# Dynamic Bargaining and Stability with Veto Players

**Vincent Anesi and John Duggan**

Since the seminal work of Baron (1996), bargaining games with an endogenous status quo have become more and more prominent in the literature on dynamic collective decision-making. In these games, each period begins with a status quo alternative inherited from the previous period, and a player is chosen randomly to propose any feasible alternative, which is then subject to an up or down vote. If the proposal is voted up, then it is implemented in that period and becomes the next period's status quo; if it is voted down, then the ongoing status quo is implemented and remains in place until the next period; this process continues ad infinitum. Anesi (Games and Economic Behavior 2010) was the first to consider the finite framework, where the set of alternatives is finite and players have strict preferences. His main goal was to provide a noncooperative interpretation for von Neumann-Morgenstern solutions (von Neumann and Morgenstern, 1944), whose rationale in the voting context had been questioned by political scientists (eg. McKelvey et al., American Political Science Review 1978). Assuming patient players, he shows that given a von Neumann-Morgenstern Solution Y for the voting rule and a sufficiently high discount factor, there is a stationary Markovian equilibrium whose set of absorbing alternatives is equal to Y. Left open is the opposite logical direction: conditions under which given a stationary Markovian equilibrium whose set of absorbing alternatives is a von Neumann-Morgenstern solution. Concentrating on pure strategy equilibria, Diermeier and Fong (Games and Economic Behavior 2012) obtain this direction by assuming, in addition to high discount factors, that the same player proposes with probability one in every period.

In this Nottingham School of Economics working paper, forthcoming in * Games and Economic Behavior*, Vincent Anesi and John Duggan contribute further to this research programme by examining the structure of (mixed-strategy) stationary Markovian equilibria in the finite framework of Anesi (2010). Their analysis relies on the characterization of the ergodic properties of equilibria. Namely, they show that when the Nakamura number of the voting rule is high relative to the number of alternatives, all ergodic sets are singleton; in particular, if there is a veto player, then the equilibrium process transitions with probability one to the set of absorbing alternatives. Moreover, they show that if there is a veto player with positive recognition probability and players are patient, then starting from any given alternative, there is a unique absorbing point (which can depend on the alternative given) to which the equilibrium process transitions. These results allow them to establish a tight connection between the set of equilibrium absorbing points and the von Neumann-Morgenstern solutions. Maintaining the assumption that players are patient and there is at least one veto player with positive recognition probability, they increase the structure of our model in two directions. First, they assume that the voting rule is oligarchical, i.e., agreement of all veto players is necessary and sufficient for a proposal to pass, so that the von Neumann-Morgenstern solution is unique. Their main result is that under these conditions, the equilibrium absorbing alternatives comprise the von Neumann-Morgenstern solution of the voting rule. Second, allowing a general voting rule, they add the assumption that there is a persistent agenda setter. They apply their analysis of ergodic properties of equilibria to show that all equilibria are essentially pure, and they again obtain the equivalence between equilibrium absorbing points and von Neumann-Morgenstern solutions. Thus, they extend the result of Diermeier and Fong (2012) by generalizing the quota rules to an arbitrary voting rule and by removing the restriction to pure strategy equilibria.

*Games and Economic Behavior*, "Dynamic Bargaining and Stability with Veto Players", by Vincent Anesi and John Duggan. doi: 10.1016/j.geb.2016.04.010

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