# A Note on Strategic Stability of Cooperative Solutions for Multistage Games.

1. IntroductionStrategic stability of cooperative solutions in a game means that the outcome of cooperation must be attained in some Nash equilibrium, which is an effective mechanism to guarantee the sustainability of a cooperative agreement. It seeks to make the cooperative solution guaranteed by an equilibrium in an associated noncooperative game. See, for example, [1-4].

How can a cooperative agreement made at the start be sustained over time is an important issue in human behavior. Besides strategic stability, time consistency and irrational-behavior-proof condition are also effective aspects to sustain cooperation.

Time consistency maybe described informally as follows: to sustain cooperation over time, each player's cooperative payoff should belong to the same optimal principle at any time along the cooperative path. The concept of time consistency and its implement ingredient imputation distribution procedure (IDP) were initially proposed in [5, 6] and further developed in [7-9].

Irrational-behavior-proof condition requires that the partners involved in the cooperation must be sure that even in the worst scenario they will not lose compared with the noncooperative behavior. In [10], the irrational-behavior-proof condition is proposed. A further investigation could be found in [11].

In their seminal paper [12], Grauer and Petrosyan studied the strategic stability of cooperative solutions in an infinite stage game (ISG). They introduced a new (general) characteristic function in such a game, which plays a role of computing the cooperative solutions and describing the deviating payoffs of players. This characteristic function plays an essential role to construct Nash or strong Nash equilibria in the ISG. However, the general characteristic function has a drawback: it could not be super-additive, which would lead to the nonexistence of cooperative solutions, e.g., the core.

Motivated by the above observations, in this paper, we shall focus on the problem of strategic stability of cooperative solutions in ISGs and present sufficient conditions in terms of discount factors and prove that Nash or strong Nash equilibria exist in such games. The deviation payoffs are given directly, which are related closely to the sufficient condition of strategic stability and avoid the loss of super-additivity of the general characteristic functions. Then we study the repeated infinite stage game (RISG). As an illustration, Nash and strong Nash equilibria are found for the repeated infinite stage Prisoner's dilemma game.

The proof technique is the trigger (penalty) strategy combined with the appropriate construction of the time-consistent IDP. When some player (coalition) deviates from the cooperative trajectory in some stage, other players (coalitions) would use the trigger strategies from the next stage. But this does not include the case, in which the trigger strategies could be used after several stages, perhaps because the information is delayed or the players in one coalition need time to coordinate their actions (see [13-16]). While in a finite stage game, to construct Nash or strong Nash equilibria, we need more strict conditions, which is the case with perfect information (see [17]).

The theory we developed could be applied to analyzing the dynamic cooperative behavior in society. A typical example is the global pollution control problem, which requires a joint effort of many countries for a long time. See, for example, [18-20].

The paper is organized as follows. In Section 2, the basic model about the ISG is introduced. In Section 3, the sufficient conditions are given and the existence of Nash and strong Nash equilibria are proved in ISGs. In Sections 4 and 5, the RISG and the repeated infinite stage Prisoner's dilemma game are studied. In Section 6, some concluding discussions are provided.

2. Formal Definitions and Terminology

In this section we introduce the basic model of ISGs (see also [12] for more details).

An infinite game tree is an infinite oriented treelike graph K = (Z, F) with the root Zq, where Z is the set of vertices and [F.sub.z] is the set of vertices following after z and F : Z [right arrow] [2.sup.Z], F(z) = [F.sub.Z] [subset] Z, z [member of] Z.

A single stage game is a simultaneous n-player game [mathematical expression not reproducible] corresponding to each vertex z [member of] Z in the tree K = (Z, F), where N = {1,2,...,n} is the set of players, [U.sup.z.sub.i] is the set of strategies of player i and [H.sup.z.sub.i]([u.sup.z.sub.1]..., [u.sup.z.sub.n]) is the payoff of player i.

A transition function is defined as [mathematical expression not reproducible] for each z [member of] Z. For each game [gamma](z), the function T determines the following stage game [GAMMA](z').

An infinite stage game G([z.sub.0]) in the tree K = (Z;F) is determined by the simultaneous games r(z) and the transition function T.

Denote a sequence of situations [mathematical expression not reproducible] by an n-tuple of strategies [mathematical expression not reproducible], where [mathematical expression not reproducible]. Define the corresponding sequence of vertices [z.sub.0],[z.sub.1],...,[z.sub.k],... as a trajectory (path) in the graph K, denoted by ([z.sub.0], [z.sub.1],..., [z.sub.k],...).

Define

[mathematical expression not reproducible] (1)

as the payoff of player i in the game G([z.sub.0]).All the payoffs in the single stage games are uniformly bounded, which guarantees the existence of sum (1).

In game G([z.sub.0]), the players possess complete information, which means they know the simultaneous game r(z) and remember all the strategies in the game history.

Suppose that players in N are playing cooperatively with objective

[mathematical expression not reproducible] (2)

Suppose there exist an n-tuple of strategies [mathematical expression not reproducible] and a trajectory [mathematical expression not reproducible] satisfying (2). Define [bar.z] as an optimal cooperative trajectory of G([z.sub.0]).

The subgame G(z) of the game G([z.sub.0]) is played in the subgraph K(z) = ([Z.sup.z], F), where [Z.sup.z] is the set of vertices of the subgraph K(z). The payoff of player i in the subgame G(z) is denoted by [mathematical expression not reproducible].

The characteristic function V(z;S), S [subset] N in the subgame G(z) is defined in a classical way:[mathematical expression not reproducible], where [mathematical expression not reproducible] is a value of zero-sum game played between coalition S acting as player 1 and coalition N \ S acting as player 2, with the payoff of coalition S equal to [mathematical expression not reproducible]. It is additionally assumed that the values V(z; S) exist for every z [member of] Z and S c N. Specially,

[mathematical expression not reproducible] (3)

Denote the pair of optimal strategies in the game [mathematical expression not reproducible]

Consider a sequence of subgames G([[bar.z].sub.k]) along an optimal cooperative trajectory [bar.z] = ([z.sub.0], [bar.z].sub.1],..., [[bar.z].sub.k],...). In each subgame G([[bar.z].sub.k]), one can obtain the imputation set L([[bar.z].sub.k]) and the core C([[bar.z].sub.k]):

[mathematical expression not reproducible] (4)

[mathematical expression not reproducible] (5)

Suppose the imputation [alpha] = ([[alpha].sub.1],..., [[alpha].sub.n]) [member of] L([z.sub.0]). Define an imputation distribution procedure (IDP) as a function [mathematical expression not reproducible], ..., such that

[mathematical expression not reproducible] (6)

For every [alpha] [member of] L([z.sub.0]), define the noncooperative infinite stage game [G.sub.[alpha([z.sub.0]), which differs from the game G([z.sub.0]) only in the payoffs along the optimal cooperative trajectory [bar.z]. Suppose under the situation u(*) = ([u.sub.1](*),..., [u.sub.n](*)) the path ([z.sub.0], [z.sub.1],..., [z.sub.k],...) is realized. Denote the payoff in game [G.sub.[alp[ha]] ([z.sub.0]) by

[mathematical expression not reproducible] (7)

where [mathematical expression not reproducible]. In a special case when [z.sub.k] = [[[bar.z].sub.k], k = 0,1,..., we have

[mathematical expression not reproducible] (8)

Let [alpha](k) [member of] L([[[bar.z].sub.k]). Game [G.sub.[alpha]] ([z.sub.0]) is called a regularized game of G([z.sub.0]) if IDP [beta] is defined in such a way that

[[beta].sub.i] (k) = [[alpha].sub.i] (k) - [delta][[alpha].sub.i] (k + 1), k = 0,1,... (9)

In particular, if [alpha](k) [member of] C([[bar.z].sub.k]), k = 0,1,..,, [G.sub.[alpha][ ([z.sub.0]) is called a strictly regularized game of G([z.sub.0]). From (9) we get

[mathematical expression not reproducible]. (10)

Now suppose that M([z.sub.0]) [subset] L([z.sub.0]) is some optimality principle in the cooperative game G([z.sub.0]), and M([[bar.z].sub.k]) [sunset] L([[bar.z].sub.k]) is the same optimality principle defined

in the subgame G([[bar.z].sub.k]) with initial conditions on the cooperative trajectory. M can be the Shapley value, the core, the nucleolus, etc. If [alpha](k) [member of] M([[bar.z].sub.k]), k = 0, 1, ..., condition (10) gives us the time consistency of the chosen imputation [alpha] (or the IDP [beta]) in game G([z.sub.0]).

An n-tuple [u.sup.*](*) = ([u.sup.*.sub.1] (*), ..., [u.sup.*.sub.n] (*)) is a Nash equilibrium of game [G.sub.[alpha]]([z.sub.0]) if and only if

[mathematical expression not reproducible] (11)

for all i [member of] N and all [u.sub.i].

An n-tuple [u.sup.*](*) = ([u.sup.*.sub.1] (*), ..., [u.sup.*.sub.n] (*)) is a strong Nash equilibrium of game [G.sub.[alpha]] ([z.sub.0]) if and only if

[mathematical expression not reproducible] (12)

for all S [subset] N and all [u.sub.s] = {[u.sub.i], i [member of] S}.

3. Existence of Nash and Strong Nash Equilibria

Consider the following inequality with respect to [delta]:

[mathematical expression not reproducible] (13)

where [mathematical expression not reproducible] is the stage payoff to player i if deviating from her cooperative strategy and playing the best response to opponents' cooperative strategies. The above inequality is reduced to the following:

[mathematical expression not reproducible] (14)

Let the value in the right-hand side in (14) be reached. Let

[mathematical expression not reproducible] (15)

We can get the following.

Theorem 1. In the regularized game [G.sub.[alpha]]([z.sub.0]), for any [delta] satisfying [delta] [less than or equal to] [bar.[delta]], the situation ([u.sup.*.sub.1](*),..., [u.sup.*n(]*)) with players' payoffs as ([[alpha].sub.1], [[alpha].sub.2],..., [[alpha].sub.n]) guaranteed by the time-consistent IDP [beta] is a Nash equilibrium.

Now suppose S [subset] N and consider another inequality with

respect to [delta]:

[mathematical expression not reproducible] (16)

where [mathematical expression not reproducible] is the stage payoff to coalition S if deviating from the cooperative strategy and playing the best response to others' cooperative strategies. The above inequality is reduced to the following:

[mathematical expression not reproducible] (17)

Let the value in the right-hand side in (17) be reached. Let

[mathematical expression not reproducible] . (18)

We can also get the following.

Theorem 2. In the strictly regularized game [G.sub.[alpha]] ([z.sub.0]),for any [delta] satisfying [mathematical expression not reproducible], the situation ([u.sup.*.sub.1](*), ..., [u.sup.*.sub.n]()) with players' payoffs as ([[alpha].sub.1], [[alpha].sub.2], ..., [[alpha].sub.n]) guaranteed by the time-consistent IDP [beta] is a strong Nash equilibrium.

Theorem 1 implies the cooperative solution (any imputation) can be strategically supported by a specially constructed Nash equilibrium in a regularized game [G.sub.[alpha]]([z.sub.0]). Theorem 2 implies the cooperative solution (any core) can be strategically supported by a specially constructed strong Nash equilibrium in a strictly regularized game [G.sub.[alpha]]([z.sub.0]). Since a strong Nash equilibrium is also a Nash equilibrium in the strictly regularized game [G.sub.[alpha]]([z.sub.0]) (z the existence of strong Nash equilibrium implies the existence of Nash equilibrium. We need only to prove Theorem 2 and the proof of Theorem 1 is similar.

Proof of Theorem 2. Consider the situation [u.sup.*](-) = ([u.sup.*.sub.1](-), ..., [u.sup.*.sub.n](*)) in the strictly regularized game [G.sub.[alpha]] ([z.sub.0]) and define the strategies of player p [member of] N as follows:

[mathematical expression not reproducible] (19)

where [[bar.z].sub.z] is the first vertex along the cooperative trajectory [[bar.z].sub.z] = ([z.sub.0], [[bar.z].sub.1], [[bar.z].sub.z],...), on which player q [member of] S [subset] N deviates from [mathematical expression not reproducible] component of strategy [mathematical expression not reproducible] in the zero-sum game [G.sub.S,N/S]([[bar.z].sub.z]).

To prove the situation [u.sup.*] (*) = ([u.sup.*.sub.i](-), ..., [u.sup.*.sub.n](-)) is a strong Nash equilibrium in the game [G.sub.[alpha]]([z.sub.0]), we have to show that

[mathematical expression not reproducible] (20)

for all S [subset] N and all [u.sub.s] = {[u.sub.q], q [member of] S}.

It is easy to see when the n-tuple [u.sup.*](*) is played the game develops along the cooperative trajectory [bar.z]. If under the situation [u.sup.*](*) [parallel] [u.sub.s](*) the trajectory Sis also realized, then (20) will be true.

Suppose the strategy [u.sub.s](?) differs from the strategy [u.sup.*.sub.s](?) in one of the single stage games [GAMMA]([[bar.z].sub.l]), k = 0,1,.... Denote the first vertex of path [bar.z] by [[bar.z].sub.s], on which [mathematical expression not reproducible]. In the situation [u.sup.*](*) [parallel] [u.sub.s](*), the deviating coalition S cannot obtain more than

[mathematical expression not reproducible] (21)

since, after deviating from [mathematical expression not reproducible], coalition N \ S will play

against coalition S in the zero-sum game [G.sub.S,N\S](z'), where [mathematical expression not reproducible].

From the time consistency of IDP [beta] and condition (17), we then obtain

[mathematical expression not reproducible] (22)

This completes the proof of Theorem 2.

4. The Case of Repeated Infinite Stage Game

In this part, we shall consider the case when G([z.sub.0]) is a repeated infinite stage game (RISG), in which a normal-form game appears infinite periods.

In each single stage game r, the characteristic function V(S),S [subset] N is defined by V(S) = Val[[GAMMA].sub.SN\S], where Val[[GAMMA].sub.SN\S] is a value of zero-sum game played between coalition S acting as player 1 and coalition N\S acting as player 2. In each game r, one can construct the imputation set L and the core C using the characteristic function V(S).

Consider a sequence of subgames G([[[bar.z].sub.k]) along the cooperative trajectory [mathematical expression not reproducible]. The value of game [G.sub.S,N\S]([[[bar.z].sub.k]) will be equal to

[mathematical expression not reproducible]. (23)

For any imputation [gamma] [member of] L or [gamma] [member of] C, define the time-consistent IDP [beta] as [[beta].sub.i](k) = [[gamma].sub.i],k = 0,1,... Then [[alpha].sub.i](k) = [mathematical expression not reproducible].

In the regularized game [G.sub.[alpha]]([[[bar.z].sub.k]), the existence of IDP is equivalent to the nonemptiness of the imputation set L([[[bar.z].sub.k]), i.e., the existence of a solution of the following inequalities:

[mathematical expression not reproducible]. (24)

It can be simplified as,

[mathematical expression not reproducible]. (25)

In the strictly regularized game [G.sub.[alpha]]([[[bar.z].sub.k]), the existence of IDP is equivalent to the nonemptiness of the core C([[[bar.z].sub.k]), i.e., the existence of a solution of the following inequalities:

[mathematical expression not reproducible] (26)

It can be simplified as

[mathematical expression not reproducible] (27)

Under the cooperative agreement, players will choose their cooperative strategies. But if some player i [member of] N deviates from her cooperative strategy at some stage s, s = 0, 1, 2, ..., she will play her best response to other players' cooperative strategies and get the payoff [mathematical expression not reproducible] at this stage. Suppose other players will choose their trigger strategies from the next stage until the end and the deviator's future payoff will be [mathematical expression not reproducible], player i will never deviate from her cooperative strategy. The above inequality can be simplified [mathematical expression not reproducible] which always holds. Let

[mathematical expression not reproducible]. (28)

The following theorem can be formulated.

Theorem 3. If the imputation set L defined by (25) is not empty and the discount factor satisfies [mathematical expression not reproducible], then in the repeated infinite stage game [G.sub.[alpha]] ([z.sub.0]), the situation ([u.sup.*.sub.1](),...,[u.sup.*.sub.n](*)) with payoffs ([[alpha].sub.1], [[alpha].sub.2], ..., [[alpha].sub.n]) guaranteed by the time-consistent IDP [beta] is a Nash equilibrium.

We can consider the similar deviation for coalitions. If some coalition S [subset] N acts individually and deviates from her cooperative strategy at some stage s, at the current stage she will get the payoff [mathematical expression not reproducible]. From the next stage, other players will form a coalition N \ S and choose the trigger strategies. So the deviating coalition's future payoff will be [delta]V(S)/(1 - [delta]). Coalition S will not deviate from her cooperative strategy, if [mathematical expression not reproducible]. It can be simplified to [mathematical expression not reproducible]. When [mathematical expression not reproducible], which always holds. Let

[mathematical expression not reproducible] (29)

The following theorem can be formulated.

Theorem 4. If the core C defined by (27) is not empty and the discount factor satisfies [mathematical expression not reproducible], then in the repeated infinite stage game [G.sub.[alpha]]([z.sub.0]), the situation ([u.sup.*.sub.1](*),..., [u.sup.*.sub.n] (*)) with payoffs ([[alpha].sub.1], [[alpha].sub.2],..., [[alpha].sub.n]) guaranteed by the time-consistent IDP [beta] is a strong Nash equilibrium.

5. Example

To illustrate the theoretical result, we consider a repeated infinite stage game G in which the two-person Prisoner's dilemma game [GAMMA] = {N; [U.sub.1], [U.sub.2]; [H.sub.1], [H.sub.2]) is played at each stage.

For each player i, [U.sub.i] consists of two strategies C and D. [H.sub.i]([u.sub.1],[u.sub.2]) is the payoff of player i defined by [mathematical expression not reproducible] Since strategy D dominates strategy C for each i, situation (D, D) is a Nash equilibrium in game r. But it is not Pareto optimal since situation (C, C) is better off for both players.

The following strategies describe the cooperative behaviors for two players: they choose [[bar.u].sub.1] = C, [[bar.u].sub.2] = C at each stage game. In Table 1, values of the characteristic function V(S) and the deviation payoffs in stage games are presented. Using these values, we find the Shapley value [y.sub.i] as the cooperative solution in each stage game, the imputation ai(k) from stage k, and the time-consistent IDP [[beta].sub.i](k), k = 0, 1, 2, ... (see Table 2).

From (28) and (29), we get [mathematical expression not reproducible]. Therefore, following Theorem 3, for all [delta] [member of] [3/8,1), the strategy profile (C,C) with players payoffs (6/(1 - [delta]), 6/(1 - [delta])) is a Nash equilibrium in the class of trigger strategies. It can be checked that condition (27) holds for the Shapley value in each stage game, which implies the Shapley value is in the core of each stage game. This is also true for the repeated infinite stage game G. Therefore, following Theorem 4, for all [delta] [member of] [3/8, 1), the strategy profile (C, C) with players payoffs (6/(1 - [delta]), 6/(1 - [delta])) is also a strong Nash equilibrium.

6. Conclusions

In this paper, the sufficient conditions are presented to guarantee the existence of Nash or strong Nash equilibria in multistage games, which guarantee the strategic stability of cooperative solutions. They are related closely to the discount factors and are simplified in the repeated infinite stage games. These conditions avoid the loss of super-additivity of a class of general characteristic functions. Furthermore, if random influences are taken into account, the considered problem is quite involved and this is one of our future research works.

https://doi.org/10.1155/2018/3293745

Data Availability

All data generated or analyzed during this study are included in this published article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by National Natural Science Foundation of China (No. 71571108), Projects of International (Regional) Cooperation and Exchanges of NSFC (Nos. 71611530712,61661136002), China Postdoctoral Science Foundation Funded Project (No. 2016M600525), and Qingdao Postdoctoral Application Research Project (No. 2016029).

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Lei Wang, (1,2) Cui Liu, (1) Juan Xue, (1) and Hongwei Gao (iD) (1,3)

(1) School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China

(2) College of Automation and Electrical Engineering, Qingdao University, Qingdao 266071, China

(3) Institute of Applied Mathematics of Shandong, Qingdao University, Qingdao 266071, China

Correspondence should be addressed to Hongwei Gao; cmgta2007@163.com

Received 12 January 2018; Accepted 14 October 2018; Published 1 November 2018

Academic Editor: Seenith Sivasundaram

Table 1: Characteristic function and deviating payoff. S {1.2} {1} {2} 0 V(S) 12 1 1 0 [mathematical expression 12 9 9 0 not reproducible] Table 2: Shapley value and IDP. Player i Player 1 Player 2 [[gamma].sub.i] 6 6 [[alpha].sub.i])(k) 6 / 1 - [delta] 6 / 1 - [delta] [[beta].sub.i](k) 6 6

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Title Annotation: | Research Article |
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Author: | Wang, Lei; Liu, Cui; Xue, Juan; Gao, Hongwei |

Publication: | Discrete Dynamics in Nature and Society |

Date: | Jan 1, 2018 |

Words: | 4095 |

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