Endogenous timing in duopolies

Abstract

In this thesis we aim to understand how does leadership emerge in duopolistic competitions. In particular, we want to know which are the key features, of the firms and the market, that explain that some interactions are simultaneous while others are sequential. In order to do so, we develop a general model of duopolistic competition in which the timing of movements is not exogenously given, but is part of the equilibrium, this is, depends on the actions of the players. More precisely, we consider two firms that are absolutely identical except for one single characteristic that makes them different. To fix ideas, it is possible to think this feature as the marginal cost or capacity of production. We interpret this difference as the consequence of different levels of investment made prior to the competition. This investment variable might be \textit{tough} or \textit{soft}, which means that the total effect of the investment on the payoff of the other player is negative or positive, respectively. After the investment, firms engage in supermodular or submodular competition. This competition can be (a priori) simultaneous or sequential, allowing us to endogenously obtain the timing of movements in equilibrium. In order to do so, we use the extension models from \cite{Hamilton1990}, namely, the Game with Observable Delay (GOD) and the Game with Action Commitment (GAC). When there is multiple equilibria, we base our refinement on the risk dominance concept from \cite{HarsanyiJ;Selten1998}. For the supermodular case, we found that simultaneous competition is never the outcome of the interaction, neither with GOD nor GAC. In the GOD extension model this result comes from the fact that the existence theorem, in our setting, predicts that only sequential play is an equilibrium. In the GAC model the result comes from the refinement process based on risk considerations and the nature of the investment. Also, our results predict that, when the investment variable is tough, the firm with the largest investment is more likely to become the risk dominant leader, for both extension models. When the investment variable is soft, we provide sufficient (but not necessary) conditions for the leadership of the firm with the largest investment. Regarding this last point, we still need to work further on finding necessary hypotheses to characterize the leadership. For the submodular case, we fully characterize which equilibrium will emerge when the extension model is GOD: simultaneous competition. This result holds regardless of the type nor level of investment. On the other hand, for the GAC extension model, we find that the simultaneous equilibrium is never the risk dominant (and therefore it should never emerge). Also, when the investment variable is tough, the firm with the largest investment is more likely to become the leader. In the case of soft investment, as with supermodular competition, we give sufficient conditions for the leadership of the player with the largest investment. Considering the results obtained in this setting, we also provide an interpretation of the differences between both extension models, GOD and GAC, based on risk considerations.