Misspecified Estimate-then-Optimize Leads to Supra-Competitive Prices

Abstract

We study whether simple algorithmic pricing systems can systematically produce collusive-like prices in multi-firm markets. We consider firms that price using a myopic estimate-then-optimize rule: each repeatedly fits a demand model to its own price and sales history and sets the price that maximizes estimated profit. This demand model is misspecified, omitting competitors' prices. We analyze the dynamics of this rule when it is initialized by an exploration phase of independent random prices. We characterize when this pipeline converges to supra-competitive prices above the Nash equilibrium, via a fluid-limit ordinary differential equation analysis. We show that supra-competitive prices arise when firms initially explore within similar price ranges on the same side of the Nash price. Moreover, prices can be substantially above the Nash price; we show that prices can reach monopoly levels under symmetric exploration. Simulations calibrated to a real multifamily rental market confirm that supra-competitive outcomes arise robustly beyond our theoretical assumptions, including under finite horizons, heterogeneous products, and nonlinear logit demand.