Notes on “Refinements and Higher Order Beliefs”
The abstract of our 1997 survey paper Kajii and Morris (1997b) on "Refinements and Higher Order Beliefs" reads:
This paper presents a simple framework that allows us to survey and relate some different strands of the game theory literature. We describe a “canonical” way of adding incomplete information to a complete information game. This framework allows us to give a simple “complete theory” interpretation (Kreps 1990) of standard normal form refinements such as perfection, and to relate refinements both to the “higher order beliefs literature” (Rubinstein 1989; Monderer and Samet 1989; Morris, Rob and Shin 1995; Kajii and Morris 1997a) and the “payoff uncertainty approach” (Fudenberg, Kreps and Levine 1988; Dekel and Fudenberg 1990).
In particular, this paper provided a unified framework to relate the notion of equilibria robust to incomplete information introduced in Kajii and Morris (1997a) [Hereafter, KM1997] to the classic refinements literature. It followed Fudenberg, Kreps and Levine (1988) and Kreps (1990) in relating refinements of Nash equilibria to a "complete theory" where behavior was rationalized by explicit incomplete information about payoffs, rather than depending on action trembles or other exogenous perturbations. It followed Fudenberg and Tirole (1991), chapter 14, in providing a unified treatment of refinements and a literature on higher-order beliefs rather than proposing a particular solution concept.
The primary purpose of the survey paper was to promote the idea of robust equilibria in KM1997 and we did not try to publish it as an independent paper. Since we wrote this paper, there have been many developments in the literature on robust equilibria, fortunately. But there has been little work emphasizing a unified perspective, and consequently this paper seems more relevant than ever. We are therefore very happy to publish it twenty years later. We provide some notes in the following on relevant developments in the literature and how they relate to the survey. These notes assume familiarity with the basic concepts introduced in the survey paper and KM1997.