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CeDEx 2015-01: Existence and Indeterminacy of Markovian Equilibria in Dynamic Bargaining Games

Existence and Indeterminacy of Markovian Equilibria in Dynamic Bargaining Games

Summary

Most formal political analysts of legislative policymaking, until recently, have used models in which legislative interaction ends once a proposal is passed. As pointed out by Baron (1996) and later by Kalandrakis (2004), however, most legislatures have the authority to change existing laws by enacting new legislation; so that laws continue in effect only in the absence of new legislation. To explore this dynamic feature of legislative policymaking, these authors have introduced an alternative model that casts the classical spatial collective-choice problem into a dynamic bargaining framework. The problem immediately encountered in this framework is that existence results for stationary Markov perfect equilibria provided in the extant game-theoretic literature do not apply. The result has been a fast growing body of literature that explicitly constructs stationary Markovian equilibria for bargaining games with an endogenous status quo, and then analyzes the properties of policy outcomes implied by this selection of equilibria. These constructions are an important development in the study of legislative dynamics; but almost all of the analysis either assumes that the space of alternatives is unidimensional, or it focuses on pie-division settings where each bargainer's utility only depends on her own share of the pie. There are no known conditions that guarantee the existence of a stationary Markovian equilibrium for more general multidimensional choice spaces.

In this Nottingham School of Economics working paper, Anesi and Duggan allow the feasible set of alternatives to be any nonempty subset of multidimensional Euclidean space. They show that when players are sufficiently patient, stationary Markov perfect equilibria in pure strategies can be constructed “close to” any alternative at which the gradients of the players’ utilities are linearly independent; i.e., every open neighborhood of an alternative satisfying this condition contains the absorbing points of a stationary Markov perfect equilibrium. In fact, the authors show that there is a continuum of distinct stationary Markov perfect equilibria with absorbing points close to that alternative. Given a set of alternatives of sufficiently high dimension, the linear independence condition holds generically outside a set of alternatives with measure zero, with the implication that equilibria typically abound in such models. Thus constructive techniques, which involve an explicit specification of a particular equilibrium and are common in the literature, implicitly rely on a restrictive selection of equilibria.

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CeDEx Discussion Paper 2015-01, Existence and Indeterminacy of Markovian Equilibria in Dynamic Bargaining Games by Vincent Anesi and John Duggan, March 2015

Authors

Vincent Anesi and John Duggan

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Posted on Friday 6th March 2015

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