The Empirical Implications of Rank in Bimatrix Games

We study the structural complexity of bimatrix games, formalized via rank, from an empirical perspective. We consider a setting where we have data on player behavior in diverse strategic situations, but where we do not observe the relevant payoff functions. We prove that high complexity (high rank) has empirical consequences when arbitrary data is considered. Additionally, we prove that, in more restrictive classes of data (termed laminar), any observation is rationalizable using a low-rank game: specifically a zero-sum game. Hence complexity as a structural property of a game is not always testable. Finally, we prove a general result connecting the structure of the feasible data sets with the highest rank that may be needed to rationalize a set of observations

On the Existence of Low-Rank Explanations for Mixed Strategy Behavior

Nash equilibrium is used as a model to explain the observed behavior of players in strategic settings. For example, in many empirical applications we observe player behavior, and the problem is to determine if there exist payoffs for the players for which the equilibrium corresponds to observed player behavior. Computational complexity of Nash equilibria is an important consideration in this framework. If the instance of the model that explains observed player behavior requires players to have solved a computationally hard problem, then the explanation provided is questionable. In this paper we provide conditions under which Nash equilibrium is a reasonable explanation for strategic behavior, i.e., conditions under which observed behavior of players can be explained by games in which Nash equilibria are easy to compute. We identify three structural conditions and show that if the data set of observed behavior satisfies any of these conditions, then it is consistent with payoff matrices for which the observed Nash equilibria could have been computed efficiently. Our conditions admit large and structurally complex data sets of observed behavior, showing that even with complexity considerations, Nash equilibrium is often a reasonable model.Comment: Updated writeup. 19 page

The Complexity of Nash Equilibria as Revealed by Data

In this paper we initiate the study of the computational complexity of Nash equilibria in bimatrix games that are specified via data. This direction is motivated by an attempt to connect the emerging work on the computational complexity of Nash equilibria with the perspective of revealed preference theory, where inputs are data about observed behavior, rather than explicit payoffs. Our results draw such connections for large classes of data sets, and provide a formal basis for studying these connections more generally. In particular, we derive three structural conditions that are sufficient to ensure that a data set is both consistent with Nash equilibria and that the observed equilibria could have been computed effciently: (i) small dimensionality of the observed strategies, (ii) small support size of the observed strategies, and (iii) small chromatic number of the data set. Key to these results is a connection between data sets and the player rank of a game, defined to be the minimum rank of the payoff matrices of the players. We complement our results by constructing data sets that require rationalizing games to have high player rank, which suggests that computational constraints may be important empirically as well