The study of game theory — the model of strategic interaction between rational agents — has a rich history dating back to the formalization of the field by John von Neumann in 1928von1959theory . The concept of equivalence, in particular strategic equivalence, between game theoretic models also enjoys a rich history, dating back to at least von Neumann and Morgenstern’s book first published in 1944 (von2007theory, , p. 245). We say that two games, and , are strategically equivalent if the optimal strategies of every player in corresponds to the optimal strategies of every player in .111See Subsection 2.1 for a more formal definition and the specific form of strategic equivalence that we study in this work.
Computing a solution in a -player finite normal-form game , typically referred to as a Nash equilibrium (NE), is one of the fundamental problems in game theory.222Nash 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 nash1951 . However, outside of some restricted classes of games, it is not clear that a NE can be efficiently computed.
In this work we focus on finite, -player bimatrix games in which the payoffs to the players can be represented as two matrices, and , and define the rank of a game as the rank of the sum of the two payoff matrices.The rank of a game is known to impact both the most suitable computation methods for determining a solution and the expressive power of the game. For example, it is well known that zero-sum games333A game is called a zero sum game if the sum of payoffs of the players is identically zero for all actions of the players.
, which are rank-0 games, can be solved via a linear programming approach. For approximate solutions,kannan2007games give a computational method for rank-k games, defined as a class of games where , for some given .
On the other hand, many operations that preserve the strategic equivalence of bimatrix games modify the rank of the game. For example, the well studied constant-sum game is strategically equivalent to a zero-sum game.444In a constant-sum game, the sum of the two payoff matrices equals a constant matrix. However, the zero-sum game has rank zero, while the constant-sum game is a rank- game. Since the rank of a game influences both the most suitable solution techniques and the expressive nature of a game, one should be particularly interested in determining if a given game is strategically equivalent to a game , where the rank of is less than the rank of .
In this contribution, we do just that. Given a game , we apply the classical theory of matrix pencils in conjunction with the Wedderburn rank reduction formula to determine whether or not is strategically equivalent to a game of lower rank. If so, we also show techniques for efficiently calculating the lower rank game.
Prior to discussing related research in this area, let us first proceed to outline the remainder of the paper. In the next section, we review some game theoretic concepts, introduce our notation, and proceed to define the specific form of strategic equivalence that we consider. Following that, we then formalize our problem and present the main result. Then, in Section 4, we briefly review the theory of matrix pencils and the Wedderburn rank reduction formula, and apply those results to derive a sequence of theorems and corollaries which thus prove the main result. Following that, we present some consequences of our results when applied to generic (random) games. We then proceed to compare our results to other results in the literature, discuss some additional immediate consequences of our results, and lightly touch on some algorithmic implications. Finally we wrap up the paper with a conclusion, and collect some related theorems and proofs in the appendix.
1.1 Prior Work
Many papers in the literature have explored the concept of equivalent classes of games. One such concept, which we focus on in this work, 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 (von2007theory, , p. 245) and constant-sum games (von2007theory, , p. 346).
In a recent work by Possieri and Hespanha, the authors consider the problem of designing strategically equivalent games possieri2017algebraic . As such, given a bimatrix game , their problem is to design a family of games that is either weakly strategically equivalent555The authors of possieri2017algebraic define weak strategic equivalence as games the have the same NE in pure strategies. or strongly strategically equivalent666The authors of possieri2017algebraic define strong strategic equivalence as games the 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 game of strictly lower rank?
More closely related to our work is the class of strategically zero-sum games defined by Moulin and Vial in moulin1978strategically . 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 (moulin1978strategically, , Theorem 2).
Around the same time, Isaacson and Millham studied a class of bimatrix games that they characterized as row-constant games isaacson1980class . 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 isaacson1980class and moulin1978strategically , 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 aumann1961almost . 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 setfriedman1983characterizing . Indeed, Aumann claims that strictly competitive games are equivalent to zero-sum games aumann1961almost . 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 made was often repeated, but without formal proof adler2009note . They then proceeded to prove that this claim does indeed hold true. Comparing the results of Adler et. al adler2009note to the characterization of strategically zero-sum games in moulin1978strategically , one can observe that Moulin and Vial were correct in asserting that strictly competitive games form a subclass of strategically zero-sum games.
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 777Some 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 strategiesand 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 nash1951 )
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 nash1951 that every bimatrix game with 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 correspondence888A correspondence is a set valued map (ali2006, , p. 555).: Given the matrices , denotes the set of all Nash equilibria of the game . Note that due to the result in nash1951 , 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.999See e.g moulin1978strategically ; kontogiannis2012mutual . 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 Result
The positive affine transformation (PAT) that we have presented in Lemma 1 has been well studied for the forward direction case. In those situations, one calculates the Nash equilibria of the game . Then, by choosing suitable parameters , one can design a family of games that share the same set of NE as , but with a different payoff structure.
Here, we consider the inverse problem. Given a game , is it possible to determine parameters , such that is strategically equivalent to via a PAT? This is particularly interesting when the game is a low rank game and is a game of higher rank. In other words, and , with .
Low rank games are an interesting class of games to study, as the rank of a game is known to impact both the most suitable computation methods for determining a solution and the expressive power of the game. For example, it is well known that zero-sum games, where , can be solved via a linear programming approach. In addition, recent work has shown computationally efficient algorithms for finding one NE adsul2011rank or all NE adsul2011rank ; theobald2009enumerating of rank-1 games. For approximate NE, kannan2007games give a computational method for rank-k games, defined as a class of games where , for some given .
As for the expressive power of a game, we consider the maximum number of NE that a game may have. For nondegenerate101010For a concise discussion of (non)degeneracy, see (von2002computing, , Section 2.6) zero-sum games, it is well-known that there is only one possible NE. As for the expressive power of rank-k games, with , the maximal number of NE is an open question. However, lower bounds, even for rank-1, are significantly higher than the zero-sum case; see, for example, (kannan2007games, , Corollary 3.1) and adsul2019 .
With the motivation for determining the true rank (whether the game is strategically equivalent to a game of lower rank) established above, we now turn our attention to analyzing the mathematical properties of the positive affine transformation.
From Lemma 1, we conclude that if there exists , , , and such that:
then is strategically equivalent to the rank- game via a positive affine transformation (PAT).
Defining , , , and letting , we rewrite (3) as:
Thus, what we have shown above is the following result:
If is strategically equivalent to through a PAT, then .
The proof follows from the preceding discussions. ∎
In what follows, we show the converse holds. Furthermore, in Section 4, we show constructive methods for obtaining a game of lower rank that is strategically equivalent to the original game.
The game satisfies
There exists a such that
There exists and such that can be decomposed into .
In Section 4, we prove Theorem 4.7 and, consequently, Theorem 3.3 above, through a series of intermediate theorems and corollaries. We will show the necessary conditions under which Assumption 3.2 holds true and devise methods to test those conditions. In addition, we will show that if either condition of Assumption 3.2 hold true, it is possible to reduce the rank of the game to some degree, although that reduction isn’t as high as the reduction that one obtains when both conditions hold true.
We can therefore decompose our problem into two parts based on Assumption 3.2, and a third part based on constructing the strategically equivalent lower-rank game . Those three parts are:
Given , determine whether or not there exists such that . If there does not exist any satisfying this condition, then set .
With determined above, determine if there exists and such that can be decomposed into
From determined above, construct the strategically equivalent rank- game
4 Proof of The Main Result
In this section, we introduce matrix pencils, discuss an existing canonical form for calculating the eigenvalues of rectangular pencils, and show how such a canonical form can be applied to obtain a bimatrix game of lower rank than the original game. Following that, we will discuss the Wedderburn rank reduction formula and apply that to further reduce the rank of a game. Finally, we conclude the section with the statement and proof of our main result.
4.1 Matrix Pencils
Let be matrices of known values, and let represent an unknown parameter.111111In general, the theory of matrix pencils is defined over a field . In a our game-theoretic context, we restrict this field to the field of real numbers. Then the set of all matrices of the form , with , define a linear matrix pencil (or just a pencil)(gantmakher1959theory, , p. 24),ikramov1993matrix .121212The literature defines both the set of matrices and as pencils. Although seems to be more common (possibly due to the connection to the standard presentation of the eigenvalue problem ), we choose to use as it more closely aligns with the problem presented in Section 3.
While not as well studied as the standard eigenproblem, , the theory of pencils still enjoys a rich history. For the square, nonsingular, case, Weierstrass investigated pencils and developed a canonical form as early as 1867. The rectangular case was later solved, with a canonical form presented, by Kronecker in 1890. His canonical form, aptly named the Kronecker Canonical Form (KCF), was popularized by Gantmacher in chapter XII of his two volume treatise on the theory of matrices (gantmakher1959theory, , Ch. 12). For a (not so short) survey on pencils, we refer the reader to ikramov1993matrix . For a discussion of the various canonical forms and computational methods, for the square case see (golub2012matrix, , Ch. 7.7), and for the more general singular/rectangular case, see demmel1993generalized .
, the authors study the relationship between the eigenvalues and eigenvectors of matrix pencils and the solution of a zero-sum game, where the game is formulated as.131313The square, traditional eigenvalue problem with is studied in weil1968game . The authors present the rectangular matrix pencil version in thompson1972roots . Their results are indeed theoretically interesting; however, as the author states in weil1968game , the relationship between eigensystems and game theory is ”tenuous”. This seems to make, at least in accordance with the current theoretical results, the study of eigensystems ill-suited as a solution concept for bimatrix games. In contrast, as we will show in this subsection, the matrix pencil problem is well-suited to the study of strategically equivalent games.
In the remainder of this section, we review the canonical form of a matrix pencil presented by Thompson and Weil in thompson1970reducing ; thompson1972roots . Although not a common terminology in the literature, we’ll refer to this canonical form as the Thompson-Weil Canonical Form (TWCF). Our motivation for studying the TWCF is two-fold. First off, the TWCF focuses on only computing those eigenvalues, if they exist, that strictly reduce the rank of the pencil . Other extraneous values, such as those computed in the KCF, are ignored. Secondly, we wish to bring renewed emphasis on existing results that connect the study of matrix pencils to game theory.
For completeness, we now restate some results from thompson1970reducing ; thompson1972roots ; dell1971algorithm . Following that, we show how to apply those results to calculate an equivalent game of lower rank.
Let rank(A)=p and rank(B)=r.
Definition 3 ((thompson1972roots, , Definition 2.1))
By a solution to the pencil , we shall mean a triple , satisfying and , that solve the set of equations
and have the property that for any that is not an element of the solution triple.
Throughout their series of works on matrix pencils, the authors of thompson1970reducing ; thompson1972roots ; dell1971algorithm refer to the solution triple in Definition 3 by various names such as pencil value, pencil roots, rank-reducing numbers, left pencil-vector, and right pencil-vector. For simplicity, we choose to use the terms eigenvalue and left\right eigenvector. We will also use the set to represent all that are in the solution triple as defined in Definition 3.141414Note to the reader who is more familiar with the KCF form. The set exactly corresponds to the eigenvalues of the Jordan block in KCF form.
Lemma 2 ((thompson1970reducing, , Lemma 1))
For all , there exists nonsingular and such that can be partitioned into:
where , is in column echelon form, , is in row echelon form, and
is in row echelon form, and. Any of may be zero, , and .
Lemma 3 ((thompson1970reducing, , Lemma 2))
If in Lemma 2, there exists nonsingular and such that can be partitioned into:
See Lemma 2 of thompson1970reducing .
Furthermore, by repeated applications of Lemmas 2 and 3, the authors define an iterative algorithm that solves for the set , including identifying if . We briefly outline the algorithm in Algorithm 1. For the full proof and implementation details, we refer the reader to thompson1970reducing ; dell1971algorithm .
As the authors note in dell1971algorithm , their algorithm may be numerically unsound for ill-conditioned problems. Indeed, while there does not appear to be any results in the literature comparing the numerical stability of the TWCF algorithm and Gantmacher’s method for computing the KCF, it seems likely that both methods may share similar numerical difficulties. Therefore, other algorithms may be better suited for ill-conditioned problems. For the square, dense matrix pencil, (whether ill-conditioned or not) the famous QZ algorithm of Moler and Stewart is likely a better option moler1973algorithm ,(golub2012matrix, , Ch. 7.7). For the rectangular case, numerical accuracy for ill-conditioned problems can likely be improved via the GUTPRI algorithm demmel1993generalized . However, even in light of this discussion, we choose to explore the TWCF as it provides insight into the mathematical structure of the pencil that applies to our problem at hand.
We now state one final definition before proceeding to state the main theorem from thompson1970reducing , which we use to prove our first main result.
For any , define the geometric multiplicity of as:
the number of Jordan blocks containing in the Jordan normal form of where is the smallest integer such that .
Theorem 4.1 ((thompson1970reducing, , Theorem))
See thompson1970reducing .
As we are only concerned with the real, strictly positive eigenvalues, let us further define the restricted set of eigenvalues as:
Consider the game with . Calculate and as in Algorithm 1. If , then select such that for all . Let and . Define and . If , then the game is strategically equivalent via a positive transform to the rank- game , where .
Since, by definition, , we have that . Therefore, the game is a rank- game. Finally, by Lemma 1, the game is strategically equivalent to with , , .
4.2 Further Reduction via the Wedderburn Rank Reduction Formula
In this subsection, we show that if Theorem 4.2 identifies a reduction to an equivalent game of lower rank, then it may be possible to reduce the rank even further. In contrast, if Theorem 4.2 does not identify a game of lower rank, if certain conditions hold we may still be able to reduce the rank of the game (by at most 2 in both cases). In what follows, we first present the Wedderburn Rank Reduction formula upon which our technique is based, and then we proceed to state and prove our next two results.
The Wedderburn Rank Reduction formula is a classical technique in linear algebra that allows one to reduce the rank of a matrix by subtracting a specifically formulated rank-1 matrix. By repeated applications of the formula, one can obtain a matrix decomposition as the sum of multiple rank-1 matrices. In contrast to other well-known matrix factorization algorithms, such as singular value decomposition, the Wedderburn rank reduction formula allows almost limitless flexibility in choosing the basis of the rank-1 matrices that are subtracted at each iteration. For further reading on the Wedderburn rank reduction formula, we refer the reader to Wedderburn’s original book(wedderburn1934lectures, , p. 69), or to the excellent treatment of the topic by Chu et al. chu1995rank .
We now proceed to state Wedderburn’s original theorem. Following that, we show how one can exploit the flexibility of the decomposition to extract specifically formulated rank-1 matrices that allow us, when certain conditions hold true, to further reduce the rank of a bimatrix game.
Theorem 4.3 ((wedderburn1934lectures, , p. 69) chu1995rank )
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 .
Consider again the game upon which one has applied Theorem 4.2. Here we use the same notation as in Theorem 4.2. If Theorem 4.2 has identified a such that , let be as defined in Theorem 4.2. Otherwise, let and . We now apply the following series of theorems and corollaries to this game .
Consider the game with , , and . If
then there exists and such that and . Let and compute using (7) as follows:
Define and . Then the bimatrix game is strategically equivalent to the rank- game .
Before stating the proof of Theorem 4.4, we first state an auxiliary lemma that we invoke in the proof.
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. ∎
Proof (Proof of Theorem 4.4)
Suppose (8) holds true, then there exists such that . In addition, implies that . Then applying Lemma 4 shows the existence of such that . With and therefore well-defined, apply (7) to obtain as in the statement of the theorem. By Theorem 4.3, is a matrix of rank-. With and we have that:
Then the game is strategically equivalent to by Lemma 1 and has rank . ∎
Apply the proof of Theorem 4.4 to , making the appropriate dimensional changes to , and the all ones vector. Then let and . ∎
Consider the game with , , and . The game is strategically equivalent to a game of rank-
Consider the game with and . Let , where . If
then there exists and such that:
Define and . Then, , where , and the bimatrix game is strategically equivalent to the rank- game .
Before stating the proof of Theorem 4.5, we first state proposition that is necessary for the proof. Motivated by (chu1995rank, , Theorem 2.1), we have the following proposition that gives conditions for when a chosen vector is in the row span of a matrix , obtained after one application of (7).
We use here the same notation as in Theorem 4.3. Let , with . Let be vectors associated with a rank-reducing process (so that ). We have if and only if and .
See Appendix B. ∎
Proof (Proof of Theorem 4.5)
By assumption (10), there exists a vector such that and . Let equal such an . Then, and Lemma 4 implies that there exists such that and . Therefore, we have . Let , then . After applying the Wedderburn rank reduction formula, we have
We now proceed to show that there exists such that . By assumption (9), we have that . Also, by assumption 10, we have that . Then, by Proposition 4.6, and implies that . Therefore, there exists such that .