A Bargaining Model Where Parties Make Errors

Abstract

IN A NOW CLASSICAL PAPER, Nash (1953) studied the following bilateral bargaining game: The two parties state their demands simultaneously. If these are compatible with a feasible agreement, each party gets the utility corresponding to his demand. Otherwise both parties get their conflict payoffs. Under quite general conditions this game has a continuum of equilibria: Any possible agreement which is both Pareto optimal and individually rational corresponds to a particular equilibrium point. Recently, the literature on sealed-bid double auctions (see, e.g., Leininger et al. (1989) and Matthews and Postlewaite (1989)) has extended Nash's model to the case where each party has incomplete information about the other party's valuation. A common feature of these models is that they lead to an even larger set of equilibria and, thus, to an aggravation of the nonuniqueness problem. In the present paper, we will show that this problem all but disappears if a different kind of uncertainty is introduced into Nash's model, viz. if one assumes that the parties make errors in choosing their actions in the bargaining process. This modification will be seen to imply the existence of an equilibrium which Pareto-dominates all other equilibria. The rationale for our assumption lies in the implausibly precise coordination needed to induce equilibrium play in Nash's original model: each (nontrivial) equilibrium consists of a pair of demands that are just compatible. Even an arbitrarily small deviation (in the wrong direction) will reduce both players to their conflict payoffs. By adding error terms to tne bids, we get rid of this discontinuity and force the players to weigh their demands against the risk of breakdown. Thus, our assumption could be seen as a way of accounting for the strategic uncertainty which seems practically unavoidable in a game where strategies are continuously variable. More fundamentally, the errors may be thought to reflect the presence of some uncertainty about the exact values of relevant parameters. Formally, the errors will be modeled by letting a player's bid result from the addition of a stochastic term to his strategy. The rules of Nash's game will also be modified by allowing a surplus to be divided between the players. While in Nash's model the players get precisely what they have demanded even when demands are more than compatible, we make the more general assumption that some fraction (ranging between zero and one) of the unclaimed surplus is divided between the parties according to a surplus partition rule. A similar assumption is made in the above-mentioned incomplete information models, but the severe indeterminacy makes it impossible to assess its influence. The above described equilibrium properties of our model hold for errors of arbitrary magnitude. Naturally, it is particularly interesting to study the properties when errors go to zero. In the special case where no surplus is divided, we find a convergence to the

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