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.

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