Bayesian Nash equilibrium; a statistical test of the hypothesis

Many economic environments are modeled as games under incomplete information. A fundamental concept in these models is the hypothesis of a Bayesian Nash equilibrium. However, in spite of the common use of this hypothesis in economic theory and econometric estimation models, no statistical methods have been suggested how to use real life data to test its validity. There are many doubts whether economic agents are sufficiently sophisticated and informed to find their way to such an equilibrium (see, for example, Rubinstein (1998) and Gilboa&Schmeidler (2001)). We show in this paper how a test of the hypothesis can be conducted in the case of a first price auction game with risk averse bidders. We derive a theoretical characterization of the distribution of equilibrium bids, and show that, with some modifications, the Kolmogorov conservative test (see Bedford and Meilijson (1997)) can be applied as a statistical tool. However, since our problem is cast as a composite hypothesis in an infinite dimensional space, it is not clear how to verify whether the theoretical characterization is satisfied. In our case, the question is whether a distribution function that satisfies a second order differential inequality can be ’squeezed’ into a corridor between two given monotone stepwise functions. The problem motivates the concept of relative concavity which we introduce as a testable characterization. The latter makes it possible to develope an algorithm that implements the test in a finite number of steps. We are grateful to E. Dekel, E. Guerre, P. Haile and I. Meilijson for helpful comments on a previous draft of this paper.