The Scientific World Journal

Volume 2014, Article ID 853128, 7 pages

http://dx.doi.org/10.1155/2014/853128

## The Owen Value of Stochastic Cooperative Game

^{1}School of Economics and Management, Yanshan University, Qinhuangdao 066004, China^{2}Liren College, Yanshan University, Qinhuangdao 066004, China

Received 31 August 2013; Accepted 31 December 2013; Published 29 April 2014

Academic Editors: F. R. B. Cruz and W. Szeto

Copyright © 2014 Cheng-Guo E et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. 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.

#### 2. Notations and Preliminaries

*Definition 1 (see [7]). *A stochastic cooperative game is described by a tuple , where is the set of agents, is the payoff function of coalition , where with finite expectation, and is the preference relation of agent over the set of stochastic payoffs with finite expectation. The class of all cooperative games with stochastic payoffs with agent set is denoted by . An allocation of a stochastic payoff to the agents in coalition is represented by a pair such that and and 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 the agents. When , agent receives money, while 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 . Then

*Definition 2. * is called the stochastic payoff vectors of the game if it satisfies and for all . Let be a coalition and and be two stochastic payoff vectors of the game . One says dominates through , , if for all and .

*Definition 3. *The set of all undominated payoffs for a stochastic cooperative game is called the core of the stochastic cooperative game and denoted by Core. That is, the payoff of stochastic cooperative game is said to be a core payoff if it satisfies for all and .

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.

#### 3. 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 on a player set is a partition of the player set ; that is, and for . Denote by a set of all coalition structures on . A coalition value is an operator that assigns a vector of payoffs to any pair of a game and a coalition structure on . More precisely, for any set of game and any set of coalition structures , a coalitional value on with a coalition structure from is a mapping that associates with each pair of a game and a coalition structure a vector , where the real number represents the payoff to the player in the game with the coalition structure .

We considers the stochastic cooperative game which induces among coalitions in . This game, which is denoted by and called the game between coalitions or intermediate game, is defined formally for every by where .

We will use the following axioms to present characterizations of Owen value.

*Definition 4. * is called Owen value on if it satisfies the following three axioms.*Axiom **1* (efficiency). For all and all ,
*Axiom **2* (additivity). For all and all , if there exists such that for all , then
for all .*Axiom **3* (null player). For all and all , if is a dummy player in the game (i.e., for each , , then
*Axiom 4* (symmetry in the unions). For all , for any , and for any , if for each such as , then
*Axiom 5* (symmetry across the unions). For all and for any , if for all which satisfies , then

Theorem 5. *Let be a set of players; then the unique Owen value of the stochastic cooperative games is
**
for all , where is such that 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.*(**1) Proof of Existence**Axiom **1* (efficiency). Consider
*Axiom **2* (additivity). It is easy to see from (8) that is a linear function for ; it obviously satisfies additivity.*Axiom **3* (null player). For and , we suppose that is a dummy player in the game ; then
for any , , and ; that is,
It is easy to see from (8) that
*Axiom **4* (symmetry in the unions). For all , for any , and for any , if for each which satisfies = , then
for any , , and .

From (8) we have that
*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 .

From (15), (16), and (17), we have that
*(**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 ; then is defined on . Any stochastic cooperative game can be presented via unanimity basis :
where , and
The Owen value in the unanimity game with a coalition structure is equal to
where is the element of the coalition structure that contains player and is equal to the number of coalitions in that have a nonempty intersection with ; that is, and . Because of its additivity property the Owen value in any stochastic cooperative game with a coalition structure can be equivalently expressed as

Let the index of a stochastic cooperative game be the minimum number of terms under summation in (20); then
where all . We proceed with the remaining part of the proof by induction on this index .

If , then is identically zero on all coalitions. All players in both games and are symmetric. Therefore, by symmetry across coalitions for all ,
But
and by efficiency
Thus, for all ,
By symmetry within coalitions it follows that, for all ,
that is

Assume now that is the Owen value whenever the index of is at most , and consider some with the index being equal to . Let and . Consider the game
Obviously, the index of is at most and
therefore, by induction hypothesis,
for all .

Using the additivity of the and the Owen value, we have
for all .

By the definition of the unanimity game, we have that
for all .

From (32), (33) and (34), we have that
for all .

If then to complete the proof it is enough to show that the last equality is true for all as well. Consider with relevance to a coalition structure and denote that
Notice that if then and all players are symmetric in the stochastic cooperative game . By symmetry among coalitions for both values and Owen value, for all players ,
Therefore, because of efficiency of both values and equality (37) it follows that
for all .

Using the equality (35), we have
for all ; then
for all .

But all players are symmetric in the game . Hence, by symmetry within coalitions, for all , ,
Thus
for all .

From (35) and (42), we have that
for all . This completes the proof.

#### 4. Conclusion

In this paper, 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. In future, we will explore the applications of Owen value of stochastic cooperative game in economy.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors were supported by the National Natural Science Foundation of China under Grants 71271187, 71301139, and 71101124, the Natural Science Foundation of Hebei Province, China, under Grants A2012203125 and F2013203136, and the Research Fund for the Doctoral Program of Higher Education under Grant 20131333120001.

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