This paper provides general conditions on the direct demand functions in a Bertrand duopoly with differentiated substitute products and constant marginal costs, that allow an unambiguous ranking of firms' equilibrium payoffs between sequential play (with both order of moves) on the one hand, and simultaneous play on the other. The main results are that (i) when prices are strategic complements, both firms prefer sequential moves (with either order) to simultaneous moves, (ii) when prices are strategic substitutes, both firms prefer simultaneous moves to moving second in sequential play, and (iii) in the mixed strategic substitute/complement case, one firm is as in (i) and the other as in (ii). Thus, sequential moves would plausibly endogenously emerge in cases (i) and (iii), with one specified leader in the latter case. The analysis relies crucially on the theory of supermodular games, and is conducted at a high level of generality, dispensing with concavity-type assumptions, and taking into account both the issues of existence and possible non-uniqueness of the different equilibria involved.
Strategic substitutes or complements? The game of where to fish