📈 economics

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

This paper demonstrates that simple algorithmic pricing systems using an explore-then-exploit pipeline with misspecified monopoly demand models can systematically converge to supra-competitive, collusive-like prices in multi-firm markets, particularly when firms explore similar price ranges on the same side of the Nash equilibrium.

Original authors: Jackie Baek, Vivek F. Farias, Farrell Wu

Published 2026-05-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jackie Baek, Vivek F. Farias, Farrell Wu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: How "Dumb" Algorithms Can Accidentally Collude

Imagine a busy street with several pizza shops. Usually, they compete fiercely, keeping prices low to steal customers from each other. This is the "Nash Equilibrium"—the fair, competitive price.

However, this paper asks a scary question: What if these shops start using simple computer programs to set their prices, and those programs accidentally make them act like a secret cartel, raising prices way above what is fair?

The authors found that you don't need complex, evil AI or secret phone calls to do this. You just need a very common, simple strategy called "Explore-then-Exploit" combined with a little bit of ignorance.

The Strategy: "Try, Learn, Then Guess"

The paper studies a specific way businesses use algorithms:

The "Explore" Phase: Imagine the pizza shops are new. They don't know what price to charge. So, for a while, they just throw darts at a board. They randomly pick prices (some high, some low) to see how many pizzas they sell.
The "Exploit" Phase: After gathering enough data, they stop guessing. They look at their own sales records and say, "Okay, when I charged \ $10, I sold 50 pizzas. When I charged \$ 12, I sold 30. I'll just use this pattern to pick the perfect price to make the most money."

The Catch (The "Misspecification"):
Here is the critical flaw. When the algorithm looks at its own sales data, it ignores the other shops. It assumes it is the only shop on the street. It thinks, "If I raise my price, I'll lose customers because people will go buy cheaper pizza." But it forgets that if everyone raises their price, there is no cheaper pizza to buy!

The Discovery: How They Get Stuck at High Prices

The authors used math to simulate what happens when all the shops do this. They found a surprising result: If the shops happen to test prices in a similar range during their "Explore" phase, they get locked into high prices forever.

The Analogy: The "Echo Chamber"

Imagine two people trying to guess the temperature of a room, but they can't talk to each other.

The Setup: They both start by guessing temperatures between 70°F and 80°F.
The Mistake: They only look at their own history. They don't realize the other person is doing the exact same thing.
The Result: Because they both started in that 70–80°F range, their internal "models" of the room get confused. They start thinking, "Hey, whenever I guess 75°F, the room feels warm. If I guess 80°F, it feels even warmer."
The Trap: Because they are all guessing in the same "neighborhood" of prices, their algorithms start reinforcing each other. They slowly drift upward, thinking they are just optimizing for themselves, but they are actually pushing the price up toward the "Monopoly Price" (the highest price possible).

The paper calls the safe zone where this happens "Best-Response Cones." Think of this as a specific region on a map. If all the shops start their experiment inside this region (which is surprisingly large), they are doomed to end up with high prices.

Key Findings in Plain English

It's Not Just "Smart" AI: You don't need advanced, learning-from-punishment AI (like the kind that learns to play chess). Even simple, "dumb" algorithms that just look at their own past data can cause this.
The "Cluster" Effect: If the shops happen to test prices that are close to each other (e.g., everyone tests between \ $10 and \$ 12), they are very likely to end up colluding. If they test wildly different prices, they might not. But since businesses often test prices near current market rates, they naturally "cluster," making this outcome very common.
The Price Gap: The prices don't just go up a little bit. They can go all the way up to the Monopoly Price—the price a single evil owner would charge if they owned all the shops.
It Happens Fast: You don't need to wait years for this to happen. The simulations show that even with short timeframes, prices shoot up quickly.
Real-World Proof: The authors tested this theory using real data from the Boston apartment rental market. They simulated landlords using these simple algorithms. The result? Even with complex, real-world factors (different apartment sizes, different tenants, non-linear demand), the prices still rose significantly above the competitive level.

The "Why" (The Mechanism)

Why does this happen? It comes down to correlation.

When the shops explore, they happen to move their prices in the same direction at the same time (because they are all reacting to similar market conditions). Because their algorithms are "blind" to each other, they misinterpret this synchronized movement.

Normal Logic: "If I raise my price, I lose customers."
Blind Algorithm Logic: "I raised my price, and sales didn't drop as much as I thought! Maybe people are willing to pay more."

Because they are all doing this at the same time, they create a false signal that "demand is strong," so they keep raising prices. They are trapped in a feedback loop where their own ignorance creates a collective illusion of high demand.

The Bottom Line

The paper warns that simple, widely used pricing algorithms can accidentally create a cartel.

It's not because the companies are trying to collude. It's because their computers are using a "myopic" (short-sighted) view of the world. They look at their own history, ignore their competitors, and if they start experimenting in a similar price range, they all get stuck in a high-price trap. This suggests that regulators and businesses need to be careful about how these simple algorithms are deployed, as they might be driving up prices without anyone realizing it.

Technical Summary: Misspecified Explore-then-Exploit Leads to Supra-Competitive Prices

1. Problem Statement

The paper investigates whether simple, model-based algorithmic pricing systems can systematically generate supra-competitive prices (prices above the Nash equilibrium) in multi-firm markets, potentially mimicking collusion without explicit communication or complex reinforcement learning strategies.

While existing literature (e.g., Calvano et al., 2020) demonstrates that complex Q-learning agents can sustain collusion through reward-and-punishment dynamics, these mechanisms are often computationally intensive and sensitive to hyperparameter tuning. In contrast, many firms employ simpler "estimate-then-optimize" pipelines: they estimate a demand curve from historical data and set prices myopically to maximize predicted revenue. A critical feature of these practical systems is misspecification: firms typically estimate a monopoly-style demand curve using only their own price and sales history, omitting competitors' prices.

The central question is whether this specific combination of an explore-then-exploit policy and a misspecified demand model can lead to robustly supra-competitive outcomes. Previous work by Cooper et al. (2015) suggested that under pure exploitation with misspecified models, limiting prices are highly path-dependent and unpredictable, depending entirely on initial conditions. This paper argues that introducing an initial exploration phase with independent random prices resolves this unpredictability, leading to systematic supra-competitive convergence under well-defined conditions.

2. Methodology

Model Setup

The authors analyze a competitive market with $N$ symmetric firms over $T$ time periods under a linear demand model with idiosyncratic shocks. The firms follow a two-phase policy:

Exploration Phase ( $t \le K$ ): Firms post prices drawn independently from a distribution with mean $\mu_i$ and covariance $\Sigma_{exp}$ .
Exploitation Phase ( $t > K$ ): Firms employ a myopic estimate-then-optimize rule. In each period, firm $i$ fits a linear demand curve $q(p) = \alpha + \beta p$ using Ordinary Least Squares (OLS) on its own historical price-quantity data. Crucially, this model omits competitors' prices. The firm then sets the next price to maximize predicted profit based on this fitted curve.

Analytical Framework: Fluid Limit and ODE

To analyze the long-run behavior, the authors consider a fluid scaling regime where both the exploration horizon $K$ and exploitation horizon $T$ grow large, with their ratio $(K+T)/K \to \alpha$ .

Convergence: The stochastic pricing process is shown to converge to a deterministic limit described by a price-moments Ordinary Differential Equation (ODE).
State Variables: The ODE tracks the running price means $U(t)$ and the accumulated price covariances $V(t)$ for all firms.
Mechanism: The ODE reveals that the misspecification introduces a bias. When firms' historical prices are positively correlated with competitors' prices (captured by off-diagonal terms in $V(t)$ ), the OLS estimator suffers from an omitted-variable bias, leading firms to set prices higher than the true best response.

Theoretical Tools

Best-Response Cones: The authors define regions in the space of exploration means ( $\mu$ ) called "best-response cones" ( $C^+$ and $C^-$ ). These regions characterize exploration profiles where, if firms were to best-respond to the average exploration price of competitors, all firms would adjust prices in the same direction.
Invariant Analysis: The proof relies on establishing that once the system enters a cone, the gap between posted prices and historical means maintains a consistent sign (e.g., prices remain above historical means in the lower cone), which drives persistent positive cross-covariances and upward price bias.

3. Key Contributions and Results

1. Characterization of Supra-Competitive Convergence

The primary theoretical result (Theorem 3.2) establishes that if the firms' exploration means $\mu$ lie within the best-response cones, the limiting prices are strictly supra-competitive.

Upper Cone ( $C^+$ ): If firms explore at prices above their best responses, supra-competitive prices arise immediately.
Lower Cone ( $C^-$ ): If firms explore below their best responses, prices eventually rise to supra-competitive levels after a finite waiting period.
Significance: This contrasts with Cooper et al. (2015), where outcomes were unpredictable. Here, the introduction of exploration creates a "basin of attraction" for high prices.

2. Quantification of Price Gaps

Under symmetric exploration (all firms explore with the same mean price $s$ ), the authors provide a sharp characterization of the final price (Theorem 3.3):

If $s$ is between the Nash price ( $p_{NE}$ ) and the monopoly price ( $p_{MNP}$ ), the limiting price converges to $s$ .
If $s$ is below $p_{NE}$ or above $p_{MNP}$ , the limiting price converges to the monopoly price (or the capped monopoly price).
Result: This demonstrates that the price gap above Nash can be substantial, potentially reaching monopoly levels, depending on the exploration strategy.

3. Probability of Entering the "Collusive" Region

The paper addresses the likelihood of firms naturally falling into these supra-competitive regimes:

Duopoly ( $N=2$ ): The best-response cones cover at least one-quarter of the feasible price space. Furthermore, the dynamics are such that limit points naturally fall into these cones (Proposition 3.1).
General $N$ : Under a "random interval prior" (where firms' exploration means cluster around a common local band), the probability of landing in a supra-competitive cone is at least $1/4$ and increases with the number of firms $N$ (Proposition 3.2). This suggests that clustered exploration, common in real markets, makes supra-competitive outcomes robustly likely.

4. Empirical Validation via Simulation

The authors validate their theoretical findings using simulations calibrated to the Greater Boston multifamily rental market (using data from Calder-Wang and Kim, 2024).

Robustness: The supra-competitive outcomes persist under realistic conditions including finite horizons, heterogeneous products, nonlinear logit demand, and product-specific costs.
Speed: The simulations show that supra-competitive rents emerge quickly (within the first 50 exploitation periods) and do not require asymptotic time horizons.
Pattern: The U-shaped relationship between exploration means and final prices (predicted by Theorem 3.3) is observed: prices are elevated when exploration is clustered away from the Nash benchmark on either side.

4. Significance and Claims

The paper claims to demonstrate that supra-competitive pricing is a robust feature of simple, widely deployed algorithmic pricing systems, not just an artifact of complex reinforcement learning or explicit collusion.

Mechanism: The mechanism does not require punishment strategies, communication, or learned threats. Instead, it arises from the interaction between exploration and misspecification. The exploration phase initializes a correlation structure in price histories; the misspecified model misinterprets this correlation as a demand signal, leading to a persistent upward bias in pricing.
Policy Implication: The findings suggest that regulators and market participants should be concerned about "naive" algorithmic pricing pipelines that ignore competitor data. Even without malicious intent or sophisticated AI, the standard "estimate-then-optimize" approach can systematically drive prices above competitive levels.
Theoretical Contribution: The paper bridges the gap between the theoretical unpredictability of misspecified learning (Cooper et al., 2015) and the empirical reality of algorithmic collusion by showing that the inclusion of an exploration phase creates a deterministic path to supra-competitive equilibria.

The authors remain modest regarding the scope, noting that their analytical results rely on linear demand and symmetric firms, but their simulations confirm that the qualitative effects extend to more complex, realistic market environments.