Hidden symmetries and focal points

Abstract

This paper provides a general formal framework to define and analyze the concepts of focal points and frames for normal form games. The information provided by a frame is captured by a symmetry structure which is consistent with the payoff structure of the game. The set of alternative symmetry structures has itself a clear structure (a lattice). Focal points are strategy profiles which respect the symmetry structure and are chosen according to some meta-norm, which is not particular to the framed game at hand. We also clarify the difference between different concepts of symmetry used in the literature. The main aim of this paper is to provide a general framework for the analysis of focal points in the tradition of Schelling (1960). Our analysis also enables us to clarify the meaning of the concept of a framed game, which is extensively used in various forms in the literature. We remark right away that the framework we develop is not limited to pure coordination games (as most papers on focal points and framed games), but encompasses arbitrary normal-form games. Games, in this paper, are truly one-shot. That is, they are not played recurrently, such as the game of which side of the road to drive on, for which conventions have been established through recurrent interaction. Rather, we assume that players are unfamiliar with the particular game at hand (and have no expectation of ever playing it again either). The game might be of a form that is recognized, but the game itself is new to the players. 1 The game, in this paper, can also come with a frame 2 (context, setting). The frame can be familiar to players or it can be unfamiliar as well. We will argue that a formal definition of a focal point of a framed game has to come in two parts. First, a careful study of the symmetry structures present in the game and its frame is required. Much of this paper is devoted to the analysis of these symmetry structures. On the basis of the induced symmetry structure of a framed game, a subset of strategy profiles is identified which is consistent (in a well-specified sense) with this symmetry structure and meets certain rationality requirements. We shall call these rational symmetric recommendations. A multiplicity problem is unavoidable, and indeed will be exacerbated the more symmetries are broken by increasingly detailed frames. Hence, the second