1. INTRODUCTION
Sequential bargaining models of complete information starting with Rubinstein, 1982,
have provided a fruitful environment for the study of the resolution of disagreements among
agents. Unlike cooperative formulations which are silent about the underlying actions of
bargaining parties and impose excessively strong equilibrium notions, these games juxtapose
equilibrium conditions based on a scrutiny of the optimality of individual actions with which
proposals emerge and agreements are crafted. In typical situations, (refined) equilibria of
these games exist allowing applications to numerous areas of social interaction. In legislative
or other political environments where the non-existence of equilibrium of the cooperative
genre is pervasive, such models have been welcomed in celebratory spirits.
Our goal in this paper is to study the structure of the set of equilibria for an important
class of these bargaining games and analyze their stability to perturbations of the parameters.
We focus on stationary subgame perfect equilibria in pure strategies (PSSP). We allow
general agreement rules, although our results are stronger for a subclass of these. We assume
players are recognized to make proposals with some probability fixed across periods. We
study both the case of discounting and fixed delay costs.
Except in special cases, the behavior of SSP equilibria for these games is not fully
understood. Banks and Duggan, 2000, studied in significant generality discounted games
such as those we analyze and showed upper-hemicontinuity of equilibria with respect to
parameters. But they also provided an example of a majority rule game that has a continuum
of pure strategy SSP and the equilibrium correspondence fails lower-hemicontinuity. Thus,
a slight perturbation of the parameters of the model may lead to a new game with no
equilibrium near that (rather, near one of the equilibria) of the original game.
This possibility is disturbing for obvious epistemological reasons, since we are seldom
able to measure a model’s parameters with precision. Also, in political applications, such sta-
bility to perturbations seems essential in order to build richer models of political interaction.
In parliamentary systems, for example, government formation following elections requires
us to append a bargaining model at the end of a game preceded by an electoral stage. In
these situations, it is important to ensure conditions such that voters, by changing their
behavior at the electoral stage, induce continuous changes in the distribution of subsequent
agreements and policy outcomes.
This property arises in certain types of unanimity games. For such n-player games with
discounting, Merlo and Wilson, 1995, have shown that stationary equilibria are unique when
bargaining emulates the division of a possibly stochastic cake and a contraction condition is
met. Thus, for the subset of games in Merlo and Wilson for which the upper-hemicontinuity
result of Banks and Duggan holds, the equilibrium simply becomes a continuous function of
the parameters.
But collective bargaining will often not emulate the division of a cake, due to the
public goods aspect of agreements or the ideological nature of disagreements. For such an
ideological space, in section 4 of this paper we provide an example of a four-player discounted
unanimity game that admits a continuum of PSSP.
The situation is less encouraging in the case of fixed delay costs. In that case, we
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Convention