Equilibrium delay and non-existence of equilibrium in unanimity bargaining games

We consider a class of perfect information unanimity bargaining games, where the players have to choose a payoff vector from a fixed set of feasible payoffs. The proposer and the order of the responding players is determined by a state that evolves stochastically over time. The probability distribution of the state in the next period is determined jointly by the current state and the identity of the player who rejects the current proposal. This protocol encompasses a vast number of special cases studied in the literature. These special cases have in common that equilibria in pure stationary strategies exist, are efficient, are characterized by the absence of delay, and converge to a unique limit corresponding to an asymmetric Nash bargaining solution. For our more general protocol, we show that subgame perfect equilibria in pure stationary strategies need not exist. When such equilibria do exist, they may exhibit delay. Limit equilibria as the players become infinitely patient need not be unique.