Markets are competitive if and only if P != NP

TL;DR

This paper establishes a fundamental link between computational complexity and market competitiveness, showing that if P=NP, collusion is sustainable, whereas if P!=NP, collusion becomes infeasible, explaining the rise of algorithmic collusion.

Contribution

It proves that market competitiveness is inherently tied to computational intractability, revealing a new complexity-theoretic foundation for market outcomes.

Findings

If P=NP, firms can detect collusion efficiently, enabling stable collusive agreements.

If P!=NP, collusion detection is infeasible, making collusion unstable.

Artificial intelligence enhances firms' computational power, increasing the likelihood of algorithmic collusion.

Abstract

I prove that competitive market outcomes require computational intractability. If P = NP, firms can efficiently solve the collusion detection problem, identifying deviations from cooperative agreements in complex, noisy markets and thereby making collusion sustainable as an equilibrium. If P != NP, the collusion detection problem is computationally infeasible for markets satisfying a natural instance-hardness condition on their demand structure, rendering punishment threats non-credible and collusion unstable. Combined with Maymin (2011), who proved that market efficiency requires P = NP, this yields a fundamental impossibility: markets can be informationally efficient or competitive, but not both. Artificial intelligence, by expanding firms' computational capabilities, is pushing markets from the competitive regime toward the collusive regime, explaining the empirical emergence of algorithmic collusion without explicit coordination.