# SFB 303 Discussion Paper No. A - 095

**Author**: Leininger, Wolfgang, Peter B. Linhart, and Roy Radner

**Title**: The Sealed-Bid Mechanism for Bargaining with Incomplete Information

**Abstract**: A bargaining procedure of particular interest is the sealed-bid mechanism, which is a special case of the
determination of price by equating supply and demand. In the standard case the market-clearing price P and
quantity Q are determined by the intersection of the two curves. In the present case of bargaining over a single
object, the quantity can be only 0 or 1. Note that the sealed-bid mechanism is a special case of a double auction,
in which there is only one bidder on each side. The current theoretical approach to the study of bargaining
mechanisms in the present context is to model the situation as a noncooperative game with incomplete
information. From the point of view of the firm as a whole, an equilibrium of the sealed-bid game (with
incomplete information) is efficient if trade occurs whenever the benefit exceeds the cost and does not occur
when the cost exceeds the benefit. One can show that, if both of these inequalities have positive probability, then
no equilibrium of the sealed-bid game is efficient. We shall concentrate most of our analysis on the special case
in which the benefit and cost are independently and identically distributed, uniformly on the unit interval. The
linear-strategy equilibrium of the sealed-bid mechanism is "second-best" (for the uniform prior to
distribution).We shall show that, for the uniform case, the sealed-bid game has a very large set of equilibria. We
have confined our search to pure-strategy equilibria. All of the step-function equilibria are "trembling-hand
perfect". We also consider the case of general independent distributions of benefits and costs. We do not give as
thorough an analysis of the set of equilibria for the case of general independent priors as we do for the uniform
case, but we do demonstrate the existence of both differentiable and step-function equilibria. Multiplicity of
equilibria is, of course, a frequent phenomenon in game theory. In the present game, however, the set of
equilibria is so large that its predictions about equilibrium behavior are extremely weak. Even worse, the
equilibria in the uniform distribution case range from second-best to worthless, so that equilibrium theory
provides no basis for recommending the sealed-bid mechanism in practice. In our opinion, this is a strong
indictment of the theory of equilibrium with incomplete information as it is currently formulated.

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**Creation-Date**: 1987

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