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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]

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 equilibriums.^[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 optimal 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.

Mathematical definition

A cooperative game is given by specifying a value for every coalition. Formally, the coalitional game consists of a finite set of players Ndisplaystyle N, called the grand coalition, and a characteristic function v:2N→Rdisplaystyle v:2^Nto mathbb R ^[4] from the set of all possible coalitions of players to a set of payments that satisfies v(∅)=0displaystyle v(emptyset )=0. 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 c:2N→Rdisplaystyle c:2^Nto mathbb R satisfying c(∅)=0displaystyle c(emptyset )=0. In this setting, players must accomplish some task, and the characteristic function cdisplaystyle c 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.

Harsanyi dividend

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 dv(S)displaystyle d_v(S) of coalition Sdisplaystyle S in game vdisplaystyle v is recursively determined by

dv(i)=v(i)dv(i,j)=v(i,j)−dv(i)−dv(j)dv(i,j,k)=v(i,j,k)−dv(i,j)−dv(i,k)−dv(j,k)−dv(i)−dv(j)−dv(k)⋮dv(S)=v(S)−∑T⊊Sdv(T)displaystyle beginalignedd_v(i)&=v(i)\d_v(i,j)&=v(i,j)-d_v(i)-d_v(j)\d_v(i,j,k)&=v(i,j,k)-d_v(i,j)-d_v(i,k)-d_v(j,k)-d_v(i)-d_v(j)-d_v(k)\&vdots \d_v(S)&=v(S)-sum _Tsubsetneq Sd_v(T)endaligned

An explicit formula for the dividend is given by dv(S)=∑T⊊S(−1)|S∖T|v(T)textstyle d_v(S)=sum _Tsubsetneq S(-1)^v(T). The function dv:2N→Rdisplaystyle d_v:2^Nto mathbb R is also known as the Möbius inverse of v:2N→Rdisplaystyle v:2^Nto mathbb R .^[6] Indeed, we can recover vdisplaystyle v from dvdisplaystyle d_v by help of the formula v(S)=∑T⊆Sdv(T)textstyle v(S)=sum _Tsubseteq Sd_v(T).

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 ϕi(v)displaystyle phi _i(v) of player idisplaystyle i in game vdisplaystyle v is given by summing up a player's share of the dividends of all coalitions that she belongs to, ϕi(v)=∑S⊂N:i∈Sdv(S)/|S|textstyle phi _i(v)=sum _Ssubset N:iin Sd_v(S)/S.

Duality

Let vdisplaystyle v be a profit game. The dual game of vdisplaystyle v is the cost game v∗displaystyle v^* defined as

v∗(S)=v(N)−v(N∖S),∀ S⊆N.displaystyle v^*(S)=v(N)-v(Nsetminus S),forall ~Ssubseteq N.

Intuitively, the dual game represents the opportunity cost for a coalition Sdisplaystyle S of not joining the grand coalition Ndisplaystyle N. A dual profit game c∗displaystyle c^* can be defined identically for a cost game cdisplaystyle c. 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).

Subgames

Let S⊊Ndisplaystyle Ssubsetneq N be a non-empty coalition of players. The subgame vS:2S→Rdisplaystyle v_S:2^Sto mathbb R on Sdisplaystyle S is naturally defined as

vS(T)=v(T),∀ T⊆S.displaystyle v_S(T)=v(T),forall ~Tsubseteq S.

In other words, we simply restrict our attention to coalitions contained in Sdisplaystyle S. Subgames are useful because they allow us to apply solution concepts defined for the grand coalition on smaller coalitions.

Properties for characterization

Superadditivity

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:

v(S∪T)≥v(S)+v(T)displaystyle v(Scup T)geq v(S)+v(T) whenever S,T⊆Ndisplaystyle S,Tsubseteq N satisfy S∩T=∅displaystyle Scap T=emptyset .

Monotonicity

Larger coalitions gain more:

S⊆T⇒v(S)≤v(T)displaystyle Ssubseteq TRightarrow v(S)leq v(T).

This follows from superadditivity. i.e. if payoffs are normalized so singleton coalitions have zero value.

Properties for simple games

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 S∈Wdisplaystyle Sin W and S⊆Tdisplaystyle Ssubseteq T imply T∈Wdisplaystyle Tin W.


• A simple game W is proper if the complement (opposition) of any winning coalition is losing, that is, if S∈Wdisplaystyle Sin W implies N∖S∉Wdisplaystyle Nsetminus Snotin W.


• A simple game W is strong if the complement of any losing coalition is winning, that is, if S∉Wdisplaystyle Snotin W implies N∖S∈Wdisplaystyle Nsetminus Sin W.
  
  • If a simple game W is proper and strong, then a coalition is winning if and only if its complement is losing, that is, S∈Wdisplaystyle Sin W iff N∖S∉Wdisplaystyle Nsetminus Snotin W. (If v is a colitional simple game that is proper and strong, v(S)=1−v(N∖S)displaystyle v(S)=1-v(Nsetminus S) 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 ⋂W:=⋂S∈WSdisplaystyle bigcap W:=bigcap _Sin WS 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 T⊆Ndisplaystyle Tsubseteq N such that for any coalition S, we have S∈Wdisplaystyle Sin W iff S∩T∈Wdisplaystyle Scap Tin W. 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^[7]):

• 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^[8]
has been made (Kumabe and Mihara, 2011^[9]),
whose results are summarized in the Table "Existence of Simple Games" below.

The restrictions that various axioms for simple games impose on their Nakamura number were also studied extensively. ^[11]
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 theory

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:^[12]

• 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.

Solution concepts

The main assumption in cooperative game theory is that the grand coalition Ndisplaystyle N will form.^{[citation needed]} The challenge is then to allocate the payoff v(N)displaystyle v(N) 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 x∈RNdisplaystyle xin mathbb R ^N 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: ∑i∈Nxi=v(N)displaystyle sum _iin Nx_i=v(N).


• Individual rationality: No player receives less than what he could get on his own: xi≥v(i),∀ i∈Ndisplaystyle x_igeq v(i),forall ~iin N.


• Existence: The solution concept exists for any game vdisplaystyle v.


• Uniqueness: The solution concept is unique for any game vdisplaystyle v.


• 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: v(S∪i)=w(S∪i),∀ S⊆N∖idisplaystyle v(Scup i)=w(Scup i),forall ~Ssubseteq Nsetminus i implies that xidisplaystyle x_i is the same in vdisplaystyle v and in wdisplaystyle w.


• Monotonicity: The payoff of a player increases if the marginal contribution of this player increase: v(S∪i)≤w(S∪i),∀ S⊆N∖idisplaystyle v(Scup i)leq w(Scup i),forall ~Ssubseteq Nsetminus i implies that xidisplaystyle x_i is weakly greater in wdisplaystyle w than in vdisplaystyle v.


• Computational ease: The solution concept can be calculated efficiently (i.e. in polynomial time with respect to the number of players |N|displaystyle .)


• Symmetry: The solution concept xdisplaystyle x allocates equal payments xi=xjdisplaystyle x_i=x_j to symmetric players idisplaystyle i, jdisplaystyle j. Two players idisplaystyle i, jdisplaystyle j are symmetric if v(S∪i)=v(S∪j),∀ S⊆N∖i,jdisplaystyle v(Scup i)=v(Scup j),forall ~Ssubseteq Nsetminus i,j; 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 vdisplaystyle v and ωdisplaystyle omega are games, the game (v+ω)displaystyle (v+omega ) 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 (v+ω)displaystyle (v+omega ) the sum of what he would receive in vdisplaystyle v and ωdisplaystyle omega .


• Zero Allocation to Null Players: The allocation to a null player is zero. A null player idisplaystyle i satisfies v(S∪i)=v(S),∀ S⊆N∖idisplaystyle v(Scup i)=v(S),forall ~Ssubseteq Nsetminus i. 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

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 vdisplaystyle v be a game and let xdisplaystyle x, ydisplaystyle y be two imputations of vdisplaystyle v. Then xdisplaystyle x dominates ydisplaystyle y if some coalition S≠∅displaystyle Sneq emptyset satisfies xi>yi,∀ i∈Sdisplaystyle x_i>y_i,forall ~iin S and ∑i∈Sxi≤v(S)displaystyle sum _iin Sx_ileq v(S). In other words, players in Sdisplaystyle S prefer the payoffs from xdisplaystyle x to those from ydisplaystyle y, and they can threaten to leave the grand coalition if ydisplaystyle y is used because the payoff they obtain on their own is at least as large as the allocation they receive under xdisplaystyle x.

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.

Properties

• 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 n−2displaystyle n-2 players. In such sets at least n−3displaystyle n-3 of the discriminated players are excluded (Owen 1995, p. 240).

The core

Let vdisplaystyle v be a game. The core of vdisplaystyle v is the set of payoff vectors

C(v)=x∈RN:∑i∈Nxi=v(N);∑i∈Sxi≥v(S),∀ S⊆N.displaystyle C(v)=leftxin mathbb R ^N:sum _iin Nx_i=v(N);quad sum _iin Sx_igeq v(S),forall ~Ssubseteq Nright.

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.

Properties

• 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 preferences

For simple games, there is another notion of the core, when each player is assumed to have preferences on a set Xdisplaystyle X of alternatives.
A profile is a list p=(≻ip)i∈Ndisplaystyle p=(succ _i^p)_iin N of individual preferences ≻ipdisplaystyle succ _i^p on Xdisplaystyle X.
Here x≻ipydisplaystyle xsucc _i^py means that individual idisplaystyle i prefers alternative xdisplaystyle x
to ydisplaystyle y at profile pdisplaystyle p.
Given a simple game vdisplaystyle v and a profile pdisplaystyle p, a dominance relation ≻vpdisplaystyle succ _v^p is defined
on Xdisplaystyle X by x≻vpydisplaystyle xsucc _v^py if and only if there is a winning coalition Sdisplaystyle S
(i.e., v(S)=1displaystyle v(S)=1) satisfying x≻ipydisplaystyle xsucc _i^py for all i∈Sdisplaystyle iin S.
The core C(v,p)displaystyle C(v,p) of the simple game vdisplaystyle v with respect to the profile pdisplaystyle p of preferences
is the set of alternatives undominated by ≻vpdisplaystyle succ _v^p
(the set of maximal elements of Xdisplaystyle X with respect to ≻vpdisplaystyle succ _v^p):

x∈C(v,p)displaystyle xin C(v,p) if and only if there is no y∈Xdisplaystyle yin X such that y≻vpxdisplaystyle ysucc _v^px.

The Nakamura number of a simple game is the minimal number of winning coalitions with empty intersection.
Nakamura's theorem states that the core C(v,p)displaystyle C(v,p) is nonempty for all profiles pdisplaystyle p of acyclic (alternatively, transitive) preferences
if and only if Xdisplaystyle X is finite and the cardinal number (the number of elements) of Xdisplaystyle X is less than the Nakamura number of vdisplaystyle v.
A variant by Kumabe and Mihara states that the core C(v,p)displaystyle C(v,p) is nonempty for all profiles pdisplaystyle p of preferences that have a maximal element
if and only if the cardinal number of Xdisplaystyle X is less than the Nakamura number of vdisplaystyle v. (See Nakamura number for details.)

The strong epsilon-core

Because the core may be empty, a generalization was introduced in (Shapley & Shubik 1966). The strong εdisplaystyle varepsilon -core for some number ε∈Rdisplaystyle varepsilon in mathbb R is the set of payoff vectors

Cε(v)=x∈RN:∑i∈Nxi=v(N);∑i∈Sxi≥v(S)−ε,∀ S⊆N.displaystyle C_varepsilon (v)=leftxin mathbb R ^N:sum _iin Nx_i=v(N);quad sum _iin Sx_igeq v(S)-varepsilon ,forall ~Ssubseteq Nright.

In economic terms, the strong εdisplaystyle varepsilon -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 εdisplaystyle varepsilon for leaving. Note that εdisplaystyle varepsilon 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 εdisplaystyle varepsilon -core will be non-empty for a large enough value of εdisplaystyle varepsilon and empty for a small enough (possibly negative) value of εdisplaystyle varepsilon . Following this line of reasoning, the least-core, introduced in (Maschler, Peleg & Shapley 1979), is the intersection of all non-empty strong εdisplaystyle varepsilon -cores. It can also be viewed as the strong εdisplaystyle varepsilon -core for the smallest value of εdisplaystyle varepsilon that makes the set non-empty (Bilbao 2000).

The Shapley value

The Shapley value is the unique payoff vector that is efficient, symmetric, and satisfies monotonicity.^[13] 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)

The kernel

Let v:2N→Rdisplaystyle v:2^Nto mathbb R be a game, and let x∈RNdisplaystyle xin mathbb R ^N be an efficient payoff vector. The maximum surplus of player i over player j with respect to x is

sijv(x)=maxv(S)−∑k∈Sxk:S⊆N∖j,i∈S,displaystyle s_ij^v(x)=max leftv(S)-sum _kin Sx_k:Ssubseteq Nsetminus j,iin Sright,

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 vdisplaystyle v is the set of imputations x that satisfy

• 
  (sijv(x)−sjiv(x))×(xj−v(j))≤0displaystyle (s_ij^v(x)-s_ji^v(x))times (x_j-v(j))leq 0, and


• (sjiv(x)−sijv(x))×(xi−v(i))≤0displaystyle (s_ji^v(x)-s_ij^v(x))times (x_i-v(i))leq 0

for every pair of players i and j. Intuitively, player i has more bargaining power than player j with respect to imputation x if sijv(x)>sjiv(x)displaystyle s_ij^v(x)>s_ji^v(x), but player j is immune to player i's threats if xj=v(j)displaystyle x_j=v(j), 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).

The nucleolus

Let v:2N→Rdisplaystyle v:2^Nto mathbb R be a game, and let x∈RNdisplaystyle xin mathbb R ^N be a payoff vector. The excess of xdisplaystyle x for a coalition S⊆Ndisplaystyle Ssubseteq N is the quantity v(S)−∑i∈Sxidisplaystyle v(S)-sum _iin Sx_i; that is, the gain that players in coalition Sdisplaystyle S can obtain if they withdraw from the grand coalition Ndisplaystyle N under payoff xdisplaystyle x and instead take the payoff v(S)displaystyle v(S).

Now let θ(x)∈R2Ndisplaystyle theta (x)in mathbb R ^2^N be the vector of excesses of xdisplaystyle x, arranged in non-increasing order. In other words, θi(x)≥θj(x),∀ i<jdisplaystyle theta _i(x)geq theta _j(x),forall ~i<j. Notice that xdisplaystyle x is in the core of vdisplaystyle v if and only if it is a pre-imputation and θ1(x)≤0displaystyle theta _1(x)leq 0. To define the nucleolus, we consider the lexicographic ordering of vectors in R2Ndisplaystyle mathbb R ^2^N: For two payoff vectors x,ydisplaystyle x,y, we say θ(x)displaystyle theta (x) is lexicographically smaller than θ(y)displaystyle theta (y) if for some index kdisplaystyle k, we have θi(x)=θi(y),∀ i<kdisplaystyle theta _i(x)=theta _i(y),forall ~i<k and θk(x)<θk(y)displaystyle theta _k(x)<theta _k(y). (The ordering is called lexicographic because it mimics alphabetical ordering used to arrange words in a dictionary.) The nucleolus of vdisplaystyle v 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 Cε(v)displaystyle C_varepsilon (v) 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.

Properties

• 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 games

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 vdisplaystyle v is supermodular:

v(S∪T)+v(S∩T)≥v(S)+v(T),∀ S,T⊆N.displaystyle v(Scup T)+v(Scap T)geq v(S)+v(T),forall ~S,Tsubseteq N.

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

v(S∪i)−v(S)≤v(T∪i)−v(T),∀ S⊆T⊆N∖i,∀ i∈N;displaystyle v(Scup i)-v(S)leq v(Tcup i)-v(T),forall ~Ssubseteq Tsubseteq Nsetminus i,forall ~iin N;

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.

Properties

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 π:N→Ndisplaystyle pi :Nto N be a permutation of the players, and let Si=j∈N:π(j)≤idisplaystyle S_i=jin N:pi (j)leq i be the set of players ordered 1displaystyle 1 through idisplaystyle i in πdisplaystyle pi , for any i=0,…,ndisplaystyle i=0,ldots ,n, with S0=∅displaystyle S_0=emptyset . Then the payoff x∈RNdisplaystyle xin mathbb R ^N defined by xi=v(Sπ(i))−v(Sπ(i)−1),∀ i∈Ndisplaystyle x_i=v(S_pi (i))-v(S_pi (i)-1),forall ~iin N is a vertex of the core of vdisplaystyle v. Any vertex of the core can be constructed in this way by choosing an appropriate permutation πdisplaystyle pi .

Similarities and differences with combinatorial optimization

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".

See also

• Consensus decision-making


• Coordination game


• Intra-household bargaining


• Hedonic game

References

Further reading

• 
  Bilbao, Jesús Mario (2000), Cooperative Games on Combinatorial Structures, Kluwer Academic Publishers 
  

• 
  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 
  

• 
  Edmonds, Jack (1970), "Submodular functions, matroids and certain polyhedra", in Guy, R.; Hanani, H.; Sauer, N.; Schönheim, J., 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., 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, 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, 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 0-521-42458-5 
  

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  Owen, Guillermo (1995), Game Theory (3rd ed.), San Diego: Academic Press, ISBN 0-12-531151-6 
  

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

• 
  Shapley, Lloyd S. (1953), "A value for ndisplaystyle n-person games", in Kuhn, H.; Tucker, A.W., 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 
  

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

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  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, Princeton: Princeton University Press 
  

• 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
  

External links

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  Hazewinkel, Michiel, ed. (2001) [1994], "Cooperative game", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4