Given a finite strategic form -player game the most basic algorithmic problem is to compute a Nash equilibrium, shown always to exist by Nash . The computational complexity of this problem was characterized in seminal work by Daskalakis, Goldberg, and Papadmitriou  and Chen and Deng  as -complete for 2-player games and by Etessami and Yannakakis  as -complete for -player games, when . Any 2-player game may be viewed as a 3-player zero-sum game by adding a dummy player, thereby making the class of 3-player zero-sum games a natural class of games intermediate between 2-player and 3-player games. The problem of computing a Nash equilibrium for a 3-player zero-sum game is clearly -hard and belongs to , but its precise complexity appears to be unknown.
Rather than settling for any Nash equilibrium, one might be interested in a Nash equilibrium that satisfies a given property, e.g. giving each player at least a certain payoff. Such a Nash equilibrium might of course not exist and therefore results in the basic computational problem of deciding existence. In the setting of 2-player games, the computational complexity of several such problems was proved to be -complete by Gilboa and Zemel . Conitzer and Sandholm  revisited these problems and showed them, together with additional problems, to be -complete even for symmetric games.
Only recently was the computational complexity of analogous problems in -player games determined, for . Schaefer and Štefankovič  obtained the first such result by proving
-completeness of deciding existence of a Nash equilibrium in which no action is played with probability larger thanby any player. Garg, Mehta, Vazirani, and Yazdanbod  used this to also show -completeness for deciding if a game has more than one Nash equilibrium, whether each player can ensure a given payoff in a Nash equilibrium, and for the two problems of deciding whether the support sets of the mixed strategies of a Nash equilibrium can belong to given sets or contain given sets. In addition, by a symmetrization construction, they show that the analogue to the latter two problems for symmetric Nash equilibria are -complete as well. Bilò and Mavronicolas [5, 6] subsequently extended the results of Garg et al. to further problems both about Nash equilibria and about symmetric Nash equilibria. They show -completeness of deciding existence of a Nash equilibrium where all players receive at most a given payoff, where the total payoff of the players is at least or at most a given amount, whether the size of the supports of the mixed strategies all have a certain minimum or maximum size, and finally whether a Nash equilibrium exists that is not Pareto optimal or that is not a strong Nash equilibrium. All the analogous problems about symmetric Nash equilibria are shown to be -complete as well.
1.1 Our Results
We revisit the problems about existence of Nash equilibria in -player games, with , considered by Garg et al. and Bilò and Mavronicolas. In a zero-sum game the total payoff of the players in any Nash equilibrium is of course 0, and any Nash equilibrium is Pareto optimal. This renders the corresponding decision problems trivial in the case of zero-sum games. We show except for these, all the problems considered by Garg et al. and Bilò and Mavronicolas remain -hard for 3-player zero-sum games. We obtain our results building on a recent more direct and simple proof of -hardness of the initial -complete problem of Schaefer and Štefankovič due to Hansen . For completeness we give also comparably simpler proofs of -hardness for the problems about total payoff and existence of a non Pareto optimal Nash equilibrium.
We next show that deciding existence of a strong Nash equilibrium in an -player game with is -complete, and likewise for the similar problem of deciding existence of a Pareto optimal Nash equilibrium. Gatti, Rocco, and Sandholm  proved earlier that deciding if a given (rational valued) strategy profile is a strong Nash equilibrium can be done in polynomial time. They then erroneously concluded that the problem of deciding existence of a strong Nash equilibrium is, as a consequence -complete. A problem with this reasoning is that if a strong Nash equilibrium exists, there is no guarantee that a rational valued strong Nash equilibrium exists. Even if one disregards a concern about irrational valued strong Nash equilibria, it is possible that even when a rational valued strong Nash equilibrium exists, any rational valued strong Nash equilibrium could require exponentially many bits to describe in standard binary notation the numerators and denominators of the probabilities of the equilibrium strategy profile. Nevertheless, our proof of -membership build on the idea behind the polynomial time algorithm of Gatti et al. Our reduction for proving -hardness produces non-zero-sum games. The case of deciding existence of a Pareto optimal Nash equilibrium is, as already noted, trivial for the case of 3-player zero-sum games. We leave the complexity of the deciding existence of a strong Nash equilibrium in 3-player zero-sum games an open problem.
In another work, Bilò and Mavronicolas  considered the problems of deciding whether an irrational valued Nash equilibrium exists and whether a rational valued Nash equilibrium exists, proving both problems to be -hard. Bilò and Mavronicolas asked if the problem about existence of an irrational valued Nash equilibria is hard for the so-called square-root-sum problem. We confirm this, showing the problem to be -hard. We relate the problem about existence of rational valued Nash equilibria to the existential theory of the rationals.
We next use a symmetrization construction similar to Garg et al. to translate all problems considered to the analogous setting of decision problems about symmetric Nash equilibria. Here we do not obtain qualitative improvements on existing results, but give for completeness the simple proofs of these results in addition to our new results.
A final problem we consider is of deciding existence of a nonsymmetric Nash equilibrium of a given symmetric game. Mehta, Vazirani, and Yazdanbod  proved that this problem is -complete for 2-player games, and Garg et al.  raised the question of the complexity for -player games with . We show this problem to be -complete.
Our hardness proofs are presented for the special case of 3-player games, but extend to -player games for any fixed , in a similar way to previous works [15, 5, 6]. For the case of nonsymmetric games this is achieved by adding dummy players with suitably chosen actions sets and payoff functions (cf. ). Zero-sum games are, of course, mainly interesting for 3-player games. For the case of symmetric games, the dummy players can be introduced prior to the symmetrization construction and this together with the reductions that follow are easily generalized to players.
2.1 Existential Theory of the Reals and Rationals
The existential theory of the reals is the set of all true sentences over of the form , where is a quantifier free Boolean formula of equalities and inequalities of polynomials with integer coefficients. The complexity class is defined  as the closure of under polynomial time many-one reductions. Equivalently, is the constant-free Boolean part of the class , which is the analogue class to in the Blum-Shub-Smale model of computation . It is straightforward to see that is -hard (cf. ) and the decision procedure by Canny  shows that belongs to . Thus it follows that .
We may similarly consider the existential theory over the rationals and likewise form the complexity class as the closure of under polynomial time many-one reductions. While it is a long-standing open problem whether is decidable, Koenigsmann  recently showed that already , consisting of true sentences in prenex form with a single block of universal quantifiers followed by a single block of existential quantifiers, is undecidable. In contrast, the entire first order theory of the reals is decidable in . Schaefer and Štefankovič  show that the problem of deciding feasibility of a system of strict inequalities is complete for . Since a system of strict inequalities that is feasible over is also feasible over , it follows that .
The basic complete problem for and for , is the problem of deciding whether a system of quadratic equations with integer coefficients has a solution over and over , respectively . We denote this problem over as Quad and the problem over as .
2.2 Strategic Form Games and Nash Equilibrium
A finite strategic form game with players is given by sets of actions (pure strategies) together with utility functions . A choice of an action for each player together form a pure strategy profile .
The game is symmetric if and for every permutation on , every and every it holds that . In other words, a game is symmetric if the players share the same set of actions and the utility function of a player depends only on the action of the player together with the multiset of actions of the other players.
denote the set of probability distributions on. A (mixed) strategy for Player is an element . The support is the set of actions given strictly positive probability by . We say that is fully mixed if . A strategy for each player together form a strategy profile . The utility functions extend to strategy profiles by letting . We shall also refer to as the payoff of Player .
Given a strategy profile we let denote the strategies of all players except Player . Given a strategy for Player , we let denote the strategy profile formed by and . We may also denote by . We say that is a best reply for Player to (or to ) if for all .
A Nash equilibrium (NE) is a strategy profile where each individual strategy is a best reply to . As shown by Nash , every finite strategic form game has a Nash equilibrium. In a symmetric game , a symmetric Nash equilibrium (SNE) is a Nash equilibrium where the strategies of all players are identical. Nash also proved that every symmetric game has a symmetric Nash equilibrium.
A strategy profile is Pareto optimal if there is no strategy profile such that for all , and for some . A Nash equilibrium strategy profile need not be Pareto optimal and a Pareto optimal strategy profile need not be a Nash equilibrium. A strategy profile that is both a Nash equilibrium and is Pareto optimal is called a Pareto optimal Nash equilibrium. The existence of a Pareto optimal Nash equilibrium is not guaranteed.
A strong Nash equilibrium  (strong NE) is a strategy profile for which there is no non-empty set for which all players can increase their payoff by different strategies assuming players play according to . Equivalently, is a strong Nash equilibrium if for every strategy profile there exist such that and . The existence of a strong Nash equilibrium is not guaranteed.
3 Decision Problems about Nash Equilibria
Below we define the decision problems under consideration with names generally following Bilò and Mavronicolas . The given input is a finite strategic form game , together with auxiliary input depending on the particular problem. We let denote a rational number, an integer, and a set of actions of Player , for every . We describe the decision problem by stating the property a Nash equilibrium whose existence is to be determined should satisfy. The problems are grouped together in four groups each of which are covered in a separate subsection below.
Except for the last four problems, it is straightforward to prove membership in by an explicit existentially quantified first-order formula. We prove membership of and in subsection 3.3 and discuss decidability of and in subsection 3.4.
|for all .|
|for all .|
|for all and .|
|is not the only NE.|
|for all .|
|for all .|
|for all .|
|for all .|
|is not Pareto optimal.|
|is not a strong NE.|
|is Pareto optimal.|
|is a strong NE.|
|for some and .|
|for all and .|
A key step (implicitly present) in the proof of the first -hardness result about Nash equilibrium in 3-player games by Schaefer and Štefankovič is a result due to Schaefer  that Quad remains -hard under the promise that either the given quadratic system has no solutions or a solution exists in the unit ball . For our purposes the following variation [18, Proposition 2] will be more directly applicable (which may easily be proved from the latter, cf. Section 3.4). Here we denote by the standard corner -simplex .
It is -hard to decide if a given system of quadratic equations in variables and with integer coefficients has a solution under the promise that either the system has no solutions or a solution exists that is in the interior of and also satisfies for all and that .
Schaefer and Štefankovič showed that is -hard for 3-player games by first proving that the following problem is -hard: Given a continuous function mapping the unit ball to itself, where each coordinate function is given as a polynomial, and given a rational number , is there a fixed point of in the ball ? The proof was then concluded by a transformation of Brouwer functions into 3-player games by Etesammi and Yannakakis . This latter reduction is rather involved and goes though an intermediate construction of 10-player games. More recently, Hansen  gave a simple and direct reduction from the above promise version of Quad to .
The first step of this as well as our reductions is to transform the given quadratic system over the corner simplex into a homogeneous bilinear system over the standard -simplex which we denote by . In short, this is done by introducing a set of new variables and new equations , replacing quadratic terms by bilinear quadratic terms , and finally homogenizing the entire system using the two equations and where and are new slack variables. Doing this we arrive at the following statement (cf. [18, Proposition 3]).
It is -complete to decide if a system of homogeneous bilinear equations , with integer coefficients has a solution . It remains -hard under the promise that either the system has no such solution or a solution exists where belong to the relative interior of and further satisfies for all .
3.1 Payoff Restricted Nash Equilibria
For proving the -hardness results we start by showing that it is -hard to decide if a given zero-sum game has a Nash equilibrium in which each player receives payoff . This is in contrast to the earlier work of Garg et al.  and Bilò and Mavronicolas [5, 6] that reduce from the problem. On the other hand we do show -hardness even under the promise that the Nash equilibrium also satisfies the condition of . The construction and proof below are modifications of proofs by Hansen [18, Theorem 1 and Theorem 2].
Definition 1 (The 3-player zero-sum game ).
Let be a system of homogeneous bilinear polynomials with integer coefficients in variables and ,
We define the 3-player game as follows. The strategy set of Player 1 is the set . The strategy sets of Player 2 and Player 3 are . The (integer) utility functions of the players are defined by
When the system is understood by the context, we simply write . We think of the strategy of Player 1 as corresponding to the polynomial together with a sign , the strategy of Player 2 as corresponding to and the strategy of Player 3 as corresponding to . We may thus identify mixed strategies of Player 2 and Player 3 with assignments to variables .
The following observation is immediate from the definition of .
Any strategy profile of Player 2 and Player 3 satisfies for every the equation
Hence when is the uniform
is the uniform distribution on. Consequentially, any Nash equilibrium payoff profile is of the form , where .
Next we relate solutions to the system to Nash equilibria in .
Let be a system of homogeneous bilinear polynomials , . If has a solution , then letting be the uniform distribution on , the strategy profile is a Nash equilibrium of in which every player receives payoff . If in addition satisfies the promise of Proposition 2, then is fully mixed, Player 2 and Player 3 use identical strategies, and no action is chosen with probability more than by any player. Conversely, if is a Nash equilibrium of in which every player receives payoff , then is a solution to .
Suppose first that is a solution to and let be the uniform distribution on . By Equation (1) the strategy profile of Player 2 and Player 3 ensures that all players receive payoff regardless of which strategy is played by Player 1, and likewise the strategy of Player 1 ensures that all players receive payoff regardless of the strategies of Player 2 and Player 3. This shows that is a Nash equilibrium of , in which by Lemma 1 every player receives payoff . If in addition satisfies that the promise of Proposition 2 we have . From this and our choice of , we have that is a fully mixed and that no action is chosen by a strategy of with probability more than .
Suppose on the other hand that is a Nash equilibrium of in which every player receives payoff . Suppose that for some . Then by Equation (1) we get that , contradicting that is a Nash equilibrium. Thus is a solution to . ∎
and are -complete, even for 3-player zero-sum games.
For a strategy profile in a zero-sum game we have that , for all , if and only if , for all , if and only if , for all .
A simple change to the game give -hardness for the two problems and . Naturally we must give up the zero-sum property of the game.
Theorem 2 (Bilò and Mavronicolas ).
and are -complete, even for 3-player games.
Define the game from with new utility functions and , and thus also , where ,, and are the utility functions of . Clearly has the same set of Nash equilibria as . Now and it follows that if and only if . By Lemma 1, any Nash equilibrium must satisfy the inequality . Thus, a Nash equilibrium satisfies the inequality if and only if . We conclude that Proposition 3 gives a reduction from the promise problem of Proposition 2 to thereby showing -hardness.
Similarly, define the game from with new utility functions and . Again, clearly has the same set of Nash equilibria as . Now and it follows that if and only if . Analogously to above we then obtain -hardness for . ∎
3.2 Probability Restricted Nash Equilibria
A key property of the game is that Player 1 may ensure all players receive payoff . We now give all players this choice by playing a new additional action . We then design the utility functions involving in such a way that the pure strategy profile is always a Nash equilibrium, and every other Nash equilibrium is a Nash equilibrium in in which all players receive payoff .
For , let be the 3-player zero-sum game where each player has the action set and the payoff vectors are given by the
entries of the following two matrices, where Player 1 selects the
matrix, Player 2 selects the row, Player 3 selects the column.
and the payoff vectors are given by the entries of the following two matrices, where Player 1 selects the matrix, Player 2 selects the row, Player 3 selects the column.
It is straightforward to determine the Nash equilibria of .
When , the only Nash equilibrium of is the pure strategy profile . When the only Nash equilibria of are the pure strategy profiles and .
Let be the probability of Player choosing the action . Consider first the case of . Then the action is strictly dominating the action for both Player 2 and Player 3. Hence any Nash equilibrium would require . The only best reply for Player 1 is then as well. Consider next the case of . In case , again the action is strictly dominating the action for both Player 2 and Player 3, and we conclude that as before. Suppose now that . In a Nash equilibrium we would have either or . The former clearly gives a Nash equilibrium whereas for the only best reply for Player 1 is . ∎
We use the game to extend the game . The action of represents selecting an action from , and the payoff vector that is the result of all players playing the action is precisely of the form of the Nash equilibrium payoff profile of .
Definition 3 (The 3-player zero-sum game ).
Let be the game obtained from as follows. Each player is given an additional action . When no player plays the action , the payoffs are the same as in . When at least one player is playing the action the payoff are the same as in , where each action different from is translated to action .
We next characterize the Nash equilibria in .
The pure strategy profile is a Nash equilibrium of . Any other Nash equilibrium in is also a Nash equilibrium of and is such that every player receives payoff .
By Lemma 1 a Nash equilibrium of induces a Nash equilibrium of , where is a Nash equilibrium payoff profile of , by letting each player play the action with the total probability of which the actions of are played. By Lemma 2, any Nash equilibrium in different from must then be a Nash equilibrium of with Nash equilibrium payoff profile as claimed. ∎
The following problems are -complete, even for 3-player zero-sum games: , , , , and .
Proposition 3 and Proposition 4 together gives a reduction from the promise problem of Proposition 2 to all of the problems under consideration when setting the additional parameters as follows. For we let , we let for , and lastly we let be the set of all actions of Player except for both of the problems and . ∎
Except for the case of , the results of Theorem 3 can also be proved with the slightly simpler construction of adding an additional action to the players in which when played by at least one player results in all players receiving payoff .
To adapt the reduction of Theorem 3 to we need to replace the trivial Nash equilibrium by a Nash equilibrium with large support.
Define the 2-player zero-sum game as follows. The two players, which we denote Player 2 and Player 3, have the same set of pure strategies . The utility functions are defined by
We omit the easy analysis of the game .
For any , in the game the strategy profile in which each action is played with probability is the unique Nash equilibrium and yields payoff to both players.
Definition 5 (The 3-player zero-sum game ).
Let be the game obtained from as follows. The action of Player 2 and Player 3 are replaced by the set of actions , , where is the maximum number of actions of a player in . The payoff vector of the pure strategy profile is , where and are the utility functions of the game . Otherwise, when at least one player plays the action , the payoff is as in , where actions of the form are translated to the action .
is -complete, even for 3-player zero-sum games.
In , the strategy profile where Player 1 plays and Player 2 and Player 3 play , with chosen uniformly at random, is a Nash equilibrium that takes the role of the Nash equilibrium in . Consider now an arbitrary Nash equilibrium in . In case all players play the action with probability less than , Player 2 and Player 3 must chose each action of the form with the same probability, since has a unique Nash equilibrium. The Nash equilibrium induces a strategy profile in , letting Player 2 and Player 3 play the action with the total probability each player placed on the actions . By definition of the payoff vector of in differs by at most 1 in each entry from the payoff vectors of . The proof of Lemma 2 and Proposition 4 still holds when changing the payoff vector of by at most 1 in each coordinate. The strategy profile induced in must therefore be a Nash equilibrium in . We conclude that in a Nash equilibrium of , either Player 2 and Player 3 use strategies with support of size or is a Nash equilibrium of , where every player uses a strategy of support size strictly less than and where every player receives payoff 0. Proposition 3 thus gives a reduction showing -hardness. ∎
3.3 Pareto Optimal and Strong Nash Equilibria
For showing -hardness for we first analyze the Strong Nash equilibria in the game .
For , the Nash equilibrium of is a strong Nash equilibrium. For , the Nash equilibrium of is not a strong Nash equilibrium.
Consider first and the Nash equilibrium . This is not a strong Nash equilibrium, since for instance Player 1 and Player 2 could both increase their payoff by playing the strategy profile . Consider next and the Nash equilibrium . Since is a zero-sum game it is sufficient to consider possible coalitions of two players. Player 2 and Player 3 are already receiving the largest possible payoff given that Player 1 is playing the strategy , and hence they do not have a profitable deviation. Consider then, by symmetry, the coalition formed by Player 1 and Player 2, and let them play with probabilities and . A simple calculation shows that to increase the payoff of Player 1 requires and to increase the payoff of Player 2 requires . Adding these gives which implies . But then . Thus is a strong Nash equilibrium. ∎
is -complete, even for 3-player zero-sum games.
In a zero-sum game, every strategy profile is Pareto optimal. Thus for showing -hardness of we consider non-zero-sum games.
For , let be the 3-player game given by the following matrices, where Player 1 selects the matrix, Player 2 selects the row, Player 3 selects the column.
When , the only Nash equilibrium of is the pure strategy profile . When the only Nash equilibria of are the pure strategy profiles and . For , is Pareto optimal. For , is not Pareto optimal.
When , clearly is a Nash equilibrium, which is Pareto dominated by . Likewise, clearly is always a Pareto optimal Nash equilibrium. When , the action is strictly dominated by the action for Player 2 and Player 3, and hence they play with probability in a Nash equilibrium. The only best reply of Player 1 is to play with probability as well. ∎
Analogously to Definition 3 we define the game to be the game extending with replacing the role of and analogously to Proposition 4 any Nash equilibrium in different from , which is Pareto optimal, must by Lemma 5 be a Nash equilibrium of with payoff profile , which is not Pareto optimal. This gives the -hardness part of the following theorem.
Theorem 6 (Bilò and Mavronicolas ).
is - complete, even for 3-player games.
We next consider the problems and . We first outline a proof of membership in , building on ideas of Gatti et al  and Hansen, Hansen, Miltersen, and Sørensen . Gatti et al. proved that deciding whether a given strategy profile of an -player game is a strong Nash equilibrium can be done in polynomial time. The crucial insight behind this result that the question of whether a coalition of players may all improve their payoff by together changing their strategies can be recast into a question in a derived game about the minmax value of an additional fictitious player that has only strategies. Hansen et al. proved that in such a game, the minmax value may be achieved by strategies of the other players that are of support at most .
Lemma 6 (Hansen et al. ).
Let be a player game and let . If there exists a strategy profile of the first players such that for all then there also exists a strategy profile of the first players in which each strategy has support size at most and for all .
We next give a generalization of the auxiliary game construction of Gatti et al. that also allows us to treat Pareto optimal Nash equilibria at the same time.
Definition 7 (cf. Gatti et al ).
Let be an -player game with strategy sets and utility functions . Let be a strategy profile of and let be a partition of the players, let and . For consider the -player auxiliary game defined as follows. For the strategy set of Player is . For the strategy set of Player is . Finally, the strategy set of Player is . The utility function of Player is defined as as follows. Let be a pure strategy profile of . Define the strategy profile of letting for and for . We then let for and for .
The following is immediate from the definition of .
There exist a strategy profile in that satisfies when , when , and when if and only if there exist and a strategy in of the first players such that for all .
The task of deciding if a strategy is Pareto optimal amounts to checking the condition of Lemma 7 for and for all and to decide whether is a strong Nash equilibrium amounts to checking the condition for all nonempty while letting .
According to Lemma 6 we may restrict our attention to strategies in of supports of size at most . Fixing such a set of supports for , we may formulate the question of existence of a strategy , with for that satisfies the conditions of Lemma 7 as an existentially quantified first-order formula over the reals. For a fixed we need only existentially quantified variables to describe and the strategy . Since this is a constant number of variables, when as in our case is a constant, the general decision procedure of Basu, Pollack, and Roy  runs in polynomial time in the bitsize of coefficients, number of polynomials, and their degrees, resulting in an overall polynomial time algorithm. Now, adding a step of simply enumerating over all nonempty and all support sets of size we obtain the result of Gatti el al. that deciding whether a given strategy profile is a strong Nash equilibrium can be done in polynomial time. The same holds in a similar way for checking that a strategy profile is a Pareto optimal Nash equilibrium.
In our case, when proving membership the only input is the game , whereas the strategy profile will be given by a block of existentially quantified variables. We then need to show how to express that is a Pareto optimal or a strong Nash equilibrium by a quantifier free formula over the reals with free variables . This will be possible by the fact that quantifier elimination, rather than just decision, is possible for the first order theory of the reals. The quantifier elimination procedure of Basu et al.  runs in time exponential in the number of free variables, so we cannot apply it directly.
Instead we express the condition of Lemma 7 for a strategy profile that is constrained by for