Decentralized price-learning commonly used in algorithmic pricing can push markets to a Conjectural Variations equilibrium driven by learning-induced bias; when firms see all rivals' prices or rival experiments are independent, the market instead converges to the standard Nash outcome. The paper proves convergence conditions and supplies finite-sample error bounds (≈T^{-1/2}).

Main Finding

When decentralized sellers run practical local price experiments and fit local (misspecified) linear demand models using only the prices they observe, the repeated pricing dynamics converge to a Conjectural Variations (CV) equilibrium. The CV conjecture matrix A is not a behavioral primitive but is endogenously generated by the interaction of (i) which rivals’ prices each seller observes (feedback structure) and (ii) the statistical dependence (correlation) in sellers’ experiment-induced price variation (experimentation structure). If the learning bias from omitted (unobserved) rival prices vanishes (e.g., full price feedback or independent experimentation of unobserved rivals), the dynamics instead converge to the Nash equilibrium. A finite-sample bound shows mean-squared price error decays at rate ~T−1/2 (up to logs) under the convergence conditions.

Key Points

• Setup and feedback:
  • Multiple sellers repeatedly post prices; demand depends on all prices plus noise.
  • Each seller observes her own price and realized demand, and may observe prices of only a subset of rivals (bandit → partial → full feedback).
• Experimentation:
  • Sellers run local batch experiments (two- or three-point or general perturbations) around a focal price, possibly synchronized or correlated across sellers.
  • Perturbation magnitudes shrink over time; updates move partially toward the fitted-model revenue-maximizer.
• Source of conjectures:
  • When a seller omits some rival prices from her regression, correlated experimentation of those unobserved rivals induces an omitted-variable bias.
  • This bias makes the seller behave as if unobserved rivals systematically co-move with her own price; these implied co-movement coefficients form the induced conjecture matrix A.
• Equilibrium selection:
  • Learning dynamics converge to the CV equilibrium determined by the induced A (not an arbitrary CV).
  • Special cases: full price feedback or asymptotically uncorrelated unobserved experimentation → A = 0 → Nash equilibrium.
• Comparative statics:
  • Under strategic complementarity, larger positive induced conjectures raise equilibrium prices relative to Nash (potentially supra-competitive).
  • Induced conjectures’ signs/magnitudes depend on exactly which rivals are observed and on correlation structure; effects can be non-monotone and market-specific.
• Stability and rates:
  • Convergence requires stability conditions separating first-order competitive/conjectural effects from curvature (local linear approximation) effects.
  • For linear demand the condition resembles diagonal dominance; for MNL it typically holds when market shares are not too large.
  • Finite-sample guarantee: mean-squared price error decays on the order of T−1/2 (up to logs), matching best-known rates in related bandit-convergence work.
• Robustness:
  • The CV mechanism arises from the learnable (observable) price–demand relationship and is not specific to linear regression; other estimation rules that use the same observed variation will inherit similar conjectural terms.
  • Even if global convergence fails, the induced conjecture matrix still governs the local direction of adjustment.

Data & Methods

• Theoretical model:
  • Repeated pricing game with n sellers. Each period: sellers set prices; demands follow an unknown function of the full price vector plus iid shocks.
  • Feedback model: each seller i observes her own demand and prices of a subset Si of rivals (Si may vary across sellers).
• Learning algorithm (per seller):
  • Batch experiments: within each batch, seller perturbs a focal price by randomized amounts (A/B, switchback, or general noise) and collects batch data (own price, own demand, observed rival prices).
  • Fits a linear (local) regression of own demand on own price and observed rival prices (omitted unobserved rival prices).
  • Computes the revenue-maximizing price implied by the fitted linear model (holding observed rival prices at batch averages) and moves partially toward it (bounded update).
• Mathematical analysis:
  • Characterize omitted-variable bias from correlated experimentation as partial linear projection coefficients; these coefficients give the entries of the induced conjecture matrix A.
  • Prove that, under regularity and stability conditions and vanishing experiment magnitudes, the stochastic iterative updates converge almost surely to the fixed point characterized by the CV equilibrium with conjectures A.
  • Provide finite-sample mean-squared error bounds for prices: error ~ eO(T−1/2).
  • Derive sufficient stability conditions in canonical demand families: linear demand (diagonal dominance–type condition) and multinomial logit (restrictions on market shares).
• Scope:
  • Analytical, proof-based; no empirical estimation of real market data in the paper. The mechanism is illustrated and made concrete via linear and MNL examples.

Implications for AI Economics

• Algorithmic design matters for equilibrium selection:
  • The micro-design choices of pricing algorithms (batching, perturbation schedules, which rivals’ prices are incorporated) and shared infrastructure that synchronizes updates can systematically shift markets away from Nash toward CV equilibria that may be supra- or sub-competitive.
  • For AI economists modeling markets with algorithmic pricing, equilibrium predictions must account for the feedback and experimentation architecture—not just primitives of demand and payoffs.
• Platform and infrastructure levers:
  • Platform APIs, rate limits, shared repricing tools, and scheduled batch jobs can induce correlated experimentation. Platforms that care about competitive outcomes can reduce induced conjectural bias by (a) increasing transparency/full price feedback, (b) de-synchronizing update windows, or (c) discouraging large common perturbations.
  • Conversely, covertly aligning update schedules or enabling synchronized campaign events (e.g., platform-wide discounts) can unintentionally facilitate supra-competitive dynamics.
• Empirical implications:
  • When estimating demand or testing for algorithmic collusion, researchers should control for correlations in price variation across sellers and for which prices each firm observes. Observed supra-competitive prices may reflect learning-induced CV equilibria rather than explicit coordination.
  • The induced conjecture matrix A is estimable in principle from observed price co-movements and feedback graphs; doing so can diagnose whether algorithmic learning is shifting markets away from competitive benchmarks.
• Policy and antitrust:
  • Regulators evaluating algorithmic collusion should consider structural sources of statistical dependence (shared software, synchronized schedules, platform designs) that can produce collusive-like outcomes without explicit communication.
  • Policy responses can target the information and experimental-design channels (e.g., requiring platforms to publish update-timing heterogeneity, limiting synchronized bulk repricing, or mandating better logging of A/B schedules) as complements to enforcement aimed at explicit coordination.
• Modeling advice for AI economists:
  • Incorporate partial feedback graphs and correlated perturbations into models of algorithmic pricing and market dynamics; treat conjectures as endogenous outputs of learning processes, not primitives.
  • When comparing market welfare under different algorithmic architectures, use the paper’s mapping from feedback+correlation → induced conjectures → equilibrium prices to evaluate design tradeoffs (welfare, seller surplus, consumer surplus).
• Limitations & further questions:
  • Results hinge on local linear learning and shrinking experiment magnitudes; global nonlinearity or persistent large perturbations can invalidate stability conditions though the induced-conjecture effect remains influential locally.
  • Open empirical work: measuring the magnitude of induced conjectures in real marketplaces and testing platform interventions (de-synchronization, transparency) to mitigate supra-competitive outcomes.