Competitive selling mechanisms: the delegation principle and farsighted stability

We analyze the problem of competitive mechanism design within the context of a model of product differentiated oligopoly. In our model, firms compete via their catalogs, that is, via the sets of products (broadly defined) and prices firms offer to the market (i.e., catalogs are the primitives, while selling mechanisms are derived). In an oligopoly setting, participation by an agent in any one firm’s catalog is endogenously determined. This fact leads naturally to a modification of the classical notion of incentive compatibility for mechanisms. We extend the classical notion of incentive compatibility to take into account endogenous participation, introducing the notion of participation incentive compatibility (PIC). Our main contribution is a characterization of all PIC selling mechanisms in terms of catalogs. In particular, we show that a selling mechanism is PIC if and only if there exists a unique, minimal catalog profile which implements the mechanisms. We call this characterization the delegation principle (Theorem 4). Using the delegation principle, we conclude that in order to solve the problem of competitive mechanism design, an essentially cooperative problem, it is sufficient to consider only the underlying noncooperative problem of catalog choice by firms. Moreover, using the delegation principle we show that corresponding to each PIC mechanism there is a unique profile of nonlinear pricing schedules which implements the mechanism - thus, extending the taxation principle to problems of competitive nonlinear pricing (Theorem 5). A second contribution is our application of the notion of farsighted stability to the problem of competitive mechanism design. We show that for any approximating finite subgame (of catalog choice), the farsightedly stable set of catalog profiles (and hence the farsightedly stable set of nonlinear pricing schedules) is nonempty (Theorem 10).