Three Toed Pete: Examining equilibria and player behavior in a high-variance game

Abstract

I study “three toed pete,” a high-variance, sequential wagering game in which players decide—over multiple rounds—whether to commit to a common pot based on private signals. I develop and compare a suite of computational methods for characterizing equilibrium behavior: (i) simulation-based grid search to identify candidate cutoff strategies; (ii) gradient- based and simulated-annealing optimizers to navigate the noisy, multi-dimensional payoff landscape; (iii) state-dependent cutoff maps that adjust to current “toe” counts and alternating move order; and (iv) backward-induction algorithms that bootstrap the t = 1 solution to solve for general t recursively. My numerical experiments confirm theoretical predictions in the two-player, one toe case, reveal how cutoff thresholds rise with increasing target toes, and demonstrate scalability to more complex, n-player settings. I also prove structural lemmas—such as the weak dominance of non-contiguous strategies—that under- pin our computational approach. Beyond game theory, my methods have direct applications to multi-round auctions, sequential bidding for large-scale contracts (e.g. Olympic host selection, pension-liability transfers, 401(k) administration), and other contexts where agents face uncertainty, risk, and dynamic strategic interaction.