Short- and long-term optimality under sustainable threats in Contest Theory models of advertising and short-run competition

We model advertising with effects on different time scales for a
duopoly in imperfect substitutes using elements from Contest Theory. Firms addionally compete in a short run strategic variable, here price or quantity, allowing simultaneous or sequential decisions, or collusion in endogenously changing stage games. Strategic variables range from ‘slow’ (advertising), over moderate’ (quantities) to ‘fast’ (prices). We find feasible rewards and equilibria for the limiting average reward criterion. Uniqueness of equilibrium is not guaranteed, and we introduce two criteria which act as natural refinements. We impose stage-game rationality, i.e., the firms play optimally in each stage game.
Furthermore, in establishing threats, we require that punishment is sustainable, i.e., the punisher must have nonnegative long term average own profits to avoid bankruptcy.