Stochastic Equilibria: Noise in Actions or Beliefs?

We introduce noisy belief equilibrium (NBE) for normal-form games in which players best respond to noisy belief realizations. Axioms restrict belief distributions to be unbiased with respect to and responsive to changes in the opponents’ behavior. The axioms impose testable restrictions both within and across games, and we compare these restrictions to those of regular quantal response equilibrium (QRE) in which axioms are placed on the quantal response function as the primitive. NBE can generate similar predictions as QRE in several classes of games. Unlike QRE, NBE is a refinement of rationalizability and invariant to affine transformations of payoffs.