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 $\mathcal{X}<sup>*$ 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.

• The paper unifies forecasting, Hotelling location, spatial voting, and Voronoi games through a constant-sum position-optimization framework based on projected target distributions.
• The paper proves that pure equilibria become extreme and cover every pseudo-target for sufficiently large n, exist constructively when n ≥ 2/p₀, and converge to the projected distribution at the first explicit O(1/n) KL-divergence rate.
• The paper establishes symmetric mixed-equilibrium existence and shows large games approximately recover target probabilities, while binary forecasting equilibria exhibit strategic extremization despite accurate aggregate forecasts.

The position-optimization game

The paper "Equilibria in Large Position-Optimization Games" (2602.15225) introduces a general class of symmetric games in which $n$ agents each choose a position $x$ in an arbitrary action space $X$, competing to capture targets $y \in Y$ distributed according to a measure $Q$. An agent's utility is the $Q$-mass of targets for which their position minimizes a proximity function $d: X \times Y \to [0,\infty]$, with ties broken evenly. The central object is the set of pseudo-targets $X^* = \bigcup_{y} x^*(y)$, where $x^*(y) = \arg\min_{x \in X} d(x,y)$, and the projection $P(x) = Q(\{y : x^*(y) = x\})$ of the target distribution onto this set.

Two conditions govern the analysis: ties must have zero $x$0-mass, and $x$1 must be finite. Under these assumptions, the game is constant-sum (total utility is 1), and the authors unify several previously disjoint models as special cases: forecasting competitions under Brier scoring ($x$2, $x$3, $x$4 proximity), finite-location and classic Hotelling games, spatial voting under plurality rule, and one-round discrete Voronoi games. This unification is the paper's principal structural contribution; prior work on these settings used similar techniques but without a shared formal framework.

Pure Nash equilibria

For large $x$5, the paper gives a tight characterization of pure equilibria. If $x$6, where $x$7, every pure equilibrium is extreme (supported entirely on $x$8) and covers all pseudo-targets. The bound is tight: at $x$9 a non-extreme equilibrium exists. Existence holds for $X$0, via a greedy constructive algorithm running in time $X$1 — improving on the finite-time-only construction of Núñez and Scarsini — and this bound is also tight, with counterexamples showing no pure equilibrium exists for $X$2 arbitrarily close to but below $X$3.

The main quantitative result concerns convergence. For any equilibrium profile, the empirical distribution $X$4 over pseudo-targets satisfies

$X$5

for $X$6. Since this bound is $X$7, the empirical distribution of agent positions converges to the projected target distribution $X$8 in KL divergence at rate $X$9 — the first convergence-rate result for any of these game families. Prior Hotelling analyses (e.g., Osborne–Pitchik for twice-differentiable consumer CDFs; Núñez–Scarsini for finite locations) established convergence only asymptotically, without rates.

Symmetric mixed equilibria

Symmetric mixed equilibria exist for all $y \in Y$0 when $y \in Y$1 is finite or compact Hausdorff, proved by combining the symmetry/constant-sum structure (which pins each agent's equilibrium utility to exactly $y \in Y$2) with Reny's payoff-security theorem for discontinuous games.

The characterization shows that if $y \in Y$3, every symmetric mixed equilibrium is extreme and satisfies $y \in Y$4 for all pseudo-targets. The proof decomposes deviation utility into three events, derives an exact lower bound via binomial sums involving the function

$y \in Y$5

and uses coupon-collector arguments to show that $y \in Y$6 is covered with probability at least $y \in Y$7 in equilibrium. Notably, the upper-bound argument encounters genuine technical difficulty: monotonicity of the analogous function $y \in Y$8 breaks when $y \in Y$9 is too large, requiring careful casework.

In the two-pseudo-target case ($Q$0 forecasting), the equilibrium is pinned down exactly as $Q$1, exhibiting an extremization effect consistent with observed strategic behavior in real forecasting competitions: whenever $Q$2, the equilibrium places more weight than $Q$3 on the more likely outcome. Even so, the aggregate distribution of reports converges to $Q$4 — a wisdom-of-the-crowds phenomenon arising despite individually non-truthful play.

Applications

Applying the general results yields several firsts. For forecasting competitions, when $Q$5 (which requires $Q$6 since $Q$7), every forecaster submits a deterministic extreme prediction in $Q$8, yet the collective report distribution matches $Q$9 up to $Q$0 KL divergence or $Q$1 total variation. This implies the competition winner is not well-calibrated but lucky, and individual reports carry no information about $Q$2 beyond what many samples provide. For finite-location Hotelling and classic Hotelling games, the corollaries extend prior asymptotic results with explicit rates, and the classic-game results give a discrete-distribution analogue of Osborne–Pitchik's differentiable-CDF theorem. For spatial voting, candidates converge to matching individual voters' ideal positions rather than the median voter — though the paper notes this regime (many candidates relative to voters) does not hold in most actual elections. Discrete Voronoi games yield analogous coverage and convergence guarantees.

A notable asymmetry in the negative results: for forecasting competitions specifically, non-existence examples require $Q$3 rather than the general threshold, because hypercube geometry forces deviators to share utility with neighboring vertices. The paper conjectures its forecasting-specific bound is tight.

Limitations and open questions

The finiteness of $Q$4 is fundamental to the approach and appears difficult to relax; the no-ties condition is more technical but also restrictive. The authors observe that Osborne and Pitchik obtained convergent behavior in the classic Hotelling game without either assumption, and they conjecture convergence to $Q$5 holds far more generally — but this remains unproven. Second, all results concern large-$Q$6 regimes; nothing is known about medium-$Q$7 behavior, where the geometry of $Q$8 plausibly matters (e.g., interior forecasts on the hypercube may outperform vertex strategies). Third, the spatial-voting application applies only when candidates outnumber voters, limiting its direct relevance to real elections. Finally, whether the forecasting-specific pure-equilibrium existence threshold can be improved is left open.

Conclusion

This paper provides a unified framework in which Hotelling-style location games, forecasting competitions, spatial voting, and Voronoi games are instances of a single constant-sum structure, and delivers the first explicit $Q$9 convergence rates for both pure and symmetric mixed equilibria to the projected target distribution. Its tight existence thresholds, constructive polynomial-time equilibrium algorithm, and exact characterization of the extremizing equilibrium in binary forecasting together substantially sharpen the state of knowledge for large-player positional competition, while leaving the infinite-pseudo-target case and small-to-medium $d: X \times Y \to [0,\infty]$0 regimes open.