Hindawi Publishing Corporation ISRN Economics Volume 2013, Article ID 645481, 10 pages http://dx.doi.org/10.1155/2013/645481 Research Article Lead, Follow or Cooperate? Sequential versus Collusive Payoffs in Symmetric Duopoly Games Marco A. Marini and Giorgio Rodano Department of Computer, Control and Management Engineering, Sapienza University of Rome, Via Ariosto 25, 00185 Rome, Italy Correspondence should be addressed to Marco A. Marini; marini@dis.uniroma1.it Received 18 August 2013; Accepted 15 September 2013 Academic Editors: M. E. Kandil and M. Tsionas Copyright © 2013 M. A. Marini and G. Rodano. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In many strategic settings comparing the payoffs obtained by players under full cooperation to those obtainable at a sequential (Stackelberg) equilibrium can be crucial to determine the outcome of the game. is happens, for instance, in repeated games in which players can break cooperation by acting sequentially, as well as in merger games in which firms are allowed to sequence their actions. Despite the relevance of these and other applications, no full-fledged comparisons between collusive and sequential payoffs have been performed so far. In this paper we show that even in symmetric duopoly games the ranking of cooperative and sequential payoffs can be extremely variable, particularly when the usual linear demand assumption is relaxed. Not surprisingly, the degree of strategic complementarity and substitutability of players’ actions (and, hence, the slope of their best replies) appears decisive to determine the ranking of collusive and sequential payoffs. Some applications to endogenous timing are discussed. 1. Introduction Standard game-theoretic settings dealing with the emergence of cooperation usually weight the stream of players’ payoffs colluding a finite or infinite number of periods to those obtained by defecting one period and then playing simultane- ously ` a la Nash (noncooperatively) aſterward. e possibility that players defect from the collusive outcome as leaders or followers in every stage game is usually not considered. An exception to this approach is contained, for instance, in Mouraviev and Rey [1], who study the role of price (or quan- tity) leadership in facilitating firm collusion in an infinitely repeated setting. ey show that, under price competition and, to a much lesser extent, under quantity competition, the possibility that players sequence their actions in every stage may help to sustain collusion. is happens because the presence of a deviating leader makes it easy for the follower to punish such defection. In general, the focus on the link between timing and collusion is not entirely new within the economic literature. For instance, in some classical contributions on cartels and mergers under oligopoly, colluding firms are assumed to act as Stackelberg leaders [2–6]. Moreover, a few recent papers on mergers and R&D agreements consider different timing structures, where groups of firms can either act as leaders or followers (e.g., [7–10]). In these and other potentially interesting economic applications, it is crucial to compare the payoff under collusion to those under noncooperative sequential play to determine the outcome of the game. While the literature comparing leader and follower’s (as well as simultaneous Nash) payoffs has a long-standing tradition (see [11, 12] or [13] for references), the number of papers that compares collusive and sequential outcomes even in a simple duopoly framework appears, at the best, scant. If, on the one hand, it has been proved that in regular symmetric duopoly games with single-valued best-replies and monotone payoffs on rivals’ actions (denoted monotone spillovers), when actions are strategic complements, the follower’s payoff dominates which of the leader and the opposite holds under strategic substitutes (e.g., [13–15]. In particular, by relaxing the assumption of complements or substitutes actions, [13] proves that either the leader’s payoff dominates that of the follower (which is also dominated by the simultaneous Nash) or, in turn, is dominated by the follower’s. Dowrick [16], Amir [17], Amir and Grilo [18], Amir et al. [15], Currarini and Marini [19, 20], and Von Stengel and Zamir [21] all present