The Empirical Implications of Rank in Bimatrix Games

TL;DR

This paper investigates how the structural complexity of bimatrix games, measured by rank, affects the empirical interpretability of observed strategic behavior, revealing limitations in testing game complexity from data.

Contribution

It demonstrates that high-rank (complex) games have observable implications in general data, but in restricted laminar data, all observations can be rationalized by low-rank (zero-sum) games.

Findings

High complexity impacts empirical data interpretation.

Laminar data can be rationalized by low-rank games.

Structural properties relate to the maximum rank needed for rationalization.

Abstract

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.