The study of game theory - the model of strategic interaction between rational agents - has a rich history dating back to the initiation of the field by John von Neumann in 1928. The relatively recent rapid growth in computing power, the Internet, and game theoretical models in engineering and computer science has led to significant interest in devising fast algorithms for computation of solutions in games. Indeed, this intersection of computer science, game theory, and economic theory has led to the creation of a new field of study called Algorithmic Game Theory .
Computing a Nash equilibrium (NE) in a -player finite normal-form game is one of the fundamental problems for Algorithmic Game Theory.111Nash equilibrium is an acceptable and a widely used solution concept for games, and we define it later in the paper. Due to the well known theorem by Nash in 1951, we know that every finite game has a solution, possibly in mixed strategies . However, computation of a NE has been shown to be computationally hard. For , [4, 5] proved that computing a NE is Polynomial Parity Argument, Directed Version (PPAD) complete. Indeed, even the seemingly simpler -player bimatrix case has been shown to be PPAD-complete . For any bimatrix game, the classic Lemke-Howson algorithm  is guaranteed to find a NE. However, this algorithm is known to have exponential running time . For surveys on computational issues of NE see [9, 10] and [2, Chapter 2].
In this work we focus on finite, -player bimatrix games in which the payoffs to the players can be represented as two matrices, and . With the hardness of computing a NE in the bimatrix case well established, identifying classes of games whose solutions can be computed efficiently is an active area of research. As an example in bimatrix games, one can compute the solution of zero-sum games222A game is called a zero sum game if the sum of payoffs of the players is identically zero for all actions of the players.
in polynomial time using the minimax theorem and a simple linear program[11, p. 152]. Another subclass of games that admits a polynomial time solution is the class of strategically zero-sum games introduced by Moulin and Vial in 1978 . They identify a class of games that are strategically-equivalent to some zero-sum game and therefore have exactly the same set of NE as the zero-sum game. For these games, Moulin and Vial propose a characterization that, in polynomial time, can check for the strategically zero-sum property and, if it exists, calculate the resulting zero-sum game. More recently, Kontogiannis and Spirakis studied mutually concave bimatrix games and provided a polynomial time characterization based on solving a convex quadratic program. Interestingly, they also conclude that the mutually concave property is equivalent to the strategically zero-sum property.
In contrast to computing an exact solution in games, other recent work has focused on devising polynomial time algorithms for computing approximate Nash equilibria of special subclasses of bimatrix games. In , the authors consider games where both payoff matrices have fixed rank and propose a quasi-polynomial time algorithm for computing approximate Nash equilibrium. In , the authors define constant rank games where and provide a polynomial time algorithm to compute an approximate Nash equilibrium. For rank-1 games,  devised an algorithm that can compute the solution in polynomial time. Here we note that rank-0 games are clearly zero-sum games. With the polynomial tractability of rank- games for established, one could hope that there exists polynomial algorithms for other low-rank games. Unfortunately, a recent result has shown that rank- games for are PPAD-hard . The case of rank- games is still unresolved.
1.1 Prior Work
Many papers in the literature have explored the concept of equivalent classes of games. One such concept is strategic equivalence, those games that share exactly the same set of NE. Indeed, for a classical example, von Neumann and Morgenstern studied strategically equivalent -person zero-sum games [11, p. 245] and constant-sum games [11, p. 346].
In a recent work by Possieri and Hespanha, the authors consider the problem of designing strategically equivalent games . As such, given a bimatrix game , their problem is to design a family of games that is either weakly strategically equivalent333The authors of  define weak strategic equivalence as games the have the same NE in pure strategies. or strongly strategically equivalent.444The authors of  define strong strategic equivalence as games that have the same NE in both mixed and pure strategies. Since this work falls into the category of mechanism design, it is incompatible with the work that we present here. Indeed, we consider the inverse problem- given a bimatrix game , does there exist a strategically equivalent zero-sum game.
More closely related to our work is the class of strategically zero-sum games defined by Moulin and Vial in . They study the class of games in which no completely mixed NE can be improved upon via a correlation strategy and come to the conclusion that these games are the class of strategically zero-sum games. For the bimatrix case, they provide a complete characterization of strategically zero-sum games [12, Theorem 2]. While the authors do not analyze the algorithmic implications of their characterization, Kontogiannis and Spirakis do analyze the approach in their work on mutually concave games. They find that the characterization in [12, Theorem 2] can determine whether a bimatrix game is strategically zero-sum, and, if so, calculate an equivalent zero-sum game in time .
Around the same time, Isaacson and Millham studied a class of bimatrix games that they characterized as row-constant games . They define row-constant games as those bimatrix games where the sum of the payoff matrices is a matrix with constant rows. In their work, they show that the NE strategies of a row-constant game can be found via solving the zero-sum game . Comparing  and , one can easily see that row-constant games form a subclass of strategically zero-sum games.
Closely related to strategically zero-sum bimatrix games are the class of strictly competitive games . In a strictly competitive game, if both players change their mixed strategies, then either the payoffs remain unchanged, or one of the two payoffs increases while the other payoff decreases. In other words, all possible outcomes are Pareto optimal. It has long been claimed that strictly competitive games share many common and desirable NE features with zero-sum games, such as ordered interchangeability, NE payoff equivalence, and convexity of the NE set555In , Friedman shows that the convexity of the NE set holds for infinite games with quasiconcave payoff functions. At the time of that writing, it seemed to be widely accepted that this was true for bimatrix strictly competitive games. In light of the results of , we see that the set of NE is convex for bimatrix strictly competitive games. . Indeed, Aumann claims that strictly competitive games are equivalent to zero-sum games . Moulin and Vial proceed to cite Aumann’s claim when arguing that strictly competitive games form a subclass of strategically zero-sum games [12, Example 2]. However, many years later Adler et. al. conducted a literature search and found that the claim of equivalence of strictly competitive games and zero-sum games was often repeated, but without formal proof . They then proceeded to prove that this claim does indeed hold true. Comparing the results of Adler et. al  to the characterization of strategically zero-sum games in , one can observe that Moulin and Vial were correct in asserting that strictly competitive games form a subclass of strategically zero-sum games.
Another class of games that shares some NE properties with zero-sum games is the class of (weakly) unilaterally competitive games. As defined by Kats and Thisse in , a game is unilaterally competitive if a unilateral change in strategy by one player results in a (weak) increase in that player’s payoff if and only if it results in a (weak) decrease in the payoff of all other players. A game is weakly unilaterally competitive if a unilateral change in strategy by one player results in a strict increase in that player’s payoff, then the payoffs of all other players (weakly) decrease. If the payoff of the player who makes the unilateral move remains unchanged, then the payoffs of all players remain unchanged. For the two-player case, strictly competitive games form a subclass of unilaterally competitive games, and unilaterally competitive games form a subclass of weakly unilaterally competitive games. For the bimatrix case, Kats and Thisse show that (weakly) unilaterally competitive games have the ordered interchangeability and NE payoff equivalence properties . However, convexity of the NE set is only proved for infinite games with quasiconcave payoff functions.
Finally, in , Kontogiannis and Spirakis formulate a quadratic program where the optimal solutions of the quadratic program constitute a subset of the correlated equilibria of a bimatrix game. Furthermore, they then show that this subset of correlated equilibria are exactly the NE of the game. In order to show polynomial tractability of the quadratic program, they define a class of mutually concave bimatrix games. Surprisingly, they find that their characterization of mutually concave bimatrix games is equivalent to Moulin and Vial’s characterization of strategically zero-sum bimatrix games [23, Corollary 2]. While the time complexity of the original quadratic program is extremely high, Kontogiannis and Spirakis propose a parameterized version of the Mangasarian and Stone quadratic program that is guaranteed to be convex for a mutually concave game and has time complexity for [23, Theorem 2]. With the problem of solving a mutually concave game shown to be tractable, the authors then proceed to show that recognizing a mutually concave game can be done in time .
1.2 Our Contribution
Given a nonzero-sum bimatrix game, , we develop the SER0 algorithm that, in time linear in the size of the game, determines whether or not the given game is strategically equivalent to some zero-sum game . If so, our approach also finds an equivalent zero-sum game in time . This equivalent zero-sum game can then be solved with one call to a linear program solver. Since the two games are strategically equivalent, the NE strategies of the equivalent zero-sum game are exactly the NE strategies of the original nonzero-sum bimatrix game .
Most of the prior work on subclasses of bimatrix games focuses on characterizing those games and proving some properties of the NE of the game [12, 18, 19, 21, 22]. However, other than , these works do not discuss tractable methods for identifying whether a game is in a particular subclass. Of course, identifying whether a game is row-constant is rather trivial. However, that isn’t true for many of the other classes discussed in Subsection 1.1. In contrast, we focus here on efficient methods for identifying strategically equivalent rank- games. In addition, as we show in Section 5, our algorithm also efficiently identifies strictly competitive games.
In comparison to prior work on strategically zero-sum games , the SER0 algorithm identifies whether or not a game is strategically zero-sum in time versus , which is a significant improvement. Compared to identifying mutually concave games , the SER0 algorithm is also significantly faster, providing a speedup of . In addition, the algorithm for mutually concave games in  does not find an equivalent zero-sum game. Instead, they find a parameter that insures mutual concavity and find a NE of the original game by solving a convex quadratic program. In contrast, the SER0 algorithm returns, in linear time, an equivalent zero-sum game that can be solved via linear programming.
1.3 Outline of the Paper
We begin by defining the notation used throughout the paper and recalling some game-theoretic definitions. Then, in Section 3 we formulate the problem and informally present the results. We defer formal statements and proofs of our results to the appendices. We first show that if there exists and (we precisely define later) such that , then the nonzero-sum game is strategically equivalent to the zero-sum game . Following that, we present an efficient method for computing and present a decomposition of the matrix in Theorem 14 based on the Wedderburn Rank Reduction Formula, which is a classical result in linear algebra. While the Wedderburn Rank Reduction Formula provides the analytical result that we desire, the complexity of directly implementing the algorithm can be high. Therefore, in Appendix C we show an efficient technique for implementing the technique for a matrix .
While the results in in Section 3 apply in general to real bimatrix games with , for the purposes of algorithmic analysis in Section 4 we constrain the problem to rational bimatrix games with . In Section 4, we provide an algorithmic analysis of a shortened version of the SER0 algorithm while presenting an in-depth discussion in Appendix D.
We then present some additional discussion on the convexity of the set of NE in a strategically zero-sum game and numerical results.
We use and
to denote, respectively, the all ones and all zeros vectors of length. All vectors are annotated by bold font, e.g , and all vectors are treated as column vectors.
is the set of probability distributions over, where
Let , , denote the vector with in the th position and ’s elsewhere.
Consider a matrix . We use to indicate the rank of the matrix . indicates the subspace spanned by the columns of the matrix , also known as the column space of the matrix . We indicate the nullspace of the matrix , the space containing all solutions to , as . In addition, we use to denote the jth column of and to denote the ith row of .
In this section, we recall some basic definitions in bimatrix games and the definition of strategic equivalence in bimatrix games.
We consider here a two player game, in which player 1 (the row player) has actions and player 2 (the column player) has actions. Player 1’s set of pure strategies is denoted by and player 2’s set of pure strategies is . If the players play pure strategies , then player 1 receives a payoff of and player 2 receives .
We let represent the payoff matrix of player 1 and represent the payoff matrix of player 2. As the two-player finite game can be represented by two matrices, this game is commonly referred to as a bimatrix game. The bimatrix game is then defined by the tuple . Define the matrix as the sum of the two payoff matrices, . We define the rank of a game as .666Some authors define the rank of the game to be the maximum of the rank of the two matrices and , but this is not the case here.
Players may also play mixed strategies, which correspond to a probability distribution over the available set of pure strategies. Player 1 has mixed strategies and player 2 has mixed strategies , where and . Using the notation introduced above, player 1 has expected payoff and player 2 has expected payoff .
2.1 Strategic Equivalence in Bimatrix Games
A Nash Equilibrium is defined as a tuple of strategies such that each player’s strategy is an optimal response to the other player’s strategy. In other words, neither player can benefit, in expectation, by unilaterally deviating from the Nash Equilibrium. This is made precise in the following definition.
Definition 1 (Nash Equilibrium ).
We refer to the pair of strategies as a Nash Equilibrium (NE) if and only if:
It is a well known fact due to Nash  that every bimatrix game with a finite set of pure strategies has at least one NE in mixed strategies. However, one can define games in which multiple NE exist in mixed strategies. Let be the Nash equilibrium correspondence777A correspondence is a set valued map [24, p. 555].: Given the matrices , denotes the set of all Nash equilibria of the game . Note that due to the result in , is nonempty for every .
We say that two games are strategically equivalent if both games have the same set of players, the same set of strategies per player, and the same set of Nash equilibria. The following definition formalizes this concept.
The 2-player finite games and are strategically equivalent iff .
We now have a well known Lemma on strategic equivalence in bimatrix games that is typically stated without proof (e.g [12, 23]). As we were unable to find a proof in the literature, we state the relatively simple proof here.
Let be two matrices. Let and where , , , and . Then the game is strategically equivalent to .
Since and , we have and . Then, we have:
Now, assume that is an NE of . Then, for player 1,
Similarly, for player 2,
Then Definition 2 is satisfied, and is an NE of if and only if is an NE of . ∎
3 Problem Formulation and Main Results
In this section we show a technique for identifying and solving games that are strategically equivalent to a zero-sum game. This technique is closely related to the strategically zero-sum games in  and the mutually concave games in .
We begin by assuming that given a nonzero-sum game , there exists a positive affine transformation such that is strategically equivalent to the zero-sum game . We later show the necessary and sufficient conditions under which this assumption holds true.
From Lemma 1, we conclude that if there exists , , , and such that:
then is strategically equivalent to . Throughout the remainder of this work we refer to this notion of strategic equivalence as a positive affine transformation (PAT). If , we call it a uniform positive affine transformation (UPAT).
Defining and , we rewrite (3) as:
Clearly, the matrix above has maximum . However, the existence of and , with is not sufficient to ensure strategic equivalence. Indeed, from (4) one can see that we are searching for a matrix , which we define as
Thus, what we have shown above if the following result:
If is strategically equivalent to through PAT, then and .
The proof follows from the preceding discussions. ∎
In what follows, we show the converse holds, and further, we devise efficient algorithms to compute and determine if lies in the subspace . In fact, the properties of the subspace play a critical role in both the analytical and algorithmic results of our work. However, we collect these properties in Appendix A.2 in order to not detract from the main results.
The game satisfies
Pick such that . Define as
The matrix , defined as , lies in the subspace .
If Assumption 3 holds, then there exists a matrix such that the bimatrix game is strategically equivalent to the zero-sum game .
Thus, our algorithm indeed ascertains whether or not a given game is strategically equivalent to a zero sum game. Further, as we show in the next section, our algorithm runs in time that is linear in the size of the game to compute . This also leads to some important implications about convexity of the set of Nash equilibria as discussed in Section 5.
In contrast to related works in , , which attempt to solve an equation (or system of equations) similar to (4) we show that it is possible to decompose the problem into a sequence of steps that allows us to reduce the overall complexity of the problem. In brief, given a game these steps are
Throughout the remainder of this section, we introduce the key results used in the steps above informally and implement these steps on a simple example of rock-paper-scissors game. For ease of exposition, we defer full proofs and other technical matter to the appendices.
3.1 Solving for
Our next result shows a computationally tractable method for solving for the parameter .
Informal Statement of Proposition (13).
If is strategically equivalent to through a PAT and , then such that we have that
3.2 Determining if a Matrix is in
Another crucial step in our approach is to determine whether the matrices are in . We present here an informal method for determining this, while leaving the full statement of the relevant propositions to Appendix C.
Informal Statement of Theorem (17).
Given , select any and let
Then, iff .
We present a short outline of the proof here. First-off, we have . Secondly, if , then represents a valid decomposition of as
So, if , is a valid decomposition, and .
3.3 A Decomposition of
We now informally state our first main result on calculating the equivalent zero-sum game .
Informal Statement of Theorem (14).
Suppose that is strategically equivalent to a zero-sum game through a PAT with . Define by (4) and assume . Select any , let and . Then, and the bimatrix game is strategically equivalent to the zero-sum game .
For an informal argument to support the claim above, we have that for any with we can write
Then is strategically equivalent to by Lemma 1 and
3.4 A Simple Example: Rock-Paper-Scissors
Consider the game matrix given in Figure (a)a and let us represent this game as . This is the classic Rock-Paper-Scissors with well known NE strategies . The game in Figure (b)b is a positive affine transformation of . Let us represent this game as . Clearly, is neither zero-sum nor constant-sum. However, by applying the process outlined above one can obtain the game in Figure (c)c, which is a zero-sum game and strategically equivalent to .
By letting , and applying (5) we have that
Then we apply Theorem 14 with to obtain the strategically zero-sum game:
The result of these calculations is the zero-sum game which is displayed in Figure (c)c and, as expected, has the NE strategies .
4 Algorithm for Strategically Equivalent Rank- Games (SER0)
We have shown that given the game , one can determine if the game is strategically equivalent to the game through a PAT. If so, then it is possible to construct a rank-0 game which is strategically equivalent to the original game. One can then efficiently solve the strategically equivalent zero-sum game via linear programming. We state the key steps in our algorithm below and show that both the determination of strategic equivalence and the computation of the strategically equivalent zero-sum game can be done in time .
The analytical results and discussions throughout this paper apply to real bimatrix games, with . However, for computational reasons, when discussing the algorithmic implementations we focus on rational bimatrix games, with .
We present here a shortened version of the SER0 algorithm that applies when . The cases of are similar, but add complexity to the presentation. As such, the full algorithms and more detailed analysis are deferred to Appendix D.
Informal Statement of Theorem (19).
The SER0 algorithm determines if a game
is strategically equivalent to a rank- game and returns the strategically equivalent zero-sum game in time .
Here we give the analysis of ShortSER0 presented in Algorithm 1.
First-off, testing whether a matrix is in is equivalent to implementing (6) and then comparing two matrices. Both these operations take time . Next, for any matrix calculating takes time for . So, calculating can be done in time if one has candidate vectors . Corollary 17.2 in Appendix C gives an algorithm for determining such that runs in time .
Forming the matrix takes multiplications and additions, and therefore has time . Finally, calculating for the case consists of scalar-matrix multiplication, vector outer product, and matrix subtraction. Therefore, it has time .
This shows that overall the algorithm can both identify whether a game is strategically equivalent to a zero-sum game through a PAT and, if so, can determine the equivalent game in time .
5 Further Discussions
Typically, a general bimatrix game may admit a set of multiple Nash equilibria that are disconnected. However, one of the well-known results in game theory is that the set of NE of a zero-sum game is a convex set in . As a result, we immediately have the following result.
Corollary 4.1 ([12, Corollary 1]).
If a bimatrix game satisfies Assumption 3, then the set of Nash equilibria is a compact convex subset of . In addition, the bimatrix game also has the ordered interchangeability property.
For another example, typically fictitious play is not guaranteed to converge in nonzero-sum games with . However, it is well known that fictitious play converges in zero-sum games . Therefore, we have:
Corollary 4.2 ([12, Corollary 3]).
If a bimatrix game satisfies Assumption 3, then fictitious play converges.
As a third immediate result, consider the class of strictly competitive games, which are a subclass of strategically zero-sum games. As characterized in , a game is strictly competitive iff for some and , . Then, the following result directly follows from Theorems 15 and 19.
The SER0 algorithm determines whether a game is strictly competitive in time .
6 Numerical Results
To evaluate the performance of the SER0 algorithm we ran several sets of experiments. For each experiment, we generated square games of size , with
and uniformly distributed payoff values in the game matrices. For each value of, we ran SER0 on different game instances.
All experiments were conducted on a standard desktop computer running Windows 7 with GB of RAM and an Intel Xeon E5-1603 processor with cores running at GHz.
For our first set of experiments, for each value of we created instances that were strategically equivalent to a zero-sum game . In all cases tested, the SER0 algorithm correctly identified the games as strategically zero-sum and calculated the equivalent game . As expected, one can observe from Figure 2 that for this set of experiments there is a clear linear relationship between the runtime of SER0 and the size of the game instance. In addition, for very large games of size SER0 found the equivalent game in an average time of seconds.
For the next set of experiments we created games which were guaranteed to have a pure strategy NE. In other words, at least one of or were in . Again, the SER0 algorithm correctly identified all of these cases as having a pure strategy NE. Similar to the first set of experiments, we observe that SER0’s runtime is linear in this case. As expected, this case is much faster than the strategically equivalent case as there is no need to calculate nor test for .
Finally, we conducted experiments on games that were not strategically equivalent to a zero-sum game via a PAT. Similar to the other two sets of experiments, the SER0 algorithm correctly identified these games as not strategically equivalent to a zero-sum game. As Figure 2 shows, this case also exhibits a linear relationship between runtime and the game size, although with a much higher standard deviation compared to the other two sets of experiments. This higher standard deviation is readily explainable by examining the SER0 algorithm. When testing whether or not a game is strategically equivalent, the test can return a negative result if either or . For cases of , the algorithm terminates and returns a negative result in much less time than it takes to calculate the matrix and test for .
In this work we’ve presented an algorithm, the SER0 algorithm, that for a bimatrix game determines in linear time whether the game either has a guaranteed pure strategy Nash equilibrium or is strategically equivalent to some zero-sum game. In the latter case, the algorithm also finds the strategically equivalent zero-sum game in linear time. This strategically equivalent game can then be solved efficiently via linear programming, and this solution provides a NE strategy of the original game. This represents a significant computational advantage compared to prior work on computing strategically equivalent zero-sum games. In comparison to mutually concave games, the SER0 algorithm recognizes an equivalent game in a computationally efficient manner, then can solve the game with a linear program instead of a convex quadratic program. Thus, our algorithm substantially expands the class of nonzero-sum games for which a computationally efficient (polynomial time) algorithm for computing Nash equilibrium is known.
One interesting direction for future research would be to explore the relationship, if there is one, between (weakly) unilaterally competitive games and strategically zero-sum games. In regards to algorithmic game theory, one main challenge to address is whether or not there exists a polynomial time algorithm that could identify strategically equivalent rank- games. Another interesting extension would be to develop an algorithm to compute approximate NE based on “almost” strategically zero-sum games.
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Appendix A Results From Linear Algebra
In this section, we collect some preliminary results from linear algebra that are used throughout the rest of the paper to prove the main result. The first subsection describes the Wedderburn rank reduction formula, which presents a method to reduce the rank of a matrix through a simple decomposition technique. The second subsection discusses certain properties of the vector space that are used later in the proofs.
a.1 Wedderburn Rank Reduction Formula
Let be an arbitrary matrix, not identically zero. Then, there exists vectors and such that . Then, setting for convenience, the matrix
has rank exactly one less than the rank of .
We first show that the null space of contains the null space of . Pick such that . Then,
Thus, is in the null space of , which implies that the null space of contains the null space of .
Next, we show that is in the null space of , thereby showing that the dimension of the null space of is one more than the dimension of the null space of (since ). Consider
Thus, the rank of is one less than the rank of .
We now have the following theorem that applies Wedderburn rank reduction formula to compute a decomposition of the matrix.
Let . If , then there exists , , such that and the following holds:
where , , and . In addition, let , where . Then, we have
Apply Theorem 5 to for iterations. ∎
a.2 The Subspace
In this subsection, we recall an essential fact from linear algebra, state an interesting corollary of Theorem 6, and introduce the the subspace , which we define as
The properties of this subspace are essential to formulating the algorithmic solution to our problem as the matrix that we are searching for must lie in this subspace.
Let us begin by stating an essential fact on the rank-1 decomposition of a matrix.
For any matrix with , one can write as a summation of rank-1 matrices, . Furthermore, is a basis for and is a basis for .
Then, combined with (9) we have
From (10) and Fact 1, it is apparent that the basis for is formed by removing from the basis of . Then, since by definition the basis vectors are linearly independent, there is no linear combination of , such that the linear combination equals for . Therefore, and for all . The same argument holds for for all .
We now proceed to prove that is indeed a subspace of the vector space of all real matrices. Following that, we state some essential properties of the subspace that will be used throughout the presentation in Appendix B.
Let be the vector space of real matrices and be the space of real matrices such that for all , . Then is a subspace of .
We show that meets the three properties of a subspace of a vector space. First, consider ,. Then, and the zero vector is in . Secondly, for all ,
Therefore, . Finally, for all and , . ∎
For any matrix with there is at least one nonzero column and one nonzero row. Let and denote such a column and row. Then, with and we have that , , , and .
The proof is straightforward and therefore omitted. ∎
For any matrix :
If , then and . In addition, for all such that ,,,.
If , then either , or or both and .