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Abstract
This paper investigates the theory of endogenous timing by taking into account a vertically-related market where an integrated firm competes with a downstream firm. Contrary to the standard results in the literature, we find that both firms play a sequential game in quantity competition and play a simultaneous game in price competition. Under mixed quantity-price competition, the firm choosing a price strategy moves first and the other firm choosing a quantity strategy moves later in equilibrium. Given that the timing of choosing actions is determined endogenously, aggregate profit (consumer surplus) is higher (lower) under price competition than under quantity competition. Lastly, social welfare is higher under quantity competition than under price competition when the degree of product substitutability is relatively low.

ABSTRACTWe consider a game of endogenous timing with observable delay in a mixed duopoly with endogenous vertical differentiation in the context of sequential quality and price choice. We find that a simultaneous play in the first opportunity at each stage turns out to be the unique subgame perfect Nash equilibrium, which contrasts with the endogenous timing in a purely private duopoly.

AbstractThe aim of this paper is to generalize the endogenous timing game proposed by Hamilton and Slutsky (Games and Economic Behavior, 1990, 2, pp. 29–46) by allowing the payoff or the marginal payoff of a player to become non‐monotonic with respect to the strategy of the opponent. We propose a taxonomy of the subgame‐perfect Nash equilibria based on the characteristics of the payoff functions proposed by Eaton (Canadian Journal of Economics, 2004, 37, pp. 805–29). We determine under which conditions of the initial payoff functions commitment has a social value and when the simultaneous‐move Nash equilibrium is commitment robust and discuss its Pareto efficiency.

We modify the price-setting version of the vertically differentiated duopoly model by Aoki (2003) by introducing an extended game in which firms noncooperatively choose the timing of moves at the quality stage. Our results show that there are multiple equilibria in pure strategies, in which firms always select sequential play at the quality stage. We also investigate the mixed-strategy equilibrium, revealing that the probability of generating outcomes out of equilibrium is higher than its complement to one. In the alternative of full market coverage, we show that the quality stage is solved in dominant strategies and therefore the choice of roles becomes irrelevant as the Nash and Stackelberg solutions coincide.

ABSTRACTWe modify the price‐setting version of the vertically differentiated duopoly model by Aoki (2003) by introducing an extended game in which firms noncooperatively choose the timing of moves at the quality stage. Our results show that there are multiple equilibria in pure strategies, whereby firms always select sequential play at the quality stage. We also investigate the mixed‐strategy equilibrium, revealing that the probability of generating outcomes out of equilibrium is higher than the probability of playing one Nash equilibria in pure strategies. In the alternative case with full market coverage, we show that the quality stage is solved in dominant strategies and therefore the choice of roles becomes irrelevant as the Nash and Stackelberg solutions coincide. With full market coverage and corner solution, the results show that the game has a unique subgame perfect equilibrium in pure strategies, where the high‐quality firm takes the lead in the quality stage.