Stackelberg-Nash-Cournot Equilibria: Characterizations and Computations

Abstract

The supply side of an oligopolistic market supplying a homogeneous product noncooperatively is modeled. In this market, there is one leader and N followers. The followers operate under the Cournot assumption of zero conjectural variation and are accordingly called Cournot firms. The leader, called a Stackelberg firm, specifically takes into account the reaction of the Cournot firms to its output. For this situation, we study the behavior and implications of the joint Cournot reaction curve as generated by plausible economic market assumptions. In particular, we study the existence and uniqueness of a Stackelberg-Nash-Cournot equilibrium. In addition, we prescribe an efficient algorithm to determine a set of equilibrating output quantities for the firms.

Quantity-setting games with a dominant firm

“…Thus, by (24) lim n→∞ n i=1 r n r n − (mc n i ) (q n i ) ≥ − r MC c (q c ) , and we have established (10).…”

“…These results indicate also that checking Matsumura's (1999) Assumptions 2 and 3 for the oligopolistic case is a far more difficult task than for the simple duopolistic case for which the results are well-known. While the case of just one leader has been investigated in the literature, for example, by Sherali, Soyster and Murphy (1983), the case of just one follower has not got, as far as we know, real attention in the literature -with the exception of the above-mentioned counterexample by Ehrenmann (2004).…”

Multi-Leader Multi-Follower Model with Aggregative Uncertainty

“…The case M = 1 and N ≥ 1 has been introduced in the oligopolistic market context in [1,26] and studied from a computational point of view in [27], where it has been called MPEC, and it has been applied in other several contexts, for example in transportation in [2].…”

“…In literature, this model appeared in the context of oligopolistic markets (see [1]) with one leader firm and several follower firms acting as Cournot competitors. Other applications can be found, for example, in transportation (see [2]), in the analysis of deregulated electricity markets (see [3]), in water management systems (see [4]), and in wireless networks (see [5] ).…”

“…Then, the players optimize the expected payoffs according the considered solution concept. De Miguel and Xu (see [7] ) extend the multiple-leader Stackelberg-Nash-Cournot model studied in [1] to the stochastic case assuming uncertainty in the demand function: leaders choose their supply levels first, knowing the demand function only in distribution and followers make their decisions after observing the leader supply levels and the realized demand function.…”