The Owen Value of Stochastic Cooperative Game

We consider stochastic cooperative game and give it the definition of the Owen value, which is obtained by extending the classical case. Then we provide explicit expression for the Owen value of the stochastic cooperative game and discuss its existence and uniqueness.


Introduction
In classical cooperative game theory, payoffs to coalitions of agents are known with certainty, but in today's business world payoffs to agents are uncertain. Charnes and Granot [1] considered cooperative games in stochastic characteristic function form. These are games where the payoff ( ) to coalition is allowed to be a random variable. Research on this subject was continued by Charnes and Granot [2,3] and Granot [4]. Suijs and Borm [5] research a different and more extensive model. They describe allocation of ( ) to the members of coalition as the sum of two parts. The first part is a monetary transfer between the agents and the second part is an allocation of fractions of ( ). Dshalalow and Ke [6] are concerned with an antagonistic stochastic game between two players A and B which finds applications in economics and warfare. Levy [7] considered the two-player zero-sum stochastic games with finite state under the assumption that one or both players observe the actions of their opponent after some time-dependent delay.
The Owen value [8] as an important solution concept in cooperative game theory has been studied by a number of researchers, which shows a vector whose elements are agents' share derived from several reasonable bases. However, the Owen value for stochastic cooperative games has not been discussed yet. In this paper, we consider the Owen value of stochastic cooperative games.
We end this section with a short overview of the rest of the paper. In Section 2 we introduce preliminaries of stochastic cooperative game. Then, in Section 3 we first introduce the notion of Owen value of classical cooperative game and Owen value of the stochastic cooperative games as payoff in Theorem 5. We conclude in Section 4 also sketch and some main lines for future research.

Notations and Preliminaries
Definition 1 (see [7]). A stochastic cooperative game is described by a tuple Γ = ( , { } ⊆ , {≿ } ∈ ), where is the set of agents, : → 1 (R) is the payoff function of coalition , where { } ∈ 1 (R) with finite expectation, and ≿ is the preference relation of agent over the set 1 (R) of stochastic payoffs with finite expectation. The class of all cooperative games with stochastic payoffs with agent set is denoted by SG( ). An allocation of a stochastic payoff to the agents in coalition is represented by a pair ( , ) ∈ R × R such that ∑ ∈ ≤ 0 and ∑ ∈ = 1 and ≥ 0 for all agents ∈ . The set of all allocations for coalition is denoted by ( ).
Given such a pair ( , ) where = { | ∈ } and = { | ∈ }, agents ∈ receive the stochastic payoff + and we can also define this payoff as ( , ) ; that is, ( , ) = + . The second part, , describes the fraction of that is allocated to agent . The first part, , describes the deterministic transfer payments between 2 The Scientific World Journal the agents. When ≥ 0, agent receives money, while < 0 means that this agent pays money. The purpose of these transfer payments is that the agents compensate among themselves for transfers of random payoffs. The set of all individual rational allocations is denoted by IR( ). Then Definition 2. ( , ) ∈ ( ) is called the stochastic payoff vectors of the game Γ if it satisfies ∑ ∈ ( + ) = and + ≿ { } for all ∈ . Let be a coalition and ( , ) and (̃,̃) be two stochastic payoff vectors of the game Γ. One says ( , ) dominates (̃,̃) through , Definition 3. The set of all undominated payoffs for a stochastic cooperative game Γ is called the core of the stochastic cooperative game Γ and denoted by Core(Γ). That is, the If an allocation is not in the core there is incentive for some agents to leave the coalition. A core solution is desirable because it is stable, but the core of a cooperative game may be empty. In addition, even when the core exists, an allocation in the core may have other undesirable characteristics. In general, it is hard to determine whether the core of a coalitional game exists or not. Even when it does, the more important question is whether the suggested value allocation scheme is actually in the core. While such issues can be important, we avoid them as unpromising in this context. In the sequel we investigate the Owen value of stochastic cooperative games.

Owen Value of Stochastic Cooperative Games
In this section we consider the Owen value for stochastic cooperative games with coalition structure that can be regarded as an expansion of the Shapley value for the situation when a coalition structure is involved. The Owen value was introduced in Owen [8] via a set of axioms it was determining. We consider games with coalition structure. A coalition structure B = { 1 , . . . , } on a player set is a partition of the player set ; that is, 1 ∪ ⋅ ⋅ ⋅ = and ∩ = 0 for ̸ = . Denote by B a set of all coalition structures on . A coalition value is an operator that assigns a vector of payoffs to any pair ( , B) of a game and a coalition structure B on . More precisely, for any set of game G ⊆ G and any set of coalition structures B ⊆ B , a coalitional value on ( , ) with a coalition structure from B is a mapping : G × B → R that associates with each pair ( , B) of a game ∈ G and a coalition structure B ∈ B a vector ( , B) ∈ R , where the real number ( , B) represents the payoff to the player in the game with the coalition structure B.
We considers the stochastic cooperative game which induces ( , ) B among coalitions in B = { 1 , . . . , }. This game, which is denoted by ( , ) B and called the game between coalitions or intermediate game, is defined formally for every ⊆ by where = {1, 2, . . . , }.
We will use the following axioms to present characterizations of Owen value.
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. , } be a set of players; then the unique Owen value of the stochastic cooperative games Γ is
for all ∈ , where is such that ∈ ∈ B and = | |, = | |, = | |.
Proof. In this proof, we will prove two key issues: (1) the existence of the Owen value and (2) the uniqueness of the Owen value.
Axiom 5 (symmetry across the unions). From the assumption of Axiom 5, we have that for all ⊆ \ { , }; In particular, when = \ { , }, then we obtain where is any permutation in and = { ∈ | ( ) < ( )} is the set of players preceding player in the permutation , for all ∈ .
(2) Proof of Uniqueness. Let be a coalitional value which have efficiency, additivity, null player, and symmetry in the unions and across the unions, and let B ∈ B ; then is defined on Γ×B . Any stochastic cooperative game ( , ) ∈ Γ can be presented via unanimity basis { } 0 ̸ = ⊆ : where ( ) is the element of the coalition structure B that contains player and is equal to the number of coalitions in B that have a nonempty intersection with ; that is, ( ) = ∈ B : ∋ and = |{ ∈ : ∩ ̸ = 0}|. Because of its additivity property the Owen value in any stochastic cooperative game ( , ) with a coalition structure B can be equivalently expressed as Let the index of a stochastic cooperative game ( , ) ∈ Γ be the minimum number of terms under summation in (20); then