Abstract

A mixed dual to the Nash equilibrium is defined for -person games in strategic form. In a Nash equilibrium every player’s mixed strategy maximizes his own expected payoff for the other players’ strategies. Conversely, in the dual equilibrium every players have mixed strategies that maximize the remaining player’s expected payoff. Hence this dual equilibrium models mutual support and cooperation to extend the Berge equilibrium from pure to mixed strategies. This dual equilibrium is compared and related to the mixed Nash equilibrium, and both topological and algebraic conditions are given for the existence of the dual. Computational issues are discussed, and it is shown that for each there exists a game for which no dual equilibrium exists.

1. Introduction

The mathematical analysis of both competition and cooperation falls within the realm of game theory, whose systematic development began with von Neumann and Morgenstern [1]. For a game with players, Nash [2] later assumed that the players are rational and hence selfish. He then defined an equilibrium in which every player’s strategy maximizes his payoff for the other players’ strategies. Modern game theory [3–5] has long required that any rational solution concept for a game should be a Nash equilibrium. However, in games known as social dilemmas [6, 7] selfish behavior conflicts with group interests; and individual players do better by cooperating. The less well-known Berge equilibrium was introduced for pure strategies in [8] and formalized in [9]. A Berge equilibrium is a pure strategy profile in which every players have strategies that maximize the remaining player’s payoff. It has been increasingly studied as a model of mutual support and cooperation as in [10–15].

In this paper we consider -person games in strategic form and extend the Berge equilibrium from pure to mixed strategies to provide a dual to the Nash equilibrium. In this dual equilibrium, every players have strategies that maximize the remaining player’s expected payoff. The Nash equilibrium and the dual equilibrium thus model opposite decision criteria for choosing the player’s actions, regardless of whether these actions are independently selected by the players, are coordinated by the players, or are even prescribed. For example, the results here are valid if an arbiter assigns actions to the players as in [16].

In Section 2 the dual equilibrium is formally defined and related to the Nash equilibrium. In Section 3 topological and algebraic conditions are established for the existence of a dual equilibrium. In Section 4 computation complexity issues are discussed. In Section 5 it is shown that for each there exists a game for which no dual equilibrium exists. Conclusions are stated in Section 6.

2. Relationships between the Dual and Nash Equilibria

The following definitions and notation are used. Let be an -person game in strategic form, where is the index set for the players, is the finite set of actions for player , and is the von Neumann-Morgenstern utility of player for a pure strategy profile . Write for and denote the set of mixed strategies for player by . A mixed-strategy profile is an -tuple of individual randomized strategies, where is the probability that player uses the pure strategy . For any , abbreviate the strategy profile by . Similarly write , , and . For , extend from to by defining the expected utility function . Writing , , and noting that a pure strategy is a special case of a mixed strategy, we can derive the following useful identities:The Nash equilibrium is now formally defined for comparison with the dual equilibrium.

Definition 1 (NE). The strategy profile is a Nash equilibrium (NE) of if and only if

Definition 2 (DE). The strategy profile is a dual equilibrium (DE) of if and only if Designating a strategy profile satisfying (6) as a dual equilibrium may be justified as follows. Equations (5) and (6) involve systems of optimization problems defining NEs and DEs. These systems may be regarded as symmetric dual systems in the sense that Definition 1 becomes Definition 2 and vice versa when the subscripts and are interchanged on the right (but not left) sides of (5) and (6). Thus if (5) is considered the primal system and (6) the dual system, then the dual system of the dual system is the primal system. This duality is manifested in the fact that in Definition 1 every player’s NE strategy selfishly maximizes his own payoff for the other players’ NE strategies, while the opposite situation occurs in Definition 2. For a DE, every players have DE strategies that maximize the remaining player’s payoff. Thus a DE exhibits the musketeer property in [17] of “all for one, one for all.” Alternately, in an NE no individual player can improve his expected payoff with a unilateral change in strategy, while in a DE no individual player’s payoff can be improved by any change in the remaining players’ DE strategies.
The DE is the mixed extension of the Berge equilibrium (BE) of [11–14] and the mutual-max outcome of [18]. For a BE, which involves only pure strategies, mutual cooperation occurs in a single game. For a mixed-strategy DE, the probability can be interpreted in two ways [19]. The first interpretation is the fraction of time in the long run that player would choose to play pure strategy in a series of repeated games in which the mutually cooperative players invoke decision criterion (6). The second interpretation of considers a mixed-strategy equilibrium as a steady state in a large population with subpopulations. In this case is the fraction of subpopulation preferring pure strategy according to decision criteria (6).
Some results relating DE and NE are now established. Let the function be one-to-one on the index set . Then is a derangement of if and only if , . Thus associated with is the deranged game , where . In other words, for the expected utility function of any player is . In addition, for a strategy profile of , let be its deranged strategy profile , where is the unique for which . Obviously if is a DE of , then the deranged strategy profile is also a DE of the deranged game .

Theorem 3. If is a dual equilibrium for any -person game , then is a Nash equilibrium for for any derangement of .

Proof. It is first proved that if is a DE of a game , then is an NE of for any derangement of . Let and be an arbitrary derangement such that . From (6)Expanding (7) gives that In (8) set for , . Hence, for ,From (9), for and any derangement of , Thus from (10) and (5), is an NE for . It follows that if is a DE of any -person game , then is an NE of for any derangement of .
We next use the fact that if is a DE of , the deranged strategy profile is also a DE of the deranged game . Then from the first part of the proof, is an NE of the original game obtained by applying the inverse derangement of to .

Corollary 4. has at least as many NEs as DEs.

Proof. Let be the cyclic derangement given by , , and . Since is one-to-one, it follows from Theorem 3 that there is a one-to-one correspondence between the DEs of and a subset of the NEs for . Hence there are at least as many NEs as DEs.

In Section 5 it is shown that there can be more NEs than DEs. The converse of Theorem 3 is true in general only for two-person games, in which case an argument similar to that for Theorem 3 yields the following result of [20], where the zero-sum case is also considered.

Corollary 5. A strategy profile is a DE for the two-person game where Player 1 has utility function and Player 2 has utility function if and only if is an NE for the dual two-person game where Player 1 has utility function and Player 2 has utility function .

Example 6. Consider , with the payoff matrix of Table 1, where Player 1 has pure strategies and Player 2 has pure strategies . For simplicity, write , for Player 1 and , for Player 2. We apply Corollary 5. Using standard procedures for finding all NEs of two-person games [3], we obtain all DEs of by determining all NEs associated with the payoff matrix obtained by interchanging the payoffs in Table 1. The unique DE has , and , , with expected payoffs (2.75, 5.60) with respect to the original payoff matrix of Table 1. By comparison, the unique NE of has , and , , with expected payoffs (2.75, 5.65). Thus, the expected payoff vector for the NE dominates that of the DE in a Pareto sense. In general, the expected DE payoff for any player in may be larger than, smaller than, or equal to the expected NE payoff for . In this regard, neither mutual cooperation nor selfishness is necessarily better than the other.

3. Existence Conditions for Dual Equilibria

In this section sufficient conditions are presented for the existence of a DE for . For any two-person game, Corollary 5 implies that a DE exists since an NE exists [2]. For , however, the existence of a DE is more complicated. The following preliminaries are needed.

Definition 7. A correspondence is a set-valued mapping such that is a subset of , . The domain of is , the range is , and the graph is .

Definition 8. Let and be topological vector spaces. The correspondence is upper semicontinuous at if and only if whenever the sequence in has , then .

Definition 9. Let be a topological vector space, and let be a nonempty convex set in . The function is quasiconcave on if and only if the level set is a convex set for all real .

Result 1 (Kakutani fixed point theorem [21]). Let be a nonempty compact convex set in a finite-dimensional topological vector space , and let . If is upper semicontinuous at and if is a nonempty convex subset of , then for which . The point is called a fixed point of on .

Topological conditions for the existence of a dual equilibrium are now derived. For each player and each , denote the best support correspondence by From [3] the sets , , and can be identified, respectively, with nonempty, convex, and compact subsets of the finite-dimensional Euclidean spaces for , , and for . For each , is continuous with respect to and , so the sets and are nonempty. Hence the correspondence given by is nonempty if and only if satisfies (6). The following lemma has thus been established.

Lemma 10. A strategy profile is a DE for the game if and only if is a fixed point of   on .

Result 1 and Lemma 10 are next used to prove a topological sufficient condition.

Theorem 11. If   is a nonempty convex set, , then there exists a DE for the game .

Proof. We show that is upper semicontinuous on the nonempty, compact, and convex set . Consider convergent sequences and in , where , , and , . For it follows from the definition of that , . But the expected utility function is continuous on , so , . Hence , , and so . Thus is upper semicontinuous on . By assumption is also a nonempty convex set of , . Hence from Result 1 there exists for which , and so is a DE by Lemma 10.

More restrictive sufficient conditions are given as a corollary.

Corollary 12. If , , is a quasiconcave function of for each fixed and if   is nonempty, , then there exists a DE for the game .

Proof. Fix and . Since is a quasiconcave on , it follows that the level set is a convex set in for all real . In particular, setting implies that , , , Thus , and is a nonempty convex set, , . Hence, for , is convex in addition to being nonempty by assumption, so Theorem 11 applies.

4. Computational Complexity

For the DE to be useful as a solution concept, it must be computable. Since NEs can be difficult to obtain [22, 23], we briefly discuss the computational complexity of DEs as compared to NEs. Let (# players) (# all possible joint actions) denote the size of the input data for the game . It is shown in [22] that is the computational complexity on a deterministic Turing machine for checking whether a pure NE exists for and then computing all such NEs. An algorithm described in [20] uses a regret matrix to compute all pure NEs of and readily extends to . A similar algorithm to compute all Berge equilibria (pure DEs) was proposed for in [24] using a disappointment matrix. Both algorithms have computational complexity . The general problem of finding a mixed NE for , which is well known to exist, involves solving the following algebraic equivalence for (5), which is proved in [25] using (1) and (2).

Result 2. The strategy profile is an NE of if and only if For a given and , (11) can be checked by total enumeration in to determine if is an NE. The problem of computing an NE for is thus a member of the complexity class NP, yet obtaining an NE from (11) by solving for may not be feasible for large . Indeed, it is shown in [23] that this problem is PPAD-complete. In other words, this problem is believed to be computationally very difficult but may not be NP-complete. The next theorem gives an algebraic equivalence for (6) similar to (11) for DEs.

Theorem 13. The strategy profile is a DE of if and only if

Proof. From (3) and (6), is a DE if and only ifFix and let maximize over . This maximum exists since is continuous in on the closed and bounded set . Then by inspection, on the right side of (13) is maximized over when for and for . Hence and (12) follows from (4) and (14).

For a given and , (12) can also be checked by total enumeration in to determine if is a DE. Hence the problem of computing a DE for is also a member of the complexity class NP. Nonetheless, for , using (12) to obtain a DE or else determine that one does not exist is computationally harder than using (11) to find an NE. For DEs, the maximization for each on the right side of (12) is over , as opposed to being over in (11) for NEs. This increased difficulty results from the fact that determining a DE from (6) requires that the components for each of the maximizations of (6) must match, whereas in (5) only the single component for all the maximizations of (5) must match. Thus it appears more likely for the DE computational problem to be NP-complete than the NE computational problem. Theorem 11 further suggests this fact since stringent conditions are required there to guarantee the existence of a DE, whereas an NE always exists. Indeed, it is next shown that a DE may not exist.

5. Mutual Cooperation Impossibility Theorem

The following result establishes that mutual cooperation in the form of a DE may be impossible when the number of players is greater than two. As noted above, a DE always exists when . For , however, existence depends on the payoff matrix.

Theorem 14. For every there exists a game with no DE.

Proof. The result is first proved for . Consider a game with the payoff matrix of Table 2, where Player 1 has pure strategies , Player 2 has pure strategies , and Player 3 has pure strategies . For simplicity, write and for Player 1, and for Player 2, and and for Player 3.
Assume that has a DE . Then it follows from (6) that there exist , , , , such thatSolving the maximization in (15) yields the only possible solutions , , , for (15) as follows.
If , the solution isIf , the solution isIf , the two solutions areas well asEquations (18)–(21) therefore require that any DE in (6) must satisfyNext utilizing (22) to solve the maximization in (16) yields the only two possible solutions , , , for (16) as follows. If and , the solution isIf and , the solution isHence (22)–(24) require that Thus from (18)–(25), the only two possible solutions , , jointly solving the maximizations in (15) and (16) areas well asBut using (25) to solve the maximization in (17) gives the only two possible solutions , solving the maximization in (17) to beas well asIt now follows that the only possible solutions (26) and (27) for (15) and (16) differ from the only possible solutions (28) and (29) for (17), in contradiction to the assumption that has a DE. Hence does not have a DE, and the theorem is true for .
For we generalize to a game with no DE. Let , , and let the von Neumann-Morgenstern utility vector on beAssume that has a DE . In a manner similar to that giving equations (26) and (27) for , it can be shown from (30) that the joint maximizations for Players in (6) yield the only possible two solutions for , andfor . However, in a manner similar to that giving (28) and (29) for , it can also be shown from (30) that the maximization for Player in (6) yields the only possible two solutionsas well asSince (31) and (32) contradict (33) and (34), has no DE, and the proof is complete.