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arxiv: 2605.16064 · v1 · pith:MDN3OWE6new · submitted 2026-05-15 · 💻 cs.GT · cs.AI· econ.TH

Misspecified Explore-then-Exploit Leads to Supra-Competitive Prices

Jackie Baek , Vivek F. Farias , Farrell Wu This is my paper

Pith reviewed 2026-05-19 18:01 UTC · model grok-4.3

classification 💻 cs.GT cs.AIecon.TH
keywords algorithmic pricingexplore-then-exploitmisspecified demandsupra-competitive pricesNash equilibriumdemand estimationfluid limitpricing dynamics
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The pith

Firms using explore-then-exploit pricing with misspecified demand models converge to prices above the Nash equilibrium.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines a simple algorithmic pricing approach in which firms first randomize prices during an exploration phase and then estimate demand from their own data to set myopic prices thereafter. The estimation uses a monopoly-style model that leaves out competitors' prices entirely. When the exploration ranges are similar and lie on the same side of the Nash price, the resulting dynamics drive prices upward, sometimes all the way to monopoly levels under symmetric exploration. A sympathetic reader cares because the result shows how routine algorithmic tools can produce persistently high prices without any coordinated intent.

Core claim

The authors establish that an explore-then-exploit pricing pipeline relying on a misspecified monopoly-style demand estimation converges to supra-competitive prices above the Nash equilibrium when firms explore within similar price ranges on the same side of the Nash price. Through a fluid-limit ordinary differential equation analysis, they 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 the theoretical assumptions, including under finite horizons, heterogeneous products, and nonlinear logit demand.

What carries the argument

Fluid-limit ordinary differential equation analysis of the explore-then-exploit pricing dynamics under misspecified monopoly demand estimation.

If this is right

  • Supra-competitive prices arise when firms explore within similar price ranges on the same side of the Nash price.
  • Prices can reach monopoly levels under symmetric exploration.
  • The outcome persists in simulations with finite horizons, heterogeneous products, and nonlinear logit demand.
  • Basic algorithmic pricing systems can systematically generate collusive-like prices without explicit coordination.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Regulators could examine whether common pricing software structures create unintended high-price equilibria across markets.
  • Firms might reduce the effect by expanding their demand models to account for observed competitor prices.
  • Analogous misspecifications in other repeated decision algorithms could produce similarly elevated equilibria in non-price settings.
  • Testing the same pipeline on markets with different demand curvatures would clarify how sensitive the supra-competitive outcome is to functional form.

Load-bearing premise

The demand estimation step uses a misspecified monopoly-style model that omits competitors' prices, and exploration occurs within similar ranges on the same side of the Nash price.

What would settle it

Observing convergence to the Nash equilibrium instead of supra-competitive prices when firms either include competitors' prices in the demand model or explore ranges on opposite sides of the Nash price.

Figures

Figures reproduced from arXiv: 2605.16064 by Farrell Wu, Jackie Baek, Vivek F. Farias.

Figure 1
Figure 1. Figure 1: Left: the shaded regions depict the best-response cones in the (µ1, µ2) plane, where µi is firm i’s average exploration price. We show that the terminal prices are supra-competitive whenever (µ1, µ2) lies in the shaded region. The angle θ depends on the demand parameters but is always at least 45◦ , so the cones cover more than one quarter of the feasible exploration-mean space. Right: Final price under sy… view at source ↗
Figure 2
Figure 2. Figure 2: Best-response cones in the duopoly case, shown in the [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: ODE-implied terminal price P ODE 1 (α; µ, σ2 expI2×2) in the duopoly case. Each panel fixes (α, Σexp) and varies the exploration means (µ1, µ2). White marks the Nash price p NE = 2/3, while red and blue indicate terminal prices above and below Nash. Thin lines mark the best-response boundaries defining the two cones. Horizons sharpen cone-like regions. As the horizon α increases (left to right), the heatma… view at source ↗
Figure 4
Figure 4. Figure 4: ODE and stochastic mean terminal-price heatmaps. The left panel is the deterministic [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: ODE map and terminal-price histograms from stochastic simulations at [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Terminal rent changes relative to Nash as the exploration mean multiplier [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Finite-time dynamics of terminal rent changes relative to Nash. We fix [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: ODE-implied mean terminal price under interval sampling. For each boundary pair [PITH_FULL_IMAGE:figures/full_fig_p042_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Mean terminal price under center–dispersion sampling. For each anchor price [PITH_FULL_IMAGE:figures/full_fig_p043_9.png] view at source ↗
read the original abstract

We study whether simple algorithmic pricing systems can systematically produce collusive-like prices in multi-firm markets. We consider firms using an explore-then-exploit pipeline: they randomize prices during an initial exploration phase, then estimate demand from their own historical data and set prices myopically thereafter. The estimation step relies on a misspecified, monopoly-style model that omits competitors' 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 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper studies the emergence of supra-competitive pricing in oligopoly markets when firms employ an explore-then-exploit strategy with a misspecified demand model that ignores competitors' prices. Using a fluid-limit ordinary differential equation (ODE) analysis, the authors show that convergence to prices above the Nash equilibrium occurs when exploration ranges are similar and lie on the same side of the Nash price. Symmetric exploration can lead to the monopoly price as the fixed point. The theoretical results are supported by simulations that extend to finite time horizons, heterogeneous products, and logit demand, calibrated to data from a real multifamily rental market.

Significance. This result is significant as it identifies a specific mechanism—misspecification in demand estimation combined with correlated exploration—through which algorithmic pricing can lead to outcomes resembling collusion without any intent to collude. The analytical approach using ODEs provides precise conditions for when this happens, and the simulations demonstrate robustness. Strengths include the parameter-free nature of the core result under the stated exploration assumptions and the connection to real-world data. This contributes to the literature on algorithmic collusion and has potential policy implications for regulating pricing algorithms.

minor comments (3)
  1. [Abstract] The abstract mentions 'supra-competitive outcomes arise robustly beyond our theoretical assumptions'; specifying one or two key extensions in the abstract would enhance impact.
  2. [§3] The transition from the discrete-time process to the fluid-limit ODE could include a brief outline of the convergence theorem used, even if standard.
  3. [Figure 2] The plot of price trajectories would be clearer with annotations indicating the Nash and monopoly prices for reference.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. The referee's description accurately reflects the paper's focus on misspecified explore-then-exploit pricing and the conditions leading to supra-competitive outcomes. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation relies on an explicit fluid-limit ODE constructed from the explore-then-exploit dynamics and the misspecified monopoly demand model. Fixed points of the ODE are solved directly from the myopic best-response mapping under the stated exploration ranges and misspecification; these are not obtained by fitting to the target supra-competitive outcome or by renaming an input. No self-citation is invoked as a load-bearing uniqueness theorem, and the analysis is self-contained against the model's own assumptions without reducing any prediction to a fitted quantity by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The analysis rests on the fluid-limit approximation and the structural assumption of misspecified monopoly demand estimation.

axioms (1)
  • domain assumption Fluid-limit ordinary differential equation approximation governs the long-run price dynamics
    Invoked to characterize convergence of the pricing process.

pith-pipeline@v0.9.0 · 5690 in / 1102 out tokens · 46171 ms · 2026-05-19T18:01:19.182846+00:00 · methodology

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