Catalog competition and stable nonlinear prices

We model strategic competition between product differentiated oligopolists in a market with privately informed buyers as an abstract game over market situations. In this game each firm's strategy space consists of a set of catalogs-and each catalog in turn consists of a set of products and prices the firm might offer to the market. Assuming that firms behave farsightedly in choosing their catalog strategies, we specify the market situation game by two objects: (i) a set of market situations, that is, a set of feasible profit-catalog profiles for firms, and (ii) a dominance relation defined on the set of market situations which reflects farsighted behavior. We show that the set of market situations is compact and we introduce two dominance relations on the set of market situations: farsighted dominance and path dominance. We then identify conditions sufficient to guarantee the existence of a nonempty set of market situations stable with respect to farsighted dominance (i.e., a nonempty largest farsightedly consistent set), as well as conditions sufficient to guarantee the existence of a nonempty set of market situations stable with respect to path dominance. Finally, we show that for any finite market situation game there exists a stable set with respect to path dominance contained in largest farsightedly consistent set. We close with an example illustrating this relationship between path dominance stability and farsighted consistency for finite market situation games.