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. This 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.

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 simultaneously

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 [

While the literature comparing leader and follower’s (as well as simultaneous Nash) payoffs has a long-standing tradition (see [

Our paper considers a class of symmetric duopoly games and shows that the ranking of cooperative and sequential payoffs can be extremely variable. In particular, when actions are

At the end of the paper we discuss some possible implications of our results. To this aim, we introduce an elementary endogenous timing game in which players can decide the timing of their cooperative or noncooperative strategies. Thus, we show that when binding agreements among players are allowed, intertemporal cooperation is in general more vulnerable to defection than cooperation occurring at just one stage.

The paper is organized as follows. The next section introduces the basic setup of the paper. Section

We assume two players

We can now define the behaviour of players in the different scenarios. Under perfect collusion, players are assumed to set cooperatively their strategy profile

It is well known that, in symmetric duopoly games with monotone spillovers and single-valued best replies, if actions are strategic substitutes (and best-replies decreasing), players’ payoffs under equilibria ((

In what follows, some of the results presented do not strictly require the monotonicity of players’ best-replies (implied, in turn, by the property of increasing or decreasing differences of players’ payoffs). However, for simplicity, our main results are characterized for two well-known classes of duopoly games in which actions are, in turn, strategic complements and substitutes. The first result is rather trivial, and it is simply based on (

In all symmetric duopoly games in which players’ actions are strategic complements (substitutes), the symmetric collusive payoff of every player must be higher than leader’s (follower’s) equilibrium payoff at the sequential game, namely,

Suppose by contradiction that, if duopoly game actions are strategic complements,

In all symmetric duopoly games in which players’ actions are strategic complements (substitutes), the following payoff ranking arises:

This is obtained by combining symmetry, the results of Proposition

However, to obtain a complete ranking of players’ payoffs, it remains to be ascertained if, in turn,

We present here some results on the relationship between players’ equilibrium strategies and their payoffs when actions are strategic complements. The next result provides a sufficient condition for the follower’s payoff to overcome the symmetric collusive payoff.

In all symmetric duopoly games with strategic complements and negative (NS) (positive (PS)) spillovers, if at the Stackelberg equilibrium the leader plays a strategy that is lower (higher) than that played under collusion, namely,

If, at the Stackelberg equilibrium, player

Duopoly game with strategic complements and negative externalities—red: coop. isoprofits; blue: leader’s isoprofit; green: follower’s isoprofit.

Duopoly Game with strategic complements and positive externalities—red: coop. isoprofits; blue: leader’s isoprofit; green: follower’s isoprofit.

Let two firms face an inverse demand function

The next proposition illustrates the relation between leader’s and follower’s strategy at any Stackelberg equilibrium in which the follower obtains a higher payoff than under collusion. Together with Proposition

Numerical example (Cournot with strategic complements)—case

Numerical example (Cournot with strategic complements)—case

In all symmetric duopoly games with strategic complements, if the follower’s payoff at the Stackelberg equilibrium is higher than the payoff obtained at the symmetric cooperative equilibrium, namely,

Assume that

The next proposition characterizes instead the order of strategies in the standard case in which the collusive agreement yields a higher payoff for a player than playing as a follower the sequential game. Again, to make things simple, we assume that Stackelberg and simultaneous Nash equilibrium differ, that is,

In all symmetric duopoly games with strategic complements, if the symmetric collusive payoff is at least as high as the follower’s payoff at the Stackelberg equilibrium; namely,

The fact that

So far, we have shown that, in duopoly games with strategic complements, two main strategy-payoff equilibrium combinations are possible in the nontrivial case in which

In all symmetric duopoly games with strategic complements (i) under negative spillovers (NS), if

It follows straightforwardly by Corollary

However, as the next example illustrates, in duopoly games with strategic complements (as, for instance, the classical price duopoly game), the case in which the follower’s payoff overcomes the collusive payoff remains a rather unusual event. When the follower’s best reply is a contraction, it is unlikely that the leader’s equilibrium strategy is so low (high) under NS (PS) for the follower to overcome the symmetric collusive payoff. To obtain this, the (increasing) best replies have to be very flat under NS (see Example

Let firm

Numerical example (price competition with strategic complements)—case

When players’ actions are strategic substitutes, it can be shown that there are only two possible rankings of equilibrium strategies and payoffs under either negative or positive externalities. These are characterized in the next proposition.

(i) In all symmetric duopoly games with strategic substitutes, the following rankings between equilibrium strategies can arise: either

(i) Similarly to Proposition

Let assume an inverse demand function given by

Numerical example (Cournot with strategic substitutes)—case

Numerical example (Cournot with strategic substitutes)—case

Numerical example—red line:

In order to check some of the potential consequences of our results, we introduce here a simple setting that extends Hamilton and Slutsky’s [

Formally, the game can be described as follows. We let, at a pre-play stage denoted

Differently from Hamilton and Slutsky [

The above rule prescribes that, if both players agree to cooperate, they can behave collusively in just one period or alternate their collusive strategy over time. However, if just one player disagrees on cooperation, both players end up playing independently in their own preferred time.

The notion used to define the stability of a given time configuration

A configuration

When a timing-configuration

The endogenous timing-cooperation game played by two symmetric players can give rise to the following cases:

(i) When players’ actions are strategic complements and

Our simple extension of Hamilton and Slutsky [

This paper has presented a first attempt to connect two usually distinct issues concerning players’ strategic interaction: one dealing with their timing of play and the other with their capacity to collude. We have shown that the nature of interaction between players in the strategic setting (duopoly game) plays a crucial role for the decision of players to sign or not a binding agreement and to sequence or not their strategies.

Let players’ payoffs be continuous and strictly quasiconcave and their strategy set be compact and convex. Then, the strategy profile

Compactness of each

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors wish to thank Christian Ewerhart, Attila Tasnadi, Jacquelin Morgan, Maria Luisa Petit, George Zaccour, and the participants at the Oligo workshop in Budapest, the MDEF workshop in Urbino, the SING7 conference in Paris, and seminar audience at CREI Roma III and Sapienza University for their useful comments and discussions.