Equilibria in Large Position-Optimization Games

Abstract

We propose a general class of symmetric games called position-optimization games. Given a probability distribution $Q$ over a set of targets $\mathcal{Y}$, the $n$ players each choose a position in a space $\mathcal{X}$. A player's utility is the $Q$-mass of targets they are closest to under some proximity measure, with ties broken evenly. Our model captures Hotelling games and forecasting competitions, among other applications. We show that for sufficiently large $n$, both pure and symmetric mixed Nash equilibria exist, and moreover are extreme: all players play on a finite set of pseudo-targets $\mathcal{X}^* \subseteq \mathcal{X}$. We further show that both pure and symmetric mixed equilibria converge to the distribution $P$ on $\mathcal{X}^*$ induced by $Q$, and bound the convergence rate in $n$. The generality of our model allows us to extend and strengthen previous work in Hotelling games, and prove entirely new results in forecasting competitions and other applications.

Figures (5)

• Figure 1: An example of spaces $\mathcal{X}$ and $\mathcal{Y} = \{y_1, y_2, y_3, y_4\}$, here both subsets of some Euclidean space with proximity function $d$. Note multiple targets ($y_1$, $y_2$) can map to the same pseudo-target, and targets in $\mathcal{X}$ are themselves pseudo-targets (see $y_3$).
• Figure 2: The finite-location Hotelling game, where $\mathcal{X}$ is a finite set of locations in a Euclidean space. Retailers choose locations from $\mathcal{X}$ (red). Consumers follow a continuous distribution, and retailers win mass inside their Voronoi cell relative to the other retailer locations.
• Figure 3: A forecasting competition with $m = 3$ binary events. Targets correspond to vertices of the cube $\{0,1\}^3$. Forecasters submit predictions (red) in the filled cube $[0,1]^3$. The outcome (blue) is drawn from distribution $Q$ over vertices, with the winner minimizing $\ell_2$ distance.
• Figure 4: All best-response dynamics for each strategy profile $\bm{x}$ in the proof of Lemma \ref{['lemma:no-pure-nash-2/n']}. The dots represent the mass of players on each position $x_1,x_2,$ or $x_3$. Because in each $\bm{x}$ some player has a profitable deviation, no such $\bm{x}$ can be an equilibrium. In all examples, $P(x_1)=\frac{2-2\epsilon}{n},$$P(x_2)=\frac{2-\epsilon}{n},$ and $P(x_3)=\frac{n-4+3\epsilon}{n}$.
• Figure 5: Plot of $\sigma_x$ as a function of $p_x$ for a symmetric strategy $\sigma$ and pseudo-target $x$, for the case $|\mathcal{X}^*| = 2$, for various values of $n$. Red lines indicate the function $\sigma_x = G^{-1}(p_x)$, and the black line indicates where $\sigma_x= p_x$; vertical lines indicate $\sigma_x=\frac{1}{n}$ and $\sigma_x=1-\frac{1}{n}$ for the different values of $n$, respectively. Observe that for increasing values of $n$, the equilibrium $\sigma_x$ given by $G^{-1}(p_x)$ hews closer and closer to $p_x$.

Theorems & Definitions (57)

• Definition 1
• Definition 2: Pure Nash equilibrium
• Definition 3: Mixed Nash equilibrium
• Definition 4: Symmetric Mixed Nash equilibrium
• Definition 5
• Definition 6
• Lemma 1
• Lemma 2
• proof
• Theorem 1