 1960–1990 by Vladimir Vapnik and Alexey Chervonenkis. The theory is a form of computational learning theory, which attempts to explain the learning process from a statistical point of view.

Game theory is the branch of mathematics in which games are studied: that is, models describing human behaviour. This is a glossary of some terms of the subject.

In cooperative game theory, the **core** is the set of feasible allocations that cannot be improved upon by a subset of the economy's agents. A coalition is said to *improve upon* or *block* a feasible allocation if the members of that coalition are better off under another feasible allocation that is identical to the first except that every member of the coalition has a different consumption bundle that is part of an aggregate consumption bundle that can be constructed from publicly available technology and the initial endowments of each consumer in the coalition.

In game theory, **folk theorems** are a class of theorems describing an abundance of Nash equilibrium payoff profiles in repeated games. The original Folk Theorem concerned the payoffs of all the Nash equilibria of an infinitely repeated game. This result was called the Folk Theorem because it was widely known among game theorists in the 1950s, even though no one had published it. Friedman's (1971) Theorem concerns the payoffs of certain subgame-perfect Nash equilibria (SPE) of an infinitely repeated game, and so strengthens the original Folk Theorem by using a stronger equilibrium concept: subgame-perfect Nash equilibria rather than Nash equilibria.

In game theory, a game is said to be a **potential game** if the incentive of all players to change their strategy can be expressed using a single global function called the **potential function**. The concept originated in a 1996 paper by Dov Monderer and Lloyd Shapley.

In game theory, an **epsilon-equilibrium**, or near-Nash equilibrium, is a strategy profile that approximately satisfies the condition of Nash equilibrium. In a Nash equilibrium, no player has an incentive to change his behavior. In an approximate Nash equilibrium, this requirement is weakened to allow the possibility that a player may have a small incentive to do something different. This may still be considered an adequate solution concept, assuming for example status quo bias. This solution concept may be preferred to Nash equilibrium due to being easier to compute, or alternatively due to the possibility that in games of more than 2 players, the probabilities involved in an exact Nash equilibrium need not be rational numbers.

In game theory, a **stochastic game**, introduced by Lloyd Shapley in the early 1950s, is a repeated game with **probabilistic transitions** played by one or more players. The game is played in a sequence of stages. At the beginning of each stage the game is in some **state**. The players select actions and each player receives a **payoff** that depends on the current state and the chosen actions. The game then moves to a new random state whose distribution depends on the previous state and the actions chosen by the players. The procedure is repeated at the new state and play continues for a finite or infinite number of stages. The total payoff to a player is often taken to be the discounted sum of the stage payoffs or the limit inferior of the averages of the stage payoffs.

The **Bondareva–Shapley theorem**, in game theory, describes a necessary and sufficient condition for the non-emptiness of the core of a cooperative game in characteristic function form. Specifically, the game's core is non-empty if and only if the game is *balanced*. The Bondareva–Shapley theorem implies that market games and convex games have non-empty cores. The theorem was formulated independently by Olga Bondareva and Lloyd Shapley in the 1960s.

In mathematics, a **cardinal function** is a function that returns cardinal numbers.

**Linear production game** is a N-person game in which the value of a coalition can be obtained by solving a linear programming problem. It is widely used in the context of resource allocation and payoff distribution. Mathematically, there are *m* types of resources and *n* products can be produced out of them. Product *j* requires amount of the *kth* resource. The products can be sold at a given market price while the resources themselves can not. Each of the *N* players is given a vector of resources. The value of a coalition *S* is the maximum profit it can achieve with all the resources possessed by its members. It can be obtained by solving a corresponding linear programming problem as follows.

In measure theory, or at least in the approach to it via the domain theory, a **valuation** is a map from the class of open sets of a topological space to the set of positive real numbers including infinity, with certain properties. It is a concept closely related to that of a measure, and as such, it finds applications in measure theory, probability theory, and theoretical computer science.

In decision theory and game theory, **Wald's maximin model** is a non-probabilistic decision-making model according to which decisions are ranked on the basis of their worst-case outcomes – the optimal decision is one with the least bad worst outcome. It is one of the most important models in robust decision making in general and robust optimization in particular.

In cooperative game theory and social choice theory, the **Nakamura number** measures the degree of rationality of preference aggregation rules, such as voting rules. It is an indicator of the extent to which an aggregation rule can yield well-defined choices.

**Jean-François Mertens** was a Belgian game theorist and mathematical economist.

Irrigation games are cooperative games which model cost sharing problems on networks. The **irrigation game** is a transferable utility game assigned to a cost-tree problem. A common example of this cost-tree problems are the irrigation networks. The irrigation ditch is represented by a graph, its nodes are water users, the edges are sections of the ditch. There is a cost of maintaining the ditch, and we are looking for the fair division of the costs among the users. The irrigation games are mentioned first by Aadland and Kolpin 1998, but the formal concept and the characterization of the game class is introduced by Márkus et al. 2011.

In game theory, an **aggregative game** is a game in which every player’s payoff is a function of the player’s own strategy and the aggregate of all players’ strategies. The concept was first proposed by Nobel laureate Reinhard Selten in 1970 who considered the case where the aggregate is the sum of the players' strategies.

In cooperative game theory, a **hedonic game** is a game that models the formation of coalitions (groups) of players when players have preferences over which group they belong to. A hedonic game is specified by giving a finite set of players, and, for each player, a preference ranking over all coalitions (subsets) of players that the player belongs to. The outcome of a hedonic game consists of a partition of the players into disjoint coalitions, that is, each player is assigned a unique group. Such partitions are often referred to as coalition structures.

- ↑ Shor, Mike. "Non-Cooperative Game - Game Theory .net".
*www.gametheory.net*. Retrieved 2016-09-15. - ↑ Chandrasekaran, R. "Cooperative Game Theory" (PDF).
- ↑ Brandenburger, Adam. "Cooperative Game Theory: Characteristic Functions, Allocations, Marginal Contribution" (PDF). Archived from the original (PDF) on 2016-05-27.
- ↑ denotes the power set of .
- ↑ Harsanyi, John C. (1982). "A Simplified Bargaining Model for the n-Person Cooperative Game".
*Papers in Game Theory*. Theory and Decision Library. Springer, Dordrecht. pp. 44–70. doi:10.1007/978-94-017-2527-9_3. ISBN 9789048183692. - ↑
*Set Functions, Games and Capacities in Decision Making | Michel Grabisch | Springer*. Theory and Decision Library C. Springer. 2016. ISBN 9783319306889. - ↑ Georgios Chalkiadakis; Edith Elkind; Michael J. Wooldridge (25 October 2011).
*Computational Aspects of Cooperative Game Theory*. Morgan & Claypool Publishers. ISBN 978-1-60845-652-9. - ↑ Peleg, B. (2002). "Chapter 8 Game-theoretic analysis of voting in committees".
*Handbook of Social Choice and Welfare Volume 1*. Handbook of Social Choice and Welfare.**1**. pp. 395–423. doi:10.1016/S1574-0110(02)80012-1. ISBN 9780444829146. - ↑ See a section for Rice's theorem for the definition of a computable simple game. In particular, all finite games are computable.
- ↑ Kumabe, M.; Mihara, H. R. (2011). "Computability of simple games: A complete investigation of the sixty-four possibilities" (PDF).
*Journal of Mathematical Economics*.**47**(2): 150–158. arXiv: 1102.4037 . Bibcode:2011arXiv1102.4037K. doi:10.1016/j.jmateco.2010.12.003. S2CID 775278. - ↑ Modified from Table 1 in Kumabe and Mihara (2011). The sixteen types are defined by the four conventional axioms (monotonicity, properness, strongness, and non-weakness). For example, type 1110 indicates monotonic (1), proper (1), strong (1), weak (0, because not nonweak) games. Among type 1110 games, there exist no finite non-computable ones, there exist finite computable ones, there exist no infinite non-computable ones, and there exist no infinite computable ones. Observe that except for type 1110, the last three columns are identical.
- ↑ Kumabe, M.; Mihara, H. R. (2008). "The Nakamura numbers for computable simple games".
*Social Choice and Welfare*.**31**(4): 621. arXiv: 1107.0439 . doi:10.1007/s00355-008-0300-5. S2CID 8106333. - ↑ Aumann, Robert J. "The core of a cooperative game without side payments." Transactions of the American Mathematical Society (1961): 539-552.
- ↑ Peters, Hans (2008).
*Game theory: a multi-leveled approach*. Springer. pp. 123. doi:10.1007/978-3-540-69291-1_17. ISBN 978-3-540-69290-4. - ↑ Young, H. P. (1985-06-01). "Monotonic solutions of cooperative games".
*International Journal of Game Theory*.**14**(2): 65–72. doi:10.1007/BF01769885. ISSN 0020-7276. S2CID 122758426.

- Bilbao, Jesús Mario (2000),
*Cooperative Games on Combinatorial Structures*, Kluwer Academic Publishers, ISBN 9781461543930

- Davis, M.; Maschler, M. (1965), "The kernel of a cooperative game",
*Naval Research Logistics Quarterly*,**12**(3): 223–259, doi:10.1002/nav.3800120303

- Driessen, Theo (1988),
*Cooperative Games, Solutions and Applications*, Kluwer Academic Publishers, ISBN 9789401577878

- Edmonds, Jack (1970), "Submodular functions, matroids and certain polyhedra", in Guy, R.; Hanani, H.; Sauer, N.; Schönheim, J. (eds.),
*Combinatorial Structures and Their Applications*, New York: Gordon and Breach, pp. 69–87

- Lovász, László (1983), "Submodular functions and convexity", in Bachem, A.; Grötschel, M.; Korte, B. (eds.),
*Mathematical Programming—The State of the Art*, Berlin: Springer, pp. 235–257

- Leyton-Brown, Kevin; Shoham, Yoav (2008),
*Essentials of Game Theory: A Concise, Multidisciplinary Introduction*, San Rafael, CA: Morgan & Claypool Publishers, ISBN 978-1-59829-593-1 . An 88-page mathematical introduction; see Chapter 8. Free online (subscription required) at many universities.

- Lucas, William F. (1969), "The Proof That a Game May Not Have a Solution",
*Transactions of the American Mathematical Society*,**136**: 219–229, doi: 10.2307/1994798 , JSTOR 1994798.

- Lucas, William F. (1992), "Von Neumann-Morgenstern Stable Sets", in Aumann, Robert J.; Hart, Sergiu (eds.),
*Handbook of Game Theory, Volume I*, Amsterdam: Elsevier, pp. 543–590

- Luce, R.D. and Raiffa, H. (1957)
*Games and Decisions: An Introduction and Critical Survey*, Wiley & Sons. (see Chapter 8). - Maschler, M.; Peleg, B.; Shapley, Lloyd S. (1979), "Geometric properties of the kernel, nucleolus, and related solution concepts",
*Mathematics of Operations Research*,**4**(4): 303–338, doi:10.1287/moor.4.4.303

- Osborne, M.J. and Rubinstein, A. (1994)
*A Course in Game Theory*, MIT Press (see Chapters 13,14,15) - Moulin, Herve (1988),
*Axioms of Cooperative Decision Making*(1st ed.), Cambridge: Cambridge University Press, ISBN 978-0-521-42458-5

- Owen, Guillermo (1995),
*Game Theory*(3rd ed.), San Diego: Academic Press, ISBN 978-0-12-531151-9

- Schmeidler, D. (1969), "The nucleolus of a characteristic function game",
*SIAM Journal on Applied Mathematics*,**17**(6): 1163–1170, doi:10.1137/0117107.

- Shapley, Lloyd S. (1953), "A value for -person games", in Kuhn, H.; Tucker, A.W. (eds.),
*Contributions to the Theory of Games II*, Princeton, New Jersey: Princeton University Press, pp. 307–317

- Shapley, Lloyd S. (1971), "Cores of convex games",
*International Journal of Game Theory*,**1**(1): 11–26, doi:10.1007/BF01753431, S2CID 123385556

- Shapley, Lloyd S.; Shubik, M. (1966), "Quasi-cores in a monetary economy with non-convex preferences",
*Econometrica*,**34**(4): 805–827, doi:10.2307/1910101, JSTOR 1910101

- Shoham, Yoav; Leyton-Brown, Kevin (2009),
*Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations*, New York: Cambridge University Press, ISBN 978-0-521-89943-7 . A comprehensive reference from a computational perspective; see Chapter 12. Downloadable free online.

- von Neumann, John; Morgenstern, Oskar (1944), "Theory of Games and Economic Behavior",
*Nature*, Princeton: Princeton University Press,**157**(3981): 172, Bibcode:1946Natur.157..172R, doi:10.1038/157172a0, S2CID 29754824

- Yeung, David W.K. and Leon A. Petrosyan. Cooperative Stochastic Differential Games (Springer Series in Operations Research and Financial Engineering), Springer, 2006. Softcover- ISBN 978-1441920942.
- Yeung, David W.K. and Leon A. Petrosyan. Subgame Consistent Economic Optimization: An Advanced Cooperative Dynamic Game Analysis (Static & Dynamic Game Theory: Foundations & Applications), Birkhäuser Boston; 2012. ISBN 978-0817682613

- "Cooperative game",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994]

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.