The Interplay Between Computational Complexity and Market Competition

Below is a comprehensive synthesis of the research paper arXiv:2602.20415, authored by Philip Z. Maymin and submitted on February 23, 2026.

📄 Document Metadata

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🧠 Core Thesis

The author posits a provocative link between the fundamental limits of computation and the nature of economic competition. Specifically, the paper argues that competitive market outcomes are only possible if certain computational problems remain intractable.

The Logic Flow

The relationship can be visualized as follows:

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🔍 Detailed Analysis

1. The Scenario where $P = NP$

If the complexity classes $P$ and $NP$ are equivalent ($P = NP$), the "collusion detection problem" becomes computationally efficient to solve.

• Mechanism: Firms can monitor complex, noisy market data to pinpoint exactly when a partner has deviated from a secret cooperative agreement.
• Outcome: Because cheating is easily detected, punishment is guaranteed, making collusion a sustainable equilibrium.

2. The Scenario where $P \neq NP$

Conversely, if $P \neq NP$, the ability to detect collusion is hindered.

• Condition: This holds true for markets that meet a "natural instance-hardness condition" regarding their demand structures.
• Mechanism: The computational cost of identifying deviations becomes prohibitive.
• Outcome: Punishment threats lose their credibility $\rightarrow$ collusion becomes unstable $\rightarrow$ the market remains competitive.

Key Insight: Competition is not merely a result of the number of firms, but a byproduct of the computational inability of those firms to police one another.

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⚖️ The Fundamental Impossibility

The author connects this finding to his previous work from 2011, creating a stark dichotomy.

The Conflict:

1. Market Efficiency $\implies$ requires $P = NP$.
2. Market Competition $\implies$ requires $P \neq NP$.

This leads to a mathematical paradox: $\text{Efficiency} \iff P = NP \iff \neg \text{Competition}$

Markets can be both perfectly efficient and perfectly competitive. $\rightarrow$ This is impossible.

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🤖 The Role of Artificial Intelligence

The emergence of AI is shifting the equilibrium of modern markets. By augmenting the computational power of firms, AI effectively narrows the gap between $P$ and $NP$ in practical terms.

The AI Transition Path:

• Increase in raw processing power.
• Advanced pattern recognition in noisy data.
• Ability to solve "hard" detection problems.
• Transition from competitive $\rightarrow$ collusive regimes.

This explains the rise of algorithmic collusion, where AI systems coordinate pricing and behavior without the need for explicit, human-led agreements.

Hypothetical Collusion Detection Logic

def check_collusion_deviation(market_data, agreement_params):
    # If P == NP, this function runs in polynomial time
    # regardless of the complexity of the demand structure.
    if solve_np_hard_problem(market_data, agreement_params):
        return "Deviation Detected: Trigger Punishment"
    return "Agreement Maintained"

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📜 Licensing and Access

The paper is available via arXiv and is distributed under the following license:

For further details, refer to the full 31-page manuscript at https://doi.org/10.48550/arXiv.2602.20415.