Optimal recommendation in two-player bargaining games

Abstract

This paper extends the traditional two-player noncooperative bargaining game by adding a recommendation stage before the regular bargaining stage. At the recommendation stage, a coordinator recommends a feasible payoff pair to the players. If both players accept the recommendation, the game ends and the recommendation is enforced. Otherwise, the game enters the bargaining stage, where each player is picked as the proposer of any bargaining round according to an exogenous probability characterizing their relative bargaining power, and both players receive their respective reservation payoffs with a certain risk of breakdown if they fail to reach an agreement in this round. We characterize the unique subgame perfect equilibrium outcome of this extended game. The equilibrium recommendation, which will be accepted by both players, is optimal for the coordinator among all acceptable recommendations. A player’s payoff in optimal recommendation increases with her bargaining power. As the risk of breakdown vanishes, the optimal recommendation converges to an asymmetric Nash bargaining solution parameterized by the probability at the bargaining stage.