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| : ''This article is about a part of game theory. For video gaming, see [[Cooperative gameplay]]. For the similar feature in some board games, see [[cooperative board game]]''
| | Hеllo! <br>I'm Portuguese male :). <br>I really like Darts!<br><br>Stop by my webpage: [http://vercandis.buzznet.com/user/journal/18807377/fap-turbo-details-fapturbo-trader/ salemsharkslive] |
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| In [[game theory]], a '''cooperative game''' is a game where groups of players ("coalitions") may enforce cooperative behaviour, hence the game is a competition between ''coalitions'' of players, rather than between individual players. An example is a [[coordination game]], when players choose the strategies by a [[consensus decision-making]] process.
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| Recreational games are rarely cooperative, because they usually lack mechanisms by which coalitions may enforce coordinated behaviour on the members of the coalition. Such mechanisms, however, are abundant in real life situations (e.g. contract law).
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| ==Mathematical definition==
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| A cooperative game is given by specifying a value for every coalition. Formally, the game ('''coalitional game''') consists of a finite set of players <math> N </math>, called the ''grand coalition'', and a ''characteristic function'' <math> v : 2^N \to \mathbb{R} </math> <ref><math>2^N</math> denotes the [[power set]] of <math>N</math>.</ref> from the set of all possible coalitions of players to a set of payments that satisfies <math> v( \emptyset ) = 0 </math>. 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''. The players are assumed to choose which coalitions to form, according to their estimate of the way the payment will be divided among coalition members.
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| Conversely, a cooperative game can also be defined with a characteristic cost function <math> c: 2^N \to \mathbb{R} </math> satisfying <math> c( \emptyset ) = 0 </math>. In this setting, players must accomplish some task, and the characteristic function <math> c </math> 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.
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| ===Duality===
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| Let <math> v </math> be a profit game. The ''dual game'' of <math> v </math> is the cost game <math> v^* </math> defined as
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| : <math> v^*(S) = v(N) - v( N \setminus S ), \forall~ S \subseteq N.\, </math>
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| Intuitively, the dual game represents the [[opportunity cost]] for a coalition <math> S </math> of not joining the grand coalition <math> N </math>. A dual profit game <math> c^* </math> can be defined identically for a cost game <math> c </math>. A cooperative game and its dual are in some sense equivalent, and they share many properties. For example, the [[Core (economics)|core]] of a game and its dual are equal. For more details on cooperative game duality, see for instance {{harv|Bilbao|2000}}.
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| ===Subgames===
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| Let <math> S \subsetneq N </math> be a non-empty coalition of players. The ''subgame'' <math> v_S : 2^S \to \mathbb{R} </math> on <math> S </math> is naturally defined as
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| : <math> v_S(T) = v(T), \forall~ T \subseteq S.\, </math>
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| In other words, we simply restrict our attention to coalitions contained in <math> S </math>. Subgames are useful because they allow us to apply [[#Solution concepts|solution concepts]] defined for the grand coalition on smaller coalitions.
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| ==Properties for characterization==
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| ===Superadditivity===
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| Characteristic functions are often assumed to be [[superadditive]] {{harv|Owen|1995|p=213}}. This means that the value of a union of [[disjoint sets|disjoint]] coalitions is no less than the sum of the coalitions' separate values:
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| <math> v ( S \cup T ) \geq v (S) + v (T) </math> whenever <math> S, T \subseteq N </math> satisfy <math> S \cap T = \emptyset </math>.
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| ===Monotonicity===
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| Larger coalitions gain more: <math> S \subseteq T \Rightarrow v (S) \le v (T) </math>. This follows from [[superadditive|superadditivity]] if payoffs are normalized so singleton coalitions have value zero.
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| ===Properties for simple games===
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| A coalitional game <math>v</math> is '''simple''' if payoffs are either 1 or 0, i.e., coalitions are either "winning" or "losing".
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| Equivalently, a '''simple game''' can be defined as a collection <math>W</math> of coalitions,
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| where the members of <math>W</math> are called '''winning''' coalitions, and the others '''losing''' coalitions.
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| It is sometimes assumed that a simple game is nonempty or that it does not contain an empty set.
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| In other areas of mathematics, simple games are also called [[hypergraph]]s or [[Boolean functions]] (logic functions).
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| *A simple game <math>W</math> is '''monotonic''' if any coalition containing a winning coalition is also winning, that is, if <math>S \in W</math> and <math>S\subseteq T</math> imply <math>T \in W</math>.
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| *A simple game <math>W</math> is '''proper''' if the complement (opposition) of any winning coalition is losing, that is, if <math>S \in W</math> implies <math>N\setminus S \notin W</math>.
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| *A simple game <math>W</math> is '''strong''' if the complement of any losing coalition is winning, that is, if <math>S \notin W</math> imples<math>N\setminus S \in W</math>.
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| **If a simple game <math>W</math> is proper and strong, then a coalition is winning if and only if its complement is losing, that is, <math>S \in W</math> iff <math>N\setminus S \notin W</math>. (If <math>v</math> is a colitional simple game that is proper and strong, <math>v(S) = 1 - v(N \setminus S)</math> for any <math>S</math>.)
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| *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 <math>W</math> is '''weak''' (''collegial'') if it has a veto player, that is, if the intersection <math>\bigcap W := \bigcap_{S\in W} S</math> of all winning coalitions is nonempty.
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| **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 game]]s in experimental economics are unrelated to this.)
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| *A '''carrier''' of a simple game <math>W</math> is a set <math>T \subseteq N</math> such that for any coalition <math>S</math>, we have <math>S \in W</math> iff <math>S\cap T \in W</math>. 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 <math>N</math> is infinite).
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| *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.
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| A few relations among the above axioms have widely been recognized, such as the following
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| (e.g., Peleg, 2002, Section 2.1<ref name=peleg02hbscw>{{cite doi | 10.1016/S1574-0110(02)80012-1}}</ref>):
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| *If a simple game is weak, it is proper.
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| *A simple game is dictatorial if and only if it is strong and weak.
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| More generally, a complete investigation of the relation among the four conventional axioms
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| (monotonicity, properness, strongness, and non-weakness), finiteness, and algorithmic ''computability''<ref>See
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| [[Rice%27s_theorem#An_analogue_of_Rice.27s_theorem_for_recursive_sets|a section for Rice's theorem]]
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| for the definition of a ''computable'' simple game. In particular, all finite games are computable.</ref>
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| has been made (Kumabe and Mihara, 2011<ref name=kumabe-m11jme>{{cite doi | 10.1016/j.jmateco.2010.12.003}}</ref>),
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| whose results are summarized in the Table "Existence of Simple Games" below.
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| {| class="wikitable"
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| |+ Existence of Simple Games<ref>Modified from Table 1 in Kumabe and Mihara (2011).
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| The sixteen '''Types''' are defined by the four conventional axioms (monotonicity, properness, strongness, and non-weakness).
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| For example, type 1110 indicates monotonic (1), proper (1), strong (1), weak (0, because not nonweak) games.
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| 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.
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| Observe that except for type 1110, the last three columns are identical.
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| </ref>
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| |-
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| ! Type | |
| ! Finite Non-comp
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| ! Finite Computable
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| ! Infinite Non-comp
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| ! Infinite Computable
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| |-
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| | 1111
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| | no
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| | yes
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| | yes
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| | yes
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| |-
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| | 1110
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| | no
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| | yes
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| | no
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| | no
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| |-
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| | 1101
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| | no
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| | yes
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| | yes
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| | yes
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| |-
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| | 1100
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| | no
| |
| | yes
| |
| | yes
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| | yes
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| |-
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| | 1011
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| | no
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| | yes
| |
| | yes
| |
| | yes
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| |-
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| | 1010
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| | no
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| | no
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| | no
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| | no
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| |-
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| | 1001
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| | no
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| | yes
| |
| | yes
| |
| | yes
| |
| |-
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| | 1000
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| | no
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| | no
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| | no
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| | no
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| |-
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| | 0111
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| | no
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| | yes
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| | yes
| |
| | yes
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| |-
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| | 0110
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| | no
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| | no
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| | no
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| | no
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| |-
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| | 0101
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| | no
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| | yes
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| | yes
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| | yes
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| |-
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| | 0100
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| | no
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| | yes
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| | yes
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| | yes
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| |-
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| | 0011
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| | no
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| | yes
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| | yes
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| | yes
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| |-
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| | 0010
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| | no
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| | no
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| | no
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| | no
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| |-
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| | 0001
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| | no
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| | yes
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| | yes
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| | yes
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| |-
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| | 0000
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| | no
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| | no
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| | no
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| | no
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| |}
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| The restrictions that various axioms for simple games impose on their Nakamura number are also studied extensively.<ref name= kumabe-m08scw>{{cite doi| 10.1007/s00355-008-0300-5}}</ref>
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| In particular, a computable simple game without a veto player has a Nakamura number greater than 3
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| only if it is proper and non-strong.
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| ==Relation with non-cooperative theory==
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| 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''.
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| * 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.
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| * 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.
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| ==Solution concepts==
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| The main assumption in cooperative game theory is that the grand coalition <math> N </math> will form. The challenge is then to allocate the payoff <math> v(N) </math> 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 <math> x \in \mathbb{R}^N </math> 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:
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| * Efficiency: The payoff vector exactly splits the total value: <math> \sum_{ i \in N } x_i = v(N) </math>.
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| * Individual rationality: No player receives less than what he could get on his own: <math> x_i \geq v(\{i\}), \forall~ i \in N </math>.
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| * Existence: The solution concept exists for any game <math> v </math>.
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| * Uniqueness: The solution concept is unique for any game <math> v </math>.
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| * Computational ease: The solution concept can be calculated efficiently (i.e. in polynomial time with respect to the number of players <math> |N| </math>.)
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| * Symmetry: The solution concept <math> x </math> allocates equal payments <math> x_i = x_j </math> to symmetric players <math> i </math>, <math> j </math>. Two players <math> i </math>, <math> j </math> are ''symmetric'' if <math> v( S \cup \{ i \} ) = v( S \cup \{ j \} ), \forall~ S \subseteq N \setminus \{ i, j \} </math>; 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.
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| * 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 <math> v </math> and <math> \omega </math> are games, the game <math> ( v + \omega ) </math> 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 <math> ( v + \omega ) </math> the sum of what he would receive in <math> v </math> and <math> \omega </math>.
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| * Zero Allocation to Null Players: The allocation to a null player is zero. A ''null player'' <math> i </math> satisfies <math> v( S \cup \{ i \} ) = v( S ), \forall~ S \subseteq N \setminus \{ i \} </math>. In economic terms, a null player's marginal value to any coalition that does not contain him is zero.
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| An efficient payoff vector is called a ''pre-imputation'', and an individually rational pre-imputation is called an [[Imputation (game theory)|imputation]]. Most solution concepts are imputations.
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| === The stable set ===
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| The stable set of a game (also known as the ''von Neumann-Morgenstern solution'' {{harv|von Neumann|Morgenstern|1944}}) was the first solution proposed for games with more than 2 players. Let <math> v </math> be a game and let <math> x </math>, <math> y </math> be two [[Imputation (game theory)|imputations]] of <math> v </math>. Then <math> x </math> ''dominates'' <math> y </math> if some coalition <math> S \neq \emptyset </math> satisfies <math> x_i > y _i, \forall~ i \in S </math> and <math> \sum_{ i \in S } x_i \leq v(S) </math>. In other words, players in <math> S </math> prefer the payoffs from <math> x </math> to those from <math> y </math>, and they can threaten to leave the grand coalition if <math> y </math> is used because the payoff they obtain on their own is at least as large as the allocation they receive under <math> x </math>.
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| A ''stable set'' is a set of [[Imputation (game theory)|imputations]] that satisfies two properties:
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| * Internal stability: No payoff vector in the stable set is dominated by another vector in the set.
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| * External stability: All payoff vectors outside the set are dominated by at least one vector in the set.
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| 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.
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| ====Properties====
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| * A stable set may or may not exist {{harv|Lucas|1969}}, and if it exists it is typically not unique {{harv|Lucas|1992}}. Stable sets are usually difficult to find. This and other difficulties have led to the development of many other solution concepts.
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| * A positive fraction of cooperative games have unique stable sets consisting of the [[Core (economics)|core]] {{harv|Owen|1995|p=240.}}.
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| * A positive fraction of cooperative games have stable sets which discriminate <math>n-2</math> players. In such sets at least <math>n-3</math> of the discriminated players are excluded {{harv|Owen|1995|p=240.}}.
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| === The core ===
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| {{main|Core (economics)}}
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| Let <math> v </math> be a game. The [[Core (economics)|''core'']] of <math> v </math> is the set of payoff vectors
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| : <math> C( v ) = \left\{ x \in \mathbb{R}^N: \sum_{ i \in N } x_i = v(N); \quad \sum_{ i \in S } x_i \geq v(S), \forall~ S \subseteq N \right\}.\, </math>
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| In words, the core is the set of [[Imputation (game theory)|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.
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| ==== Properties ====
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| * The [[Core (economics)|core]] of a game may be empty (see the [[Bondareva–Shapley theorem]]). Games with non-empty cores are called ''balanced''.
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| * If it is non-empty, the core does not necessarily contain a unique vector.
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| * The [[Core (economics)|core]] is contained in any stable set, and if the core is stable it is the unique stable set (see {{harv|Driessen|1988}} for a proof.)
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| === The core of a simple game with respect to preferences ===
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| For simple games, there is another notion of the core, when each player is assumed to have preferences on a set <math>X</math> of alternatives.
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| A ''profile'' is a list <math>p=(\succ_i^p)_{i \in N}</math> of individual preferences <math>\succ_i^p</math> on <math>X</math>.
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| Here <math>x \succ_i^p y</math> means that individual <math>i</math> prefers alternative <math>x</math>
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| to <math>y</math> at profile <math>p</math>.
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| Given a simple game <math>v</math> and a profile <math>p</math>, a ''dominance'' relation <math>\succ^p_v</math> is defined
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| on <math>X</math> by <math>x \succ^p_v y</math> if and only if there is a winning coalition <math>S</math>
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| (i.e., <math>v(S)=1</math>) satisfying <math>x \succ_i^p y</math> for all <math>i \in S</math>.
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| The ''core'' <math>C(v,p)</math> of the simple game <math>v</math> with respect to the profile <math>p</math> of preferences
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| is the set of alternatives undominated by <math>\succ^p_v</math>
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| (the set of maximal elements of <math>X</math> with respect to <math>\succ^p_v</math>):
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| :<math>x \in C(v,p)</math> if and only if there is no <math>y\in X</math> such that <math>y \succ^p_v x</math>.
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| The ''Nakamura number'' of a simple game is the minimal number of winning coalitions with empty intersection.
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| ''Nakamura's theorem'' states that the core <math>C(v,p)</math> is nonempty for all profiles <math>p</math> of ''acyclic'' (alternatively, ''transitive'') preferences
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| if and only if <math>X</math> is finite ''and'' the cardinal number (the number of elements) of <math>X</math> is less than the Nakamura number of <math>v</math>.
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| A variant by Kumabe and Mihara states that the core <math>C(v,p)</math> is nonempty for all profiles <math>p</math> of preferences that have a ''maximal element''
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| if and only if the cardinal number of <math>X</math> is less than the Nakamura number of <math>v</math>. (See [[Nakamura number]] for details.)
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| === The strong epsilon-core ===
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| Because the [[Core (economics)|core]] may be empty, a generalization was introduced in {{harv|Shapley|Shubik|1966}}. The ''strong <math> \varepsilon </math>-core'' for some number <math> \varepsilon \in \mathbb{R} </math> is the set of payoff vectors
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| : <math> C_\varepsilon( v ) = \left\{ x \in \mathbb{R}^N: \sum_{ i \in N } x_i = v(N); \quad \sum_{ i \in S } x_i \geq v(S) - \varepsilon, \forall~ S \subseteq N \right\}. </math>
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| In economic terms, the strong <math> \varepsilon </math>-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 <math> \varepsilon </math> for leaving. Note that <math> \varepsilon </math> may be negative, in which case it represents a bonus for leaving the grand coalition. Clearly, regardless of whether the [[Core (economics)|core]] is empty, the strong <math> \varepsilon </math>-core will be non-empty for a large enough value of <math> \varepsilon </math> and empty for a small enough (possibly negative) value of <math> \varepsilon </math>. Following this line of reasoning, the ''least-core'', introduced in {{harv|Maschler|Peleg|Shapley|1979}}, is the intersection of all non-empty strong <math> \varepsilon </math>-cores. It can also be viewed as the strong <math> \varepsilon </math>-core for the smallest value of <math> \varepsilon </math> that makes the set non-empty {{harv|Bilbao|2000}}.
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| === The Shapley value ===
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| {{main|Shapley value}}
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| The ''Shapley value'' is the unique payoff vector that is efficient, symmetric, additive, and assigns zero payoffs to dummy players. It was introduced by [[Lloyd Shapley]] {{harv|Shapley|1953}}. The Shapley value of a [[superadditive]] game is individually rational, but this is not true in general. {{harv|Driessen|1988}}
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| ===The kernel===
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| Let <math> v : 2^N \to \mathbb{R} </math> be a game, and let <math> x \in \mathbb{R}^N </math> be an efficient payoff vector. The ''maximum surplus'' of player ''i'' over player ''j'' with respect to ''x'' is
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| : <math> s_{ij}^v(x) = \max \left\{ v(S) - \sum_{ k \in S } x_k : S \subseteq N \setminus \{ j \}, i \in S \right\}, </math>
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| 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 <math>v</math> is the set of [[Imputation (game theory)|imputations]] ''x'' that satisfy
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| * <math> ( s_{ij}^v(x) - s_{ji}^v(x) ) \times ( x_j - v(j) ) \leq 0 </math>, and
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| * <math> ( s_{ji}^v(x) - s_{ij}^v(x) ) \times ( x_i - v(i) ) \leq 0 </math>
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| for every pair of players ''i'' and ''j''. Intuitively, player ''i'' has more bargaining power than player ''j'' with respect to [[Imputation (game theory)|imputation]] ''x'' if <math>s_{ij}^v(x) > s_{ji}^v(x)</math>, but player ''j'' is immune to player ''i'''s threats if <math> x_j = v(j) </math>, because he can obtain this payoff on his own. The kernel contains all [[Imputation (game theory)|imputations]] where no player has this bargaining power over another. This solution concept was first introduced in {{harv|Davis|Maschler|1965}}.
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| === The nucleolus ===
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| Let <math> v : 2^N \to \mathbb{R} </math> be a game, and let <math> x \in \mathbb{R}^N </math> be a payoff vector. The ''excess'' of <math> x </math> for a coalition <math> S \subseteq N </math> is the quantity <math> v(S) - \sum_{ i \in S } x_i </math>; that is, the gain that players in coalition <math> S </math> can obtain if they withdraw from the grand coalition <math> N </math> under payoff <math> x </math> and instead take the payoff <math> v(S) </math>.
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| Now let <math> \theta(x) \in \mathbb{R}^{ 2^N } </math> be the vector of excesses of <math> x </math>, arranged in non-increasing order. In other words, <math> \theta_i(x) \geq \theta_j(x), \forall~ i < j </math>. Notice that <math> x </math> is in the [[Core (economics)|core]] of <math> v </math> if and only if it is a pre-imputation and <math> \theta_1(x) \leq 0 </math>. To define the nucleolus, we consider the lexicographic ordering of vectors in <math> \mathbb{R}^{ 2^N } </math>: For two payoff vectors <math> x, y </math>, we say <math> \theta(x) </math> is lexicographically smaller than <math> \theta(y) </math> if for some index <math> k </math>, we have <math> \theta_i(x) = \theta_i(y), \forall~ i < k </math> and <math> \theta_k(x) < \theta_k(y) </math>. (The ordering is called lexicographic because it mimics alphabetical ordering used to arrange words in a dictionary.) The ''nucleolus'' of <math> v </math> is the lexicographically minimal [[Imputation (game theory)|imputation]], based on this ordering. This solution concept was first introduced in {{harv|Schmeidler|1969}}.
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| Although the definition of the nucleolus seems abstract, {{harv|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 <math> C_\varepsilon( v ) </math> 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.
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| ==== Properties ====
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| * Although the definition does not explicitly state it, the nucleolus is always unique. (See Section II.7 of {{harv|Driessen|1988}} for a proof.)
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| * If the core is non-empty, the nucleolus is in the core.
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| * 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 {{harv|Driessen|1988}} for details.)
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| ==Convex cooperative games==
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| Introduced by [[Lloyd Shapley|Shapley]] in {{harv|Shapley|1971}}, convex cooperative games capture the intuitive property some games have of "snowballing". Specifically, a game is ''convex'' if its characteristic function <math> v </math> is [[supermodular]]:
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| : <math> v( S \cup T ) + v( S \cap T ) \geq v(S) + v(T), \forall~ S, T \subseteq N.\, </math>
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| It can be shown (see, e.g., Section V.1 of {{harv|Driessen|1988}}) that the [[supermodular]]ity of <math> v </math> is equivalent to
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| : <math> v( S \cup \{ i \} ) - v(S) \leq v( T \cup \{ i \} ) - v(T), \forall~ S \subseteq T \subseteq N \setminus \{ i \}, \forall~ i \in N;\, </math>
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| that is, "the incentives for joining a coalition increase as the coalition grows" {{harv|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]].
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| ===Properties===
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| Convex cooperative games have many nice properties:
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| * [[Supermodularity]] trivially implies [[superadditivity]].
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| * Convex games are ''totally balanced'': The [[Core (economics)|core]] of a convex game is non-empty, and since any subgame of a convex game is convex, the [[Core (economics)|core]] of any subgame is also non-empty.
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| * A convex game has a unique stable set that coincides with its [[Core (economics)|core]].
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| * The [[Shapley value]] of a convex game is the center of gravity of its [[Core (economics)|core]].
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| * An [[extreme point]] (vertex) of the [[Core (economics)|core]] can be found in polynomial time using the [[greedy algorithm]]: Let <math> \pi: N \to N </math> be a [[permutation]] of the players, and let <math> S_i = \{ j \in N: \pi(j) \leq i \} </math> be the set of players ordered <math> 1 </math> through <math> i </math> in <math> \pi </math>, for any <math> i = 0, \ldots, n </math>, with <math> S_0 = \emptyset </math>. Then the payoff <math> x \in \mathbb{R}^N </math> defined by <math> x_i = v( S_{\pi(i)} ) - v( S_{\pi(i) - 1} ), \forall~ i \in N </math> is a vertex of the [[Core (economics)|core]] of <math> v </math>. Any vertex of the [[Core (economics)|core]] can be constructed in this way by choosing an appropriate [[permutation]] <math> \pi </math>.
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| ===Similarities and differences with combinatorial optimization===
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| [[Submodular]] and [[supermodular]] set functions are also studied in [[combinatorial optimization]]. Many of the results in {{harv|Shapley|1971}} have analogues in {{harv|Edmonds|1970}}, where [[submodular]] functions were first presented as generalizations of [[matroid]]s. In this context, the [[Core (economics)|core]] of a convex cost game is called the ''base polyhedron'', because its elements generalize base properties of [[matroid]]s.
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| However, the optimization community generally considers [[submodular]] functions to be the discrete analogues of convex functions {{harv|Lovász|1983}}, because the minimization of both types of functions is computationally tractable. Unfortunately, this conflicts directly with [[Lloyd Shapley|Shapley's]] original definition of [[supermodular]] functions as "convex".
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| == See also ==
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| * [[Consensus decision-making]]
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| * [[Coordination game]]
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| * [[Intra-household bargaining]]
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| == External links ==
| |
| * {{springer|title=Cooperative game|id=p/c026450}}
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| {{Game theory}}
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| [[Category:Cooperative games| ]]
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| [[Category:Game theory]]
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