On Hierarchical Spatial Competition

Abstract

In this paper we consider a model of spatial electoral competition with two dominant players (incumbents) and one entrant. The incumbents engage in a non-cooperative game against each other and act as Stackelberg leaders with respect to a vote-maximizing entrant. We prove that the equilibrium of this game, called a equilibrium, exists and is unique for an arbitrary single-peaked distribution of voters' ideal points. Moreover, we fully characterize the set of equilibrium strategies and show its equivalence to the set of strategies generated by a perfect-foresight Most of the recently developed models of entry in oligopolistic markets and the theory of political competition deal with the case of dominant firms or parties which face a threat of potential entry. These models possess a number of features on which they may be compared and contrasted. The most important of these features are the ways in which the existing parties or firms compete against each other for votes or profits, and types of expectations the incumbents have about the responses of potential entrants to changes in their decisions. In most of the cases an electoral or oligopolistic competition is represented by some type of a non-cooperative game played by parties or firms. In this paper we also consider a non-cooperative model of electoral race where the established parties (incumbents) compete against each other by offering positions (ideologies) to the population of voters. Given the profile of voters' preferences over the space of ideologies (or, issue space), each of the established parties attempts to maximize its vote share. In making their choices, the incumbents, however, anticipate that a new party (an entrant) will join the electoral race. Consistent with the other parties' objectives, the entrant is assumed to maximize its share of voters while taking the incumbents' choices as given. The resulting non-cooperative game possesses, therefore, a structural hierarchy: the established parties behave as Nash players with respect to each other, whereas acting as Stackelberg leaders with respect to the entrant and correctly anticipating its vote-maximizing response to their choices in the issue space. The main purpose of this paper is to study the properties of the game we described and, in particular, to characterize its equilibrium, which we will call hierarchical equilibrium. Naturally, some assumptions are needed in order to guarantee the existence of the We assume that there are two established parties, the issue space I is the uni-dimensional interval, all voters have symmetric single peaked-peaked preferences over I and the distribution of the voters' ideal points is represented by a quasi-concave continuous density function. Under these rather mild assumptions, we are able to prove the existence and uniqueness of the equilibrium and to completely