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arxiv: 2606.05363 · v2 · pith:LFVQRAV7new · submitted 2026-06-03 · 💻 cs.GT · cs.LG· econ.TH· math.OC

Should Demand Models Incorporate Competitor Prices? Oblivious Learning and Algorithmic Collusion

Yuhang Wu , Assaf Zeevi This is my paper

Pith reviewed 2026-06-28 03:12 UTC · model grok-4.3

classification 💻 cs.GT cs.LGecon.THmath.OC
keywords algorithmic collusiondemand estimationoblivious learningpricing algorithmscompetitive marketsNash equilibriummarket dynamics
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The pith

Oblivious sellers converge to competitive prices while informed ones earn more, making all-informed 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 whether pricing algorithms should incorporate competitor prices when learning demand in competitive markets. It finds that ignoring competitors requires more exploration but leads to convergence to competitive prices rather than sustained collusion. Informed models that include competitor prices allow sellers to earn more than oblivious ones. The modeling choice forms a game with a unique equilibrium at all sellers being informed. This suggests that deliberate obliviousness does not reliably produce collusion in this setting.

Core claim

In a stylized competitive market with unknown noisy demand estimated via iterated least squares, oblivious sellers explore more aggressively than a monopolist. When all sellers are oblivious, prices converge to the competitive outcome under sufficient exploration, but a continuum of pseudo-equilibria arises when exploration decays. An excursion phenomenon produces transient collusive patterns that dissipate over time. In mixed markets, informed sellers strictly out-earn oblivious ones, and the unique Nash equilibrium of the modeling choice game is the all-informed market where prices converge efficiently to competition.

What carries the argument

The oblivious versus informed demand modeling choice, where oblivious ignores competitor prices and informed incorporates them, driving the market dynamics and equilibrium analysis.

Load-bearing premise

The analysis assumes a stylized market where demand is estimated solely through iterated least squares and the modeling choice is the sole strategic decision.

What would settle it

If prices in a simulated or real multi-seller market with oblivious algorithms and decaying exploration remain at collusive levels instead of converging to competition, the claim would be falsified.

Figures

Figures reproduced from arXiv: 2606.05363 by Assaf Zeevi, Yuhang Wu.

Figure 1
Figure 1. Figure 1: Three sample paths of the prices for two oblivious sellers under cumulative exploration of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Empirical stress test of Theorem 4: seed-averaged MSE(p˜n) on log–log axes. Left: asymmetric markets at N ∈ {3, 5, 10}; right: symmetric N = 5 markets sweeping the cross-price coefficient γ; both panels sweep exploration variance ν 2 ∈ {0.05, 0.10, 0.20}. Across all 18 configu￾rations the sufficient condition (8) is violated by roughly two orders of magnitude, yet every MSE trajectory decays at the asympto… view at source ↗
Figure 3
Figure 3. Figure 3: Positive (panels (a), (b)) and negative (panels (c), (d)) excursions of the mean price [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Running-mean prices (panels (a), (c)) and per-period revenues (panels (b), (d)) of [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Empirical corroboration of Theorem 6 in the special case ηi = 1/2. All three panels overlay seed-averaged price-MSE E||pn −p NE||2 2 on log–log axes. Panel (a): symmetric all-informed markets at N ∈ {2, 5}. Panel (b): asymmetric all-informed markets at N ∈ {3, 5}. In both, regularity (14) holds and the empirical slope matches the predicted −1/2. Panel (c): asymmetric N = 3 market designed so every cell vio… view at source ↗
Figure 6
Figure 6. Figure 6: Empirical verification of Theorem 7: seed-averaged price MSE E||p˜n − p NE||2 2 on log–log axes. Panel (a): symmetric N = 5 with |Iob| = 2, |Iin| = 3. Panel (b): asymmetric markets at N ∈ {3, 5, 10} with one oblivious and N − 1 informed sellers. In both panels κ <¯ 0, so conditions (ii) and (iv) fail jointly. Panel (c): symmetric N = 5 with |Iob| = 4, |Iin| = 1, primitives chosen so κ >¯ 0 (condition (ii) … view at source ↗
Figure 7
Figure 7. Figure 7: Cross-seed scatter of the seed-by-seed running-mean prices [PITH_FULL_IMAGE:figures/full_fig_p057_7.png] view at source ↗
read the original abstract

On a platform with many sellers, should a pricing algorithm explicitly model competitors' prices when learning demand? Classical learning arguments suggest an affirmative answer: ignoring competitors induces model misspecification and inefficiency. In contrast, recent work on algorithmic collusion suggests that strategic obliviousness -- deliberately ignoring competitor prices -- may facilitate collusive outcomes and improve profits. We study this modeling choice in a stylized competitive market with unknown noisy demand, in which multiple sellers repeatedly set prices and estimate demand via iterated least squares, and either incorporate competitors' prices into their demand models (informed) or ignore them (oblivious). We first show that, relative to a monopolist, an oblivious seller in a competitive market must explore more aggressively to compensate for the loss of dynamic competitor information. Building on this insight, we characterize market dynamics when all sellers are oblivious and show that prices converge to the competitive outcome under sufficient exploration, while a continuum of pseudo-equilibria arises when exploration decays. Analyzing the resulting price trajectories, we uncover an excursion phenomenon that gives rise to transient collusive patterns that dissipate as learning progresses. In markets with both oblivious and informed sellers, the informed strictly out-earn the oblivious. Read as a strategy game, the modeling choice has a unique Nash equilibrium: the all-informed market, in which prices converge to the competitive outcome efficiently. Overall, our results indicate that collusive patterns are not robust and are not sustained by oblivious modeling; therefore, incorporating competitor information, together with sufficient price exploration, remains a reliable strategy for sellers in competitive markets.

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

2 major / 1 minor

Summary. The paper studies whether pricing algorithms in a competitive market with unknown noisy demand should incorporate competitor prices into demand models when learning via iterated least squares. It shows that oblivious sellers require more aggressive exploration than a monopolist; all-oblivious markets converge to the competitive outcome under sufficient exploration but admit a continuum of pseudo-equilibria when exploration decays; an excursion phenomenon produces transient collusive patterns that dissipate; informed sellers strictly out-earn oblivious ones; and the modeling choice game has a unique Nash equilibrium at the all-informed market, which converges efficiently to competitive prices. The overall conclusion is that collusive patterns are not robust and are not sustained by oblivious modeling.

Significance. If the derivations hold, the work supplies a stylized counter-example to concerns about sustained algorithmic collusion via oblivious demand models, instead showing that informed modeling plus exploration is both privately and socially preferable. The characterization of pseudo-equilibria, the excursion phenomenon, and the reduction of the modeling choice to a normal-form game are concrete contributions. The paper ships a clear, falsifiable prediction that collusion dissipates under iterated least squares with decaying but positive exploration.

major comments (2)
  1. [Abstract; dynamics and Nash sections] The convergence, out-earning, and unique-Nash claims are derived specifically for iterated least squares estimation of (presumably linear) demand in a stationary noisy environment. The manuscript provides no analysis or discussion of whether the same qualitative outcomes (dissipation of pseudo-equilibria, transient excursions, informed dominance) survive under nonparametric estimation, RL-based learners, or non-stationary demand; this assumption is load-bearing for the claim that collusive patterns are not robust.
  2. [Nash analysis] The result that informed sellers strictly out-earn oblivious ones (and therefore that all-informed is the unique Nash) treats the modeling choice as the sole strategic variable. It is unclear whether this remains the unique equilibrium when sellers can jointly optimize both model specification and exploration schedules; the paper should state the strategy space explicitly.
minor comments (1)
  1. [Abstract] The abstract states convergence and outperformance results without derivation details, error bounds, or robustness checks; the full manuscript should ensure every convergence claim is accompanied by precise conditions and rates.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the insightful comments. We address each major point below, agreeing where revisions are needed to clarify scope and strategy space, and have updated the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract; dynamics and Nash sections] The convergence, out-earning, and unique-Nash claims are derived specifically for iterated least squares estimation of (presumably linear) demand in a stationary noisy environment. The manuscript provides no analysis or discussion of whether the same qualitative outcomes (dissipation of pseudo-equilibria, transient excursions, informed dominance) survive under nonparametric estimation, RL-based learners, or non-stationary demand; this assumption is load-bearing for the claim that collusive patterns are not robust.

    Authors: We agree that all convergence, out-earning, and Nash results are derived specifically for iterated least squares on linear demand in a stationary noisy environment. The paper does not analyze or claim robustness under nonparametric estimation, RL-based learners, or non-stationary demand. We will add a dedicated limitations subsection (in the conclusion) that explicitly states the scope of the assumptions, notes that the dissipation result is tied to iterated least squares with decaying but positive exploration, and flags extensions to other learners as future work. This addresses the load-bearing nature of the assumption without overstating generality. revision: yes

  2. Referee: [Nash analysis] The result that informed sellers strictly out-earn oblivious ones (and therefore that all-informed is the unique Nash) treats the modeling choice as the sole strategic variable. It is unclear whether this remains the unique equilibrium when sellers can jointly optimize both model specification and exploration schedules; the paper should state the strategy space explicitly.

    Authors: The strategy space is defined as the binary modeling choice (informed vs. oblivious), with each seller's exploration schedule endogenously determined by its model choice via the least-squares analysis in Sections 3 and 4. We will revise the Nash section (and the opening of Section 5) to state this strategy space explicitly, including a sentence clarifying that independent choice of exploration rates is outside the current game. Under this definition the unique Nash at all-informed remains; a richer game allowing separate exploration optimization would require a different model and is noted as beyond scope. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained with no circular reductions

full rationale

The paper constructs a stylized competitive market model with unknown noisy demand estimated via iterated least squares, then derives convergence to competitive outcomes, transient excursion phenomena, out-earning results for informed sellers, and a unique Nash equilibrium at all-informed directly from the model dynamics and exploration assumptions. No load-bearing step reduces by construction to a fitted parameter renamed as prediction, a self-definitional loop, or a self-citation chain whose content is unverified within the paper. The analysis remains independent of external benchmarks and does not invoke uniqueness theorems or ansatzes from prior self-work as forcing mechanisms.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the stylized market, noisy demand, and iterated least squares are background modeling choices whose details are not supplied.

pith-pipeline@v0.9.1-grok · 5820 in / 1085 out tokens · 38099 ms · 2026-06-28T03:12:29.321957+00:00 · methodology

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Reference graph

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