Popularity in location games

TL;DR

This paper investigates a two-firm location game with a bandwagon-style popularity externality, showing that the added popularity term can create a multiplicity of market equilibria and drastically alter strategic behavior. By separating market equilibrium resolution from firms’ location decisions, it analyzes optimistic, neutral, and pessimistic attitudes, deriving NE existence and structure in each case and quantifying welfare via price of anarchy and price of stability. Key findings include no NE under optimism, a unique symmetric NE for neutrals when $a\le\tfrac{1}{2}$, and potentially differentiated symmetric or asymmetric equilibria for pessimists (with distance $|x_2-x_1|\le a$ and share asymmetry bounded by $a/(1-a)$). Welfare impli­cations are governed by a threshold at $a=\tfrac{1}{4}$: below it, a symmetric social optimum prevails, while above it, a centralized outcome dominates and popularity crowds out differentiation, with nonmonotonic PoA/PoS behavior depending on behavior type. These results highlight how popularity effects reshape spatial competition and welfare in markets and elections.

Abstract

We study a variant of the Hotelling-Downs model of spatial competition between firms where consumer choices are influenced by their individual preferences as well as the popularity of the firms. In general, a multiplicity of market equilibria might exist due to the popularity effect. To elucidate firm decision-making, we explore three distinct behavioral attitudes towards this multiplicity of equilibria: optimistic, neutral, and pessimistic. For each behavior, we characterize the set of Nash equilibria and measure the impact of the selfish behavior on the social welfare by means of the price of anarchy and price of stability.

Figures (5)

• Figure 1: Possible symmetric equilibria $(x_1,1-x_1)$ with respect to $a$.
• Figure 2: Set of equilibria for $a=\frac{1}{2}$.
• Figure 3: Price of anarchy and price of stability for neutral firms as a function of $a$, for $\theta=1$.
• Figure 4: Price of anarchy as a function of $a$ for $\theta=1$.
• Figure 5: Price of stability as a function of $a$ for $\theta=1$.

Theorems & Definitions (31)

• Definition 2.1: Market equilibrium
• Example
• Definition 2.2
• Proposition 3.1
• proof
• Corollary 3.2
• proof
• Proposition 3.3
• proof
• Proposition 3.4