The Complexity of Computational Problems about Nash Equilibria in Symmetric Win-Lose Games

07/24/2019 ∙ by Vittorio Bilò, et al. ∙ University of Cyprus 0

We revisit the complexity of deciding, given a bimatrix game, whether it has a Nash equilibrium with certain natural properties; such decision problems were early known to be NP-hard <cit.>. We show that NP-hardness still holds under two significant restrictions in simultaneity: the game is win-lose (that is, all utilities are 0 or 1) and symmetric. To address the former restriction, we design win-lose gadgets and a win-lose reduction; to accomodate the latter restriction, we employ and analyze the classical GHR-symmetrization <cit.> in the win-lose setting. Thus, symmetric win-lose bimatrix games are as complex as general bimatrix games with respect to such decision problems. As a byproduct of our techniques, we derive hardness results for search, counting and parity problems about Nash equilibria in symmetric win-lose bimatrix games.

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1 Introduction

1.1 Framework and Motivation

1.1.1 Nash Equilibria, Win-Lose Bimatrix Games and Symmetric Games

Among the most fundamental computational problems in Algorithmic Game Theory are those concerning the Nash equilibria [29, 30] of a game, where no player could unilaterally deviate to increase her expected utility. Such problems have been studied extensively [1, 4, 7, 9, 10, 11, 12, 13, 14, 20, 25, 26] for 2-player games with rational utilities given by a bimatrix. There are two prominent special cases of such general bimatrix games: win-lose and symmetric.

  • Utilities are taken from in win-lose bimatrix games, originally put forward in [12].

  • In a symmetric game [29, 30], players have identical strategy sets and the utility of a player is determined by the multiset of strategies chosen by her and the other players, with no discrimination. By a classical result of Nash, every symmetric game has a symmetric Nash equilibrium, where all players are playing the same mixed strategy [29, 30]. A symmetrization transforms a given bimatrix game into a symmetric one; the target is that a Nash equilibrium for the original bimatrix game can be reconstructed efficiently from a (symmetric) Nash equilibrium for the symmetric bimatrix game (cf. [8, 17, 21]).

1.1.2 The Search Problem

The fundamental theorem of Nash [29, 30] that a Nash equilibrium is guaranteed to exist for a finite game makes its search problem total; hence, the search problem is not -hard unless  [27, Theorem 2.1]. In a series of breakthrough papers culminating in [9, 14], it was established that, even for bimatrix games, the search problem is complete for  [31], a complexity class capturing the computation of discrete fixed points; under suitable formulations, the problem is -complete for games with more than two players [16, Theorem 18].

Abbott, Kane and Valiant [1] presented a polynomial time transformation of a general bimatrix game into a win-lose bimatrix game, accompanied with a polynomial time map returning a Nash equilibrium for the general game when given one for the win-lose game. So the search problem is -hard for win-lose bimatrix games, which suggests that hardness is not due to the rational utilities. The search problem for a symmetric Nash equilibrium in a symmetric bimatrix game is also -hard, thanks to the two symmetrizations from 1950 due to Brown and von Neumann [8] and due to Gale, Kuhn and Tucker [17], respectively; the complexity of the search problem for any Nash equilibrium in a symmetric bimatrix game has been mentioned as an open problem by Papadimitriou [32, Section 2.3.1].

1.1.3 Decision, Counting and Parity Problems

Decision problems arise naturally by twisting the search problem in simple ways that deprive it from its existence guarantee for a Nash equilibrium. Here is a non-exhaustive list of such natural decision problems (see Section 2.3 for formal statements): given a game, does it have:

  • At least Nash equilibria for some fixed integer ? (The special case with was introduced in [20]; the cases with are considered here for the first time.)

  • A Nash equilibrium where each player has expected utility at least (resp., at most) a given number? [20]

  • A Nash equilibrium where the total expected utility of players is at least (resp., at most) a given number? [13]

  • A Nash equilibrium where the players’ supports contain (resp., are contained in) a given set of strategies? [20]

  • A Nash equilibrium where the players’ supports have sizes greater (resp., smaller) than a given integer? [20]

  • A Nash equilibrium where each player is using uniform probabilities on her support? 

    [7]

  • A Nash equilibrium where some player is using non-uniform probabilities on her support?

  • A symmetric Nash equilibrium?

  • A non-symmetric Nash equilibrium? (cf. [32, Section 2.3.1])

Some of these problems were first shown -hard for symmetric bimatrix games in the seminal paper by Gilboa and Zemel [20, Section 1.2]. Later Conitzer and Sandholm [13, Section 3] gave a unifying polynomial time reduction from CNF SAT to show in one shot -hardness results, encompassing those from [20], for symmetric bimatrix games; their reduction yields games with Nash equilibria mirroring satisfiability parsimoniously. McLennan and Tourky [25, Theorem 1] refined some of these -hardness results for imitation bimatrix games, where the utility of the imitator is if and only if she chooses the same strategy as the mover [26]. The problems (vii) and (viii) are considered here for the first time; (viii) is trivial for symmetric games.

The present authors introduced the decision problem about the equivalence of the sets of Nash equilibria of two given games, which they proved co--hard [4, Theorem 1]:

  • Do the two given games have the same sets of Nash equilibria?

An additional decision problem, which is trivial for bimatrix games, becomes -hard already for 3-player games [4, Theorem 2]:

  • A Nash equilibrium where all probabilities are rational? [4]

It is natural to ask whether these problems remain -hard when restricted to win-lose games. To the best of our knowledge, this important question has been addressed only in [7, 12]. It was shown in [12, Theorem 1] that the decision problem (i) with is -hard for win-lose bimatrix games; there so is a variant of (ii) for imitation win-lose bimatrix games. The decision problem (vi) was shown -hard for imitation win-lose bimatrix games in [7, Theorem 1].

The counting problem and the parity problem for Nash equilibria ask for the number and for the parity of the number of Nash equilibria, respectively, for a given game. To each decision problem there corresponds a counting problem and a parity problem, asking for the number and the parity of the number of Nash equilibria with the corresponding property, respectively. By the parsimonious property of the reduction in [13], these counting (resp., parity) problems are -hard (resp., -hard) for general bimatrix games; the -hardness (resp., -hardness) is inherited from the -hardness [35] (resp., -hardness [33]) of computing the number (resp., the parity of the number) of satisfying assignments for a CNF SAT formula.

1.1.4 State-of-the-Art and Statement of Results

The polynomial time transformation of a general bimatrix game into a win-lose bimatrix game from [1] gave no guarantee on the preservation of properties of Nash equilibria; so, it had no implication on the complexity of deciding the properties for win-lose bimatrix games. Thus, the composition of a polynomial time reduction from an -hard problem to a decision problem about Nash equilibria for general bimatrix games (cf. [13, 20]) with the polynomial time transformation from [1] does not yield a polynomial time reduction from the -hard problem to the decision problems for win-lose bimatrix games, and their complexity remained open.

In this work, we settle the complexity of the decision problems about Nash equilibria [4, 7, 12, 13, 20, 25, 26, 28] for symmetric win-lose bimatrix games. Our main result is that these problems are -hard for symmetric win-lose bimatrix games (Theorems 7.1 and 7.7). Further, deciding the existence of a symmetric Nash equilibrium is -hard for win-lose bimatrix games (Theorem 7.9), and of a rational Nash equilibrium [4] is -hard for win-lose 3-player games (Theorem 7.10). As a byproduct, we derive, for symmetric win-lose bimatrix games, the -hardness of the search problem (Theorem 6.7), the -hardness of the counting problem and the -hardness of the parity problem (Theorem 6.8), and the -hardness and the -hardness of the counting and the parity version, respectively, of each decision problem (Theorem 7.1).

1.2 The Techniques

We combine three powerful techniques: (1) Gadget games (Section 4). (2) A win-lose reduction (Section 5). (3) The classical -symmetrization  [21], with a new analysis tailored to win-lose bimatrix games (Section 6). All three techniques involve the positive utility property: the property is enjoyed by the gadget games, which form a key component of the reduction, and it is required for the new analysis of the -symmetrization. The property requires that each player may, in response to the choices of the other players, always choose a strategy to make her utility strictly positive; it is strictly weaker than the strictly positive utilities property, assumed for the analysis of the -symmetrization in [22]. We prove that computing a Nash equilibrium for a win-lose bimatrix game with the property is as hard as computing a Nash equilibrium for a win-lose bimatrix game (Proposition 3.1); hence, it is -hard. So assuming the positive utility property for symmetric win-lose bimatrix games does not simplify the search problem.

1.2.1 Gadget Games

A gadget game is a fixed game with a small number of players, which is void by design of the property associated with some decision problem about Nash equilibria (from Section 2.3). We present win-lose gadget games covering all such properties. For example, the win-lose 3-player irrational game (Section 4.2) has a single irrational Nash equilibrium, dismatching the problem (x); the win-lose non-uniform game (Section 4.3) has no uniform Nash equilibrium, dismatching the problem (vi); the win-lose non-symmetric game (Section 4.4) has no symmetric Nash equilibrium, dismatching the problem (viii); for a given integer , the win-lose diagonal game (Section 4.5) has exactly Nash equilibria, dismatching the problem (i).

1.2.2 The Win-Lose Reduction

The technical backbone of the complexity results for decision, counting and parity problems is a win-lose reduction we design, taking a fixed win-lose gadget game with the positive utility property as a parameter (Section 5). The reduction transforms a given 3SAT formula into a win-lose game ; and have the same number of players. We prove that each Nash equilibrium for is always a Nash equilibrium for (Lemma 5.3 (Condition (1))), while has additional Nash equilibria if and only if is satisfiable; those are related parsimoniously to the satisfying assignments of . More important, each additional Nash equilibrium enjoys properties that do not depend on (Propositions 5.13 and 5.14). So a decision problem associated with some particular property reduces to deciding the inequivalence of a win-lose game with a fixed win-lose gadget game dismatching the property, and their equivalence is co--hard.

1.2.3 The -Symmetrization

To extend hardness results from win-lose to symmetric win-lose bimatrix games, we seek win-lose symmetrizations transforming a win-lose bimatrix game into a symmetric one. The -symmetrization [21] is the single win-lose symmetrization we know of; for emphasis, we shall call it the win-lose -symmetrization.***The symmetrization due to Brown and von Neumann [8] involves the sum of the two matrices, which may increase the number of utility values. The symmetrization due to Gale, Kuhn and Tucker [17] introduces utilities . So they both result in more than two values for the utilities, and none of the symmetrizations is win-lose. We use tools from [22] to provide a new analysis of the -symmetrization, tailored to win-lose bimatrix games with the positive utility property (Section 6), which yields a tight characterization of the Nash equilibria for the resulting symmetric win-lose bimatrix game : they may only result, albeit in a non-parsimonious way, as balanced mixtures [22] involving Nash equilibria for (Theorems 6.5 and 6.6).The balanced mixture was introduced in [22] for the analysis of the -symmetrization; it was later used in [23], where it was called the -product, for the analysis of the -symmetrization [17]. By Propositions 5.13 and 5.14, the set of balanced mixtures is determined by the satisfiability of ; hence, the “cascade” of the win-lose reduction and the win-lose symmetrization is a non-parsimonious reduction from 3SAT to decision problems about Nash equilibria in symmetric win-lose bimatrix games.

1.3 Three-Steps Plan of the -Hardness Proof

  • Step 1: For a property of Nash equilibria for symmetric win-lose bimatrix games, fix a gadget game: a win-lose bimatrix game whose Nash equilibria dismatch the property.

  • Step 2: Apply the win-lose reduction with parameter on the formula to get the win-lose bimatrix game .

    When is unsatisfiable, the Nash equilibria for are those for (Proposition 5.13); hence, they dismatch the property. When is satisfiable, there are additional Nash equilbria for , which satisfy the property (Proposition 5.14). (It follows that the properties of the Nash equilibria for and of the additional ones for are “conflicting”.) has a Nash equilibrium matching the property if and only if is satisfiable. The associated decision problem is -hard for win-lose bimatrix games.

    Note that Step 2 only allows proving the -hardness of decision problems associated with properties matched by the additional Nash equilibria for the win-lose game .

  • Step 3: Apply the win-lose -symmetrization on to get the symmetric win-lose bimatrix game , whose Nash equilibria may only result as balanced mixtures involving Nash equilibria for . Some balanced mixtures preserve the properties of the Nash equilibria for , while other may dismatch them. This allows incorporating properties either matched or dismatched by the additional Nash equilibria for . We show that the associated decision problem is -hard for symmetric win-lose bimatrix games by establishing a suitable equivalence to the satisfiability of as follows:

    There are three possible cases in achieving the equivalence between the existence of a Nash equilibrium matching the property for and the satisfiablity of : The existence of a Nash equilibrium matching the property for the win-lose game is equivalent to the satisfiability of , and two extra conditions hold: When is unsatisfiable, every balanced mixture dismatches the property. When is satisfiable, there is a balanced mixture matching the property. In this case, the equivalence is preserved by the win-lose -symmetrization. There is no Nash equilibrium for matching the property, no matter whether is satisfiable or not, and two extra conditions hold: When is unsatisfiable, every balanced mixture dismatches the property. When is satisfiable, there is a balanced mixture matching the property. Hence, the existence of a Nash equilibrium matching the property for is equivalent to the satisfiability of . When is unsatisfiable, there is a balanced mixture matching the property. When is satisfiable, there is an additional balanced mixture matching the property. Hence, this excludes the equivalence between the existence of a Nash equilibrium matching the property for and the satisfiability of . We extend to by “embedding” the symmetric win-lose gadget as a subgame so that the balanced mixtures arising when is unsatisfiable are “destroyed’, while the balanced mixtures arising when is satisfiable are not “destroyed”. This induces the equivalence of the existence of a Nash equilibrium matching the property for and the satisfiability of . Since equivalence to the satisfiability of holds in all cases, the -hardness of the associated decision problem for symmetric win-lose games follows.








1.4 The Complexity Results and Significance

1.4.1 Decision Problems

The three-steps proof plan from Section 1.3 is used to yield, as our main result, the -hardness of deciding a handful of properties for symmetric win-lose bimatrix games (Theorems 7.1 and 7.7). These complexity results imply that symmetric win-lose bimatrix games are as complex as general bimatrix games with respect to the handful of decision problems about Nash equilibria considered before in [4, 7, 12, 13, 20, 25, 26, 28].

While the win-lose reduction applies to games with any number of players, the -symmetrization is specific to bimatrix games. Hence, Step 1 and Step 2 suffice on their own for proving the -hardness of decision problems about Nash equilibria which either remain trivial for symmetric games (such as deciding the existence of a symmetric Nash equilibrium) or become non-trivial for win-lose games with more than two players (such as deciding the existence of a rational Nash equilibrium):

  • Choosing as the win-lose bimatrix game with no symmetric Nash equilibrium yields the -hardness of deciding the existence of a symmetric Nash equilibrium for win-lose bimatrix games (Theorem 7.9). So win-lose bimatrix games are as complex as general bimatrix games with respect to deciding the existence of a symmetric Nash equilibrium.

  • Choosing as the win-lose 3-player game with a single irrational Nash equilibrium yields the -hardness of deciding the existence of a rational Nash equilibrium for win-lose 3-player games (Theorem 7.10). So win-lose 3-player games are as complex as general 3-player games with respect to deciding the existence of a rational Nash equilibrium.

These results represent an analog of the earlier result that win-lose bimatrix games are as complex as general bimatrix games with respect to the search problem for a Nash equilibrium [1].

1.4.2 Search, Counting and Parity Problems

We show that computing a Nash equilibrium for a symmetric win-lose bimatrix game is -hard (Theorem 6.7); the proof appeals to the characterization of the Nash equilibria for the win-lose -symmetrization of a win-lose bimatrix game (Theorem 6.6).

Recall that the reduction used for the -hardness proof is non-parsimonious. Hence, the counting and parity problems about the number and the parity of the number of Nash equilibria for a symmetric win-lose bimatrix game, as well as the counting and parity versions of decision problems about Nash equilibria, are not immediately -hard and -hard, respectively. Nevertheless, the -hardness proof yields -hardness and -hardness results as well:

  • We show that computing the number (resp., the parity of the number) of Nash equilibria for a symmetric win-lose bimatrix game is -hard (resp., -hard) (Theorem 6.8). The proof draws from simple formulas for the numbers of Nash equilibria for and in terms of the numbers of Nash equilibria for and of satisfying assignments for , denoted as ; the formulas follow from properties of the gadget games, the win-lose reduction and the win-lose -symmetrization. Solving the formula for yields the -hardness; computing the parity of , denoted as , from the formula yields the -hardness.

  • We examine the balanced mixtures of Nash equilibria for matching each property to derive simple formulas for the numbers of Nash equilibria for matching the property. Hence, the counting versions of these decision problems are -hard (Theorem 7.1).

    Furthermore, all but two of the formulas are parity-preserving: the parity of the number of Nash equilibria for matching the property yields ; this implies the -hardness of all but one of the parity versions (Theorem 7.1), the exception made by a trivial parity problem. In this sense, the “cascade” of the win-lose reduction and the win-lose -symmetrization is parity-preserving: it yields the parity of the number of satisfying assignments. As far as we know, this is the first non-parsimonious, yet parity-preserving, reduction to decision problems about Nash equilibria.

1.5 Evaluation and Comparison to Related Work

None of the works [4, 7, 12, 13, 20, 25, 26, 28] on the complexity of decision and counting problems about Nash equilibria in bimatrix games considered the two restrictions to win-lose bimatrix and symmetric bimatrix games in simultaneity; neither did the works [1, 9, 10, 14] on the complexity of the search problem. This work encompasses all of the decision problems, together with their counting and parity versions, in the common framework composed of the gadget games, the win-lose reduction and the win-lose -symmetrization. So, problem-specific reductions and techniques, such as the regular subgraphs technique from [7] or the good assignments technique from [12], are not necessary.

1.5.1 Complexity Results

Theorems 7.1, 7.7 and 7.10 improve and extend previous -hardness and -hardness results for the decision problems from [4, 7, 12, 13, 20, 25, 26, 28] as follows:

  • The -hardness of the handful of decision problems improves the results in [13, 20], which applied to general symmetric bimatrix games, and extends their refinements in [25, 26], which applied to imitation bimatrix games.

  • The -hardness of deciding the existence of a uniform Nash equilibrium for a symmetric win-lose bimatrix game extends [7, Theorem 1], which applied to imitation win-lose bimatrix games.

  • The -hardness of deciding the existence of Nash equilibria, with , for a symmetric win-lose bimatrix game improves [12, Theorem 1], which addressed the special case and applied to win-lose bimatrix games.

  • The -hardness of deciding the existence of a non-symmetric Nash equilibrium for a symmetric win-lose bimatrix game improves [28, Theorem 3], which applied to general symmetric bimatrix games.

  • The -hardness of deciding the Nash-equivalence of a given symmetric win-lose lose bimatrix game with the -symmetrization of a fixed win-lose gadget game improves [4, Theorem 1], which established the Nash-equivalence of a given general bimatrix game with a fixed general gadget game.

  • The -hardness of deciding the existence of a rational Nash equilibrium for a win-lose 3-player game improves [4, Theorem 2], which applied to general 3-player games.

Corresponding extensions and improvements follow for the -hardness of the counting versions of these decision problems. In particular:

  • The -hardness of counting the number of non-symmetric Nash equilibria for a symmetric win-lose bimatrix game improves [28, Theorem 5], which applied to general symmetric bimatrix games.

Of particular interest is the -hardness (resp., -hardness) of computing, for a given symmetric win-lose bimatrix game, the number (resp., the parity of the number) of symmetric Nash equilibria; recall that the corresponding decision problem, asking for the existence of a symmetric Nash equilibrium for a given symmetric game, is trivial by the early result of Nash [29, 30].

To the best of our knowledge, the proof in [36] that computing the parity of the number of satisfying assignments for a read-twice formula, where each variable occurs at most twice, is -hard is the only proof of -hardness employing a reduction from a corresponding -completeness proof which, although non-parsimonious, is parity-preserving.

Figure 4 tabulates the presented complexity results for decision problems in comparison to those in [4, 7, 12, 13, 20, 28]. Theorem 6.7 improves the -hardness of the search problem for win-lose bimatrix games from [1] to symmetric win-lose bimatrix games. The -hardness of computing the number of Nash equilibria for a symmetric win-lose bimatrix game in Theorem 6.8 improves [13, Corollary 12], which applied to general symmetric bimatrix games.

1.5.2 The Win-Lose Reduction

The unifying reduction from CNF SAT, introduced in [13] and further developed in [4, 24], is inadequate to cover win-lose games. (See Section 5.1.1 for a technical discussion.) New ideas and technical constructs, such as pair variables, were needed for the win-lose reduction, which thus improves vastly in yielding a win-lose game while still preserving the equivalence between the existence of additional Nash equilibria for the game and the satisfiability of . We needed the finer structure of a 3SAT formula in order to guarantee that certain deviations in the game constructed by the win-lose reduction are non-profitable. Specifically, the proofs for Propositions 5.13 and 5.14 rely on choosing as a 3SAT formula in an essential way.

The win-lose reduction generalizes the reduction from [13], which yielded a bimatrix game, to yield an -player game, with ; it is this generalization that has enabled showing complexity results about decision problems, such as deciding the existence of a rational Nash equilibrium (Theorem 7.10), which are trivial for bimatrix games but become -hard for -player games.

We note that the reduction in [13] yielded the -hardness of approximate versions of some of the decision problems over general bimatrix games. We anticipate that the presented composition of the win-lose reduction and the win-lose -symmetrization yields corresponding -hardness results over symmetric win-lose bimatrix games.

1.5.3 The -Symmetrization

Our analysis of the -symmetrization [21] yields the first complete characterization of the Nash equilibria for the symmetric game resulting from a win-lose symmetrization. For its previous analysis in [22], it was assumed that all utilities in the original bimatrix game were strictly positive in order to guarantee that the utilities added for the -symmetrization are strictly less than any utility; thus, the original bimatrix game were not a win-lose game. Recall that shifting and scaling the utilities does not alter the set of Nash equilibria for a game. Thus, a bimatrix game with at most two values for the (strictly positive) utilities could be transformed into an equivalent win-lose game by shifting and scaling the utilities. (Note that the case where all utilities in the original bimatrix games are equal is degenerate.) But then it is no more the case that the utilities added for the -symmetrization are strictly less than both and , and the analysis of the -symmetrization from [22] is no more applicable.

To circumvent this difficulty, we need a different assumption on the utilities in the original bimatrix game which is not too restrictive to make the algorithmic problems easier, while it is strong enough to yield a tight characterization of the Nash equilibria for the -symmetrization; this is the positive utility property. We analyze the -symmetrization for win-lose bimatrix games (Section 6), replacing the assumption of strictly positive utilities from [22] with the positive utility property. The resulting characterization of Nash equilibria for the -symmetrization of a win-lose bimatrix game with the positive utility property is similar to the characterization of Nash equilibria for the -symmetrization of a bimatrix game with strictly positive utilities in [22]: Cases (C’.2) and (C’.3) from Theorem 6.6 correspond to the cases addressed in [22, Theorem 4.1]; [22, Theorem 4.2] and [22, Theorem 4.3] correspond to Theorems 6.5 and 6.6, respectively.

1.5.4 Tractable Cases

Bilò and Fanelli [3, Section 4]

consider a Linear Programming formulation, denoted as

LR, of Nash equilibria in bimatrix games, which is a relaxation of a corresponding Quadratic Programming formulation they propose. They show [3, Theorem 1] that any feasible solution to LR is a Nash equilibrium when the bimatrix game is regular [3, Definition 4]: the sum of corresponding entries in the two matrices remains constant either accross rows or across columns; so, regular bimatrix games are the rank-1 bimatrix games, encompassing zero-sum bimatrix games. As a consequence of [3, Theorem 1], a Nash equilibrium optimizing any objective function involving the players’ utilities and meeting any set of constraints expressible through Linear Programming can be computed in polynomial time (through solving LR). Since nearly all decision problems about Nash equilibria studied in this work are so expressible, it follows that they are polynomial time solvable when restricted to regular bimatrix games. (A notable exception is problem (v) from Section 1.1.3, which remains -hard even when restricted to zero-sum games [20].) This positive result stands in contrast to the established -hardness for symmetric win-lose bimatrix games. Other positive results on the search problem for a Nash equilibrium in planar, sparse and and minor-free and minor-closed win-lose bimatrix games appear in [2][11] and [15], respectively.

1.6 Paper Organization

Section 2 introduces the game-theoretic framework. Section 3 considers the positive utility property for win-lose bimatrix games. The win-lose gadget games are presented in Section 4. Section 5 treats the win-lose reduction and its properties. The win-lose -symmetrization and its properties are analyzed in Section 6. Section 7 presents the complexity results. We conclude, in Section 8, with a discussion of the results and some open problems.

2 Framework and Preliminaries

Games, Nash equilibria and their decision problems are treated in Sections 2.12.2 and 2.3, respectively.

2.1 Games

A game is a triple , where: (i) is a finite set of players with , and (ii) for each player , is the set of strategies for player , and the utility function is a function for player . The game is win-lose (resp., general) if for each player , the utility function is a function (resp., ). For each player , denote ; denote . For an integer , - is the set of -player games; so, is the set of all games.

A profile is a tuple of strategies, one for each player. For a profile

, the vector

is called the utility vector. For a profile and a strategy of player , denote as the profile obtained by substituting for in . A partial profile is a tuple of strategies, one for each player other than . We define:

Definition 2.1

The game has the positive utility property if for each player and each partial profile , there is a strategy such that .

For win-lose bimatrix games, the positive utility property means that the utility matrix of the row (resp., column) player has no all-zeros column (resp., row).

A bimatrix game is a 2-player game with player 1, or row player, and player 2, or column player, with ,The assumption that the two players have equal numbers of strategies is without loss of generality since equality can be achieved, without altering the positive utility property, by adding ”dummy” strategies, which are never played by a utility-maximizing player. which is represented as the pair of matrices , where for each profile , and . The game is symmetric if : exchanging players and strategies preserves utilities; so, . For a constant , the game is -sum if for each profile , ; so, . For a player in the game , we shall denote as the player other than ; so .

A mixed strategy for player

is a probability distribution

on her strategy set : a function such that . Denote as the set of strategies such that . The mixed strategy is pure, and player is pure, if for some strategy ; player is mixed if she is not pure. The mixed strategy is fully mixed, and player is fully mixed, if ; so, player puts non-zero probability on all strategies in her set of strategies . The mixed strategy is uniform if for each pair of strategies , . The mixed strategy is rational if all values of are rational numbers.

A mixed profile is a tuple of mixed strategies, one for each player. So, a profile is the degenerate case of a mixed profile where all probabilities are either or . A partial mixed profile is a tuple of mixed strategies, one for each player other than . For a mixed profile and a mixed strategy of player , denote as the mixed profile obtained by substituting for in . A mixed profile is uniform (resp., fully mixed) if all of its mixed strategies are uniform (resp., fully mixed); else it is non-uniform (resp., non-fully mixed). A mixed profile is symmetric if all mixed strategies are identical; else it is non-symmetric. A mixed profile is rational if all of its mixed strategies are rational; else it is irrational.

The mixed profile induces a probability measure on the set of profiles in the natural way; so, for a profile , . Say that the profile is supported in the mixed profile , and write , if . Under the mixed profile

, the utility of each player becomes a random variable. So, associated with the mixed profile

is the expected utility for each player : the expectation according to of her utility; so,

Recall that in a -sum bimatrix game , for each mixed profile , . Also, in a symmetric bimatrix game, the mixed exchangeability property holds: exchanging players and mixed strategies preserves expected utilities in the sense that for any mixed profile , .§§§Indeed, .

2.2 Nash Equilibria

A pure Nash equilibrium is a profile such that for each player and for each strategy , . A mixed Nash equilibrium, or Nash equilibrium for short, is a mixed profile such that for each player and for each mixed strategy , . Note that the mixed exchangeability property of symmetric bimatrix games implies that (mixed) Nash equilibria are preseved in a symmetric bimatrix game: is a Nash equilibrium if and only if is. Denote as (resp., ) the set of Nash equilibria (resp., symmetric Nash equilibria) for the game . For each game ,  [29, 30]; for each symmetric game ,  [29, 30]. Two -player games and are Nash-equivalent [4] if ; that is, they have the same set of Nash equilibria. We shall make extensive use of the following basic property of Nash equilibria.

Lemma 2.1

A mixed profile is a Nash equilibrium if and only if for each player , (1) for each strategy , , and (2) for each strategy , .

Given a partial mixed profile for some player , a best-response for player to is a pure strategy such that . Lemma 2.1 (Condition (2)) immediately implies that in a Nash equilibrium , for each player and strategy , only if is a best-response for player to . Lemma 2.1 and the positive utility property immediately imply:

Lemma 2.2

Fix a win-lose game with the positive utility property. Then, in a Nash equilibrium , for each player , .

  • Assume, by way of contradiction, that . Choose a partial profile supported in ; so, . By the positive utility property, there is a strategy with . Since is win-lose,

    A contradiction to Lemma 2.1.  

We conclude with a simple property of Nash equilibria for win-lose games.

Lemma 2.3

Fix a win-lose bimatrix game with a Nash equilibrium such that for some player . Then, has a pure Nash equilibrium.

  • For simpler notation, assume, without loss of generality, that . Denote

    so, is the strategy in that incurs the maximum possible utility to player over all strategies from chosen by player . Choose an arbitrary strategy . We shall prove that the profile is a pure Nash equilibrium for . Since is a Nash equilibrium for , Lemma 2.1 (Condition (2)) implies that for each strategy , . Since is win-lose, it follows that , which implies that for each strategy , for all strategies . Since , it follows that for each strategy , . In particular, , so that player cannot improve by deviating from . By the definition of , player cannot improve either by deviating from . Hence, is a pure Nash equilibrium for .  

2.3 Decision Problems about Nash Equilibria

We shall assume some basic familiarity of the reader with the basic notions of -completeness, as outlined, for example, in [18]. All decision problems will be stated in the style of [18], where I. and Q. stand for Instance and Question, respectively. They are categorized in six groups.

Group I Question about cardinality:

NASH (with )

I.: A game .
Q.: Does have at least Nash equilibria?

Group II Questions about the expected utilities:

NASH WITH LARGE UTILITIES

I.: A game and a number .
Q.: Is there a Nash equilibrium such that for each player , ?

NASH WITH SMALL UTILITIES

I.: A game and a number .
Q.: Is there a Nash equilibrium such that for each player , ?

NASH WITH LARGE TOTAL UTILITY

I.: A game and a number .
Q.: Is there a Nash equilibrium such that ?

NASH WITH SMALL TOTAL UTILITY

I.: A game and a number .
Q.: Is there a Nash equilibrium such that ?

Group III Questions about the supports:

NASH WITH LARGE SUPPORTS

I.: A game and an integer .
Q.: Is there a Nash equilibrium such that for each player , ?

NASH WITH SMALL SUPPORTS

I.: A game and an integer .
Q.: Is there a Nash equilibrium such that for each player , ?

NASH WITH RESTRICTING SUPPORTS

I.: A game and a subset of strategies for each player .
Q.: Is there a Nash equilibrium such that for each player , ?

NASH WITH RESTRICTED SUPPORTS

I.: A game and a subset of strategies for each player .
Q.: Is there a Nash equilibrium such that for each player , ?

Group IV Questions about refinements of Nash equilibrium:

A Nash equilibrium is Pareto-Optimal if for each mixed profile where there is a player with , there is a player such that ; so, loosely speaking, there is no other mixed profile where at least one player is strictly better off and every player is at least as well off. Denote as the set of players with different mixed strategies in the mixed profiles and . A Nash equilibrium is Strongly Pareto-Optimal if for each mixed profile with a player such that , there is a player such that ; so, loosely speaking, there is no other mixed profile where at least one player is strictly better off and every player with a different mixed strategy is strictly better off.

PARETO-OPTIMAL NASH

I.: A game .
Q.: Is there a Pareto-Optimal Nash equilibrium?

PARETO-OPTIMAL NASH

I.: A game .
Q.: Is there a Nash equilibrium which is not Pareto-Optimal?

STRONGLY PARETO-OPTIMAL NASH

I.: A game .
Q.: Is there a Strongly Pareto-Optimal Nash equilibrium?

STRONGLY PARETO-OPTIMAL NASH

I.: A game .
Q.: Is there a Nash equilibrium which is not Strongly Pareto-Optimal?

Group V Questions about the probabilities:

NASH WITH SMALL PROBABILITIES

I.: A game .
Q.: Is there a Nash equilibrium such that for each player , ?

UNIFORM NASH

I.: A game .
Q.: Does have a uniform Nash equilibrium?

UNIFORM NASH

I.: A game .
Q.: Does have a non-uniform Nash equilibrium?

SYMMETRIC NASH

I.: A game .
Q.: Does have a symmetric Nash equilibrium?

SYMMETRIC NASH

I.: A game .
Q.: Does have a non-symmetric Nash equilibrium?

RATIONAL NASH

I.: A game .
Q.: Does have a rational Nash equilibrium?

Restricted to general bimatrix games, all previous decision problems belong to : given the polynomial-length supports for a mixed profile, it is polynomial time verifiable that it is a Nash equilibrium satisfying the property associated with the decision problem. RATIONAL NASH is trivial for general bimatrix games, and SYMMETRIC NASH is trivial for symmetric games. Note that UNIFORM NASH and RATIONAL NASH belong to for any number of players.

Each of the previous decision problems has a corresponding cardinality or counting version: the problem of computing the number of Nash equilibria witnessing the validity of Question. NASH has no counting version as it is already defined with a cardinality question. Note that for general bimatrix games, the counting versions of all previous decision problems belong to (thanks again to the efficient verifiability property of Nash equilibria). Furthermore, each of the decision problems has a parity version: the problem of computing the parity of the number of Nash equilibria witnessing the validity of Question. The parity versions of all previous decision problems belong to ., read as Parity , is the complexity class formalizing the question of the parity of the number of solutions to a combinatorial problem. Formally, is the class of sets

such that there is a nondeterministic Turing machine which, on input

, has an odd number of accepting computations if and only if

. We shall adopt a definition of -completeness using polynomial-time many-to-one reductions. The development of the theory of -complete parity problems is rather limited — see the discussion in [36]. We shall use (resp., ) in the place of to denote the counting (resp., parity) versions; for example, RATIONAL NASH (resp., RATIONAL NASH) denotes the counting (resp., parity) version of RATIONAL NASH. Clearly, by the mixed exchangeability property of symmetric bimatrix games, the number of non-symmetric Nash equilibria in a symmetric bimatrix game is even; this is because is a non-symmetric Nash equilibrium if and only if is. Hence, SYMMETRIC NASH for symmetric bimatrix games is in .

Group VI Equivalence property:

NASH-EQUIVALENCE

I.: Two games and from -, for some integer .
Q.: Are and Nash-equivalent?

For a fixed game , called the gadget game, a parameterized restriction of NASH-EQUIVALENCE with a single input (the game ) results.

NASH-EQUIVALENCE()

I.: A game from - (where is from -).
Q.: Are and Nash-equivalent?

Restricted to general bimatrix games, NASH-EQUIVALENCE and NASH-EQUIVALENCE belong to co-: given the polynomial-length supports for a mixed profile, it is polynomial time verifiable that it is a Nash equilibrium for exactly one of the two games.

Matching - and co--hardness results for the decision problems above are later summarized in Figure 4.

3 Win-Lose Bimatrix Games with the Positive Utility Property

We now prove that assuming the positive utility property does not simplify the search problem for a win-lose bimatrix game. Fix a win-lose bimatrix game . We start with a preliminary definition. For a player , a strategy is an all-zeros counter-strategy against player if for every profile with , ; that is, an all-zeros counter-strategy against the row (resp., column) player is a column (resp., row) of (resp., ) made up only of zero entries. Clearly, the positive utility property excludes all-zeros counter-strategies. We prove:

Proposition 3.1

Fix a win-lose bimatrix game . Then, one of two conditions holds:

  1. has a pure Nash equilibrium.

  2. There is a polynomial time constructible win-lose bimatrix game with the positive utility property such that .

  • If Condition (C.1) holds, then we are done. So assume that Condition (C.1) does not hold. Note that if at least one of and is the null matrix, then has a pure Nash equilibrium and Condition (C.1) holds. It follows that none of and is the null matrix. Denote as and the number of all-zeros counter-strategies against the row and the column player, respectively, in the game . Note that has the positive utility property if and only if . Renumber now the players’ strategies so that the first strategies for player are the all-zeros counter-strategies against the column player, and the first strategies for player are the all-zeros counter-strategies against the row player. Construct from a win-lose bimatrix game as follows:

    If , then . Otherwise, add a new strategy to the strategy set of each player and set: . for , or for . for , or for .

    Clearly, this is a polynomial time construction. We prove:

    Lemma 3.2

    The game has the positive utility property.

    • Note that in Case (0), the game has the positive utility property by definition. So assume we are in Case (1). Consider the row matrix ; we need to prove that it does not contain an all-zeros column. By Step (1.2) in the construction, the column of has an entry equal to if and only if ; the latter holds since is not the null matrix. So it remains to consider the columns of that are inherited from . By the definition of , every column of with is not all-zeros; for a column of , by Step (1.1) in the construction, and we are done. We use corresponding arguments to prove that the column matrix does not contain an all-zeros row.  

    Furthermore, Condition (C.2) holds vacuously in Case (0). Note also that in Case (1), has no pure Nash equilibrium in which some player chooses strategy . We proceed by case analysis. Assume first that there is a Nash equilibrium for such that at least one player plays with positive probability an all-zeros counter-strategy against player . By Lemma 2.1, choosing the strategy with probability is a best-response of player to ; so, player cannot improve her utility when choosing the strategy with probability . Now, by the definition of an all-zeros counter-strategy, player gets utility in the mixed profile , which she could not improve by switching to another mixed strategy. Hence, is a Nash equilibrium with . By Lemma 2.3, it follows that has a pure Nash equilibrium . Since has no pure Nash equilibrium in which some player plays strategy , is also a pure Nash equilibrium for . A contradiction.

    Assume now that in each Nash equilibrium for , no player plays an all-zeros counter-strategy against player ; that is, and . We first prove that no Nash equilibrium for is “destroyed” due to adding strategy .

    Lemma 3.3

    It holds that .

    • Assume, by way of contradiction, that there is a Nash equilibrium such that . By the construction of the game , this implies that there is player with . Since the game is win-lose, this implies that . Since and , it follows that any profile supported in falls into either Case (1.1) with and or into Case (1.2) with and . By construction, this implies that for each player . A contradiction.  

    We now prove that no new Nash equilibrium for is “created” due to adding strategy .

    Lemma 3.4

    It holds that .

    • Assume, by way of contradiction, that there is a Nash equilibrium . Since , it must be that . Assume that and consider player switching to strategy