and Tulkens (1986) exhibit asymmetric tax rates for identical member states.
As a second motivation, the present paper is an attempt to develop a unifying
approach to understanding symmetry-breaking mechanisms in general classes
of two-player games, encompassing many of the cited studies. These two-stage
models share two key features that are critical for the symmetry-breaking ar-
guments they present. The first is a fundamental nonconcavity in the payoffs,
which may be confined to the diagonal in action space or hold globally, and the
second is some form of strategic substitutes in first-period actions, possibly of
an abstract sort (more on this in Section 4). While there is quite some variation
in the precise manner versions of these two features are present and interact
across all the models, we will be able to capture most of them in three separate
general results, which though quite distinct at first sight, nonetheless bear some
definite relationship at an abstract level.
The paper is organized as follows. Section 2 contains the overall set-up. Sec-
tion 3 provides the results on the exclusive existence of asymmetric equilibria
for submodular payoff functions. Section 4 deals with nonsubmodular payoff
functions and Section 5 presents the results for games with convex payoffs. Sec-
tion 6 discusses an extension to ordinal complementarity and substitutability
conditions. Each section provides a summary of the relevant applications the re-
sults pertain to. The appendix provides a brief overview of the supermodularity
notions and results.
2 Setup
This section lays out the general notation for use throughout the paper. The
nooncooperative game described below may be a simple one-shot game or it may
represent the payoffs of a two-stage game as a function of the first period actions,
where the unique second stage pure-strategy equilibrium has been substituted
in. In the latter case, which actually covers most of the applications of this
paper, we obviously restrict consideration to subgame-perfect equilibria and
analyze the resulting one-shot game.