Abstract
We study bargaining models in discrete time with a finite number of players, stochastic selection of the proposing player, endogenously determined sets and orders of responders, and a finite set of feasible alternatives. The standard optimality conditions and system of recursive equations may not be sufficient for the existence of a subgame perfect equilibrium in stationary strategies (SSPE) in case of costless delay. We present a characterization of SSPE that is valid for both costly and costless delay. We address the relationship between an SSPE under costless delay and the limit of SSPEs under vanishing costly delay. An SSPE always exists when delay is costly, but not necessarily so under costless delay, even when mixed strategies are allowed for. This is surprising as a quasi SSPE, a solution to the optimality conditions and the system of recursive equations, always exists. The problem is caused by the potential singularity of the system of recursive equations, which is intimately related to the possibility of perpetual disagreement in the bargaining process.
Highlights
Strategic bargaining theory, boosted by the contribution of Rubinstein (1982), has contributed significantly to our understanding of negotiation processes
We present a characterization of subgame perfect equilibria in stationary strategies (SSPE) for bargaining models that is valid for both the cases of costly and costless delay
The standard characterization of an SSPE under costly delay consists of optimality conditions corresponding to the one-stage deviation principle together with a system of recursive equations to pin down equilibrium utilities
Summary
Strategic bargaining theory, boosted by the contribution of Rubinstein (1982), has contributed significantly to our understanding of negotiation processes. We present a characterization of subgame perfect equilibria in stationary strategies (SSPE) for bargaining models that is valid for both the cases of costly and costless delay. The standard characterization of an SSPE under costly delay consists of optimality conditions corresponding to the one-stage deviation principle together with a system of recursive equations to pin down equilibrium utilities This characterization provides necessary but not sufficient conditions in case delay is costless. A necessary and sufficient condition such that the solution to the system of recursive equations corresponds to the utilities generated by a strategy profile is that the solution is equal to zero at all states which lead to perpetual disagreement. A stationary best response is characterized by the optimality conditions and the system of recursive equations for the responding player together with the requirement that utilities are equal to zero in all states that lead to perpetual disagreement.
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