One-dimensional bargaining with Markov recognition probabilities

We study a process of bargaining over alternatives represented by points in the unit interval. The identity of the proposer is determined by a general Markov process and the acceptance of a proposal requires the approval of it by all the players. We show that for every value of the continuation probability below one the subgame perfect equilibrium in stationary strategies is essentially unique and equal to what we call the bargaining equilibrium. We provide a general characterization of the bargaining equilibrium. We consider next the asymptotic behavior of the equilibrium proposals when the continuation probability approaches one. We give a complete characterization of the limit of the equilibrium proposals. We show that the limit equilibrium proposals of all the players are the same if the proposer selection process satisfies an irreducibility condition, or more generally, has a unique absorbing set. In general, the limit equilibrium proposals depend on the partition of the set of players in absorbing sets and transient states of the proposer selection process. We fully characterize the limit equilibrium proposals as the unique generalized fixed point of a particular function. This function depends in a simple way on the stationary distribution related to the proposer selection process. We compare the proposal selected according to our bargaining model to the ones corresponding to the median voter theorem and the Nash bargaining solution. We provide conditions under which the limit equilibrium proposals are closer to the middle than the median voter outcome. We argue that there is no general relationship between the limit equilibrium proposals and the Nash bargaining solution. JEL classification code: C78.