A new epistemic characterization of ε-proper rationalizability

For a given ε>0, the concept of ε-proper rationalizability (Schuhmacher, 1999) is based on two assumptions: (1) every player is cautious, i.e., does not exclude any opponent's choice from consideration, and (2) every player satisfies the ε-proper trembling condition, i.e., the probability he assigns to an opponent's choice a is at most ε times the probability he assigns to b whenever he believes the opponent to prefer b to a. In this paper we show that a belief hierarchy is ε-properly rationalizable in the complete information framework, if and only if, there is an equivalent belief hierarchy within the incomplete information framework that expresses common belief in the events that (1) players are cautious, (2) the players' beliefs about the opponent's utilities are “centered around the original utilities” in some specific way parametrized by ε, and (3) players rationalize each opponent's choice by a utility function that is as close as possible to the original utility function.