Resumen de Mediation: Incomplete information bargaining with filtered communication

Xavier Jarque Ribera , Clara Ponsatí Obiols , Jozsef Sakovics Domolky

• In this paper, we present an in-depth analysis of mediation in bargaining.

  Mediation being a wide-spread practice �from international negotiations to divorce proceedings� this is empirically relevant in its own right. Our concurrent intention is more theoretical though: to provide an alternative approach to the study of bargaining with incomplete information. The difficulties encountered by this literature are mostly due to the issues arising from unrestricted �or, arbitrarily restricted� conditional belief structures, which, unfortunately, arise most naturally in dynamic games of incomplete information. Instead of constraining the ways in which beliefs can be updated, we work with an extensive form that exogenously restricts the generation of the information events, which would trigger the formation of new posteriors. In particular, we assume that all offers that do not lead to agreement are unobservable. The easiest way to think of this situation is to say that it is a dynamic double auction: the players keep sending offers to an auctioneer, who only reveals them once they are compatible, and therefore an agreement has been reached. Within this context, we are able to give a complete characterization, as well as a number of interesting insights.

  We analyze this continuous-time bilateral double auction in the presence of two-sided incomplete information and a smallest money unit. A distinguishing feature of our model is that intermediate concessions are not observable by the adversary: they are only communicated to a passive auctioneer. An alternative interpretation is that of mediated bargaining.

  We show that an equilibrium (ex post inefficient) using only the extreme agreements always exists (War of Attrition equilibrium) and display the necessary and sufficient condition for the existence of (perfect Bayesian) equilibra which yield intermediate agreements. In any case, the efficient equilibrium is given by a particular solution of a system of differential equations belonging to the stable manifold of a generalized saddle point. Moreover when there is unique possible compromise we give the sufficient condition to ensure that there is no other ex post inefficient equilibriums.

  For the symmetric case with uniform type distribution we numerically calculate the equilibria. To do so, we numerically integrate the stable manifold of the relevant singular point of the differential equation which define the equilibrium.

  The aim of this study is to evaluate the relative efficiency of the equilibria differing in the number of agreements used and in the support of the type distribution.

  A priori, it is not clear what is the specific role of the number of possible agreements relative to the social welfare. On the one hand, as this number is larger there are more players getting to agreements � that is, ex post efficiency increases� but, on the other hand, these agreements show up later.

  We find that the equilibrium which does not use compromise agreements is the least efficient, however, the rest of the equilibria yield the lower social welfare the higher number of compromise agreements are used.

  We have shown that reducing the information flow between negotiators is a powerful tool not only conceptually but technically as well. We were able to provide a full characterization, and the nature of the multiplicity of our equilibria �it is directly related to the number of agreements used� makes it very easy to resolve: the mediator can simply announce the prices he is willing to accept (which should be the ones that maximize social welfare).

  Our results build on the combination of a finite set of equilibrium agreements with a continuous time structure that allows viewing the bargaining process as a game of timing. While the continuous time structure can be relaxed, the assumption that the set of agreements used in equilibrium is finite is technically important and cannot be dispensed with in our proofs. At the same time, the assumption is quite realistic and, in fact it is reinforced by our results. Recall, that we have shown that the more agreements are used in equilibrium the less efficient the equilibrium is (and the less likely it is to exist). Therefore, the negotiators are naturally driven towards employing �few� prices. While this reduces ex post efficiency, at no point in time could the mediator induce a Pareto improvement by allowing more prices.

  At the limit, when the set of potential agreements is an interval, things are different. While the equilibria that we have characterized are still supported as such, equilibria where a continuous set of agreements are attained with positive probability may exist as well. Addressing the problem from a point of view complementary to ours, Copic and Ponsati (2001) show that such equilibria do indeed exist. Instead of assuming that concessions must be discrete they impose that strategies are differentiable patterns of concessions. Their results stand in sharp contrast to ours too: a unique symmetric equilibrium prevails, it is always ex-post efficient and independent of the distribution of types