Equilibrium selection in a bargaining problem with transaction costs

It is the purpose of the paper to analyse a bargeining situation with the help of the equilibrium selection theory of John C. Harsanyi and Reinhard Selten. This theory selects one equilibrium point in every finite non-cooperative game. The bargaining problem is the following one: the two bargainers — player 1 and player 2 — simultaneously and independently propose a payoffx of player 1 in the interval 〈0, 1〉. If agreement is reached player 2's payoffs is 1?x. Otherwise both receive zero. Each playeri has a further alternativeW _i , namely not to bargain at all (i=1, 2). Thereby he avoids transaction costsc andd of bargaining which arise whether an agreement is reached or not. One may think of an illegal deal where bargaining involves a risk of being punished — independently whether the deal is made or not. The model has the form of a (K+1)×(K+1)-bimatrix game. It is assumed that there is an indivisable smallest money unit. The game hasK+1 pure strategy equilibrium points.K of them correspond to an agreement and the last one is the strategy pair where both players refuse to bargain. Each of theK+1 equilibrium points can be the solution of the game. The aim of the Harsanyi-Selten-theory is to select in a unique way one of these equilibrium points by an iterative process of elimination (by payoff dominance and risk dominance relationships) and substitution. For each parameter combination (c, d) a sequence of candidate sets arises which becomes smaller and smaller until finally a candidate set with exactly one equilibrium point — the solution of the game — is found. For the sake of shortness the paper will report results without detailed proofs, which can be found elsewhere [Leopold-Wildburger].