Abstract
We study a Stackelberg game with multiple leaders and a continuum of followers that are coupled via congestion effects. The followers’ problem constitutes a nonatomic congestion game, where a population of infinitesimal players is given and each player chooses a resource. Each resource has a linear cost function which depends on the congestion of this resource. The leaders of the Stackelberg game each control a resource and determine a price per unit as well as a service capacity for the resource influencing the slope of the linear congestion cost function. As our main result, we establish existence of pure-strategy Nash–Stackelberg equilibria for this multi-leader Stackelberg game. The existence result requires a completely new proof approach compared to previous approaches, since the leaders’ objective functions are discontinuous in our game. As a consequence, best responses of leaders do not always exist, and thus standard fixed-point arguments á la Kakutani (Duke Math J 8(3):457–458, 1941) are not directly applicable. We show that the game is C-secure (a concept introduced by Reny (Econometrica 67(5):1029–1056, 1999) and refined by McLennan et al. (Econometrica 79(5):1643–1664, 2011), which leads to the existence of an equilibrium. We furthermore show that the equilibrium is essentially unique, and analyze its efficiency compared to a social optimum. We prove that the worst-case quality is unbounded. For identical leaders, we derive a closed-form expression for the efficiency of the equilibrium.
Highlights
We consider a Stackelberg game with multiple leaders N = {1, . . . , n}, n ≥ 2, and a continuum of followers represented by the interval [0, 1]1
With our characterization of best response correspondences at hand, we show that the considered Stackelberg pricing game fulfills the conditions of [29] and admits pure Nash(–Stackelberg) equilibrium (PNE)
Johari et al [21] study existence, uniqueness and worst-case quality of PNE assuming that the demand of the followers is elastic: The volume of followers participating in the congestion game decreases with increasing combined cost of congestion and price
Summary
The leaders of the game each control one resource, and decide about a price which is charged to the followers for the usage of the resource, as well as a capacity which influences the slope of the congestion cost of her resource. Zn) with i∈N zi > 0, i.e., there is at least one resource in the followers’ congestion game, and a price vector p = Note that for given capacities z = 0 and prices p, there is exactly one x ∈ P satisfying the Wardrop equilibrium conditions (see, e.g., [10]). Pi xi (z, p) − γi zi , 0, for else, i∈N zi > 0, where γi > 0 is a given installation cost parameter for leader i This completes the description of our game, which we denote a Stackelberg pricing game. The subsection summarizes our results in terms of existence, uniqueness and quality of PNE for Stackelberg pricing games
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