Lead, follow or cooperate? Endogenous timing and cooperation in duopoly games

The aim of this paper is to extend Hamilton and Slutsky’ (1990) endogenous timing game to include the possibility for players to cooperate. At an initial stage players are assumed to announce both their purpose to play early or late a given duopoly game as well as their intention to cooperate or not with their rival. The cooperation and timing formation rule is rather simple: when both players agree to cooperate and play with a given timing, they end up playing their strategies coordinately and simultaneously. Otherwise, they play independently with the timing as prescribed by their own announcement. We check for the existence of a subgame perfect Nash equilibrium (in pure strategies) of such a cooperation-timing duopoly game. Two main results on the emergence of cooperation are provided. If players’strategies in the symmetric duopoly game are strategic substitutes and there is no discount, cooperating early is a subgame perfect equilibrium of the extended timing-cooperation game. Conversely, cooperating late (at period two) represents an equilibrium when players’strategies are strategic complements. Other equilibria are also possible. Most importantly, our model shows that, in general, the success of cooperation is a¤ected by the endogenous timing of the game. Moreover, the slope of players’best-replies appears crucial for the success of cooperation as well as for the choice of players to sequence their strategies.