A quantum protocol for sampling correlated equilibria unconditionally and without a mediator

A correlated equilibrium is a fundamental solution concept in game theory that enjoys many desirable properties. However, it requires a trusted mediator, which is a major drawback in many practical applications. A computational solution to this problem was proposed by Dodis, Halevi and Rabin. They extended the original game by adding an initial communication stage and showed that any correlated strategy for 2-player games can be achieved, provided that the players are computationally bounded. In this paper, we show that if the players can communicate via a quantum channel before the game, then any correlated equilibrium for 2-player games can be achieved, without a trusted mediator and unconditionally. This provides another example of a major advantage of quantum information processing. More precisely, we prove that for any correlated equilibrium p of a strategic game G, there exists an extended game (with a quantum communication initial stage) Q with an efficiently computable approximate Nash equilibrium q, such that the expected payoff for both players in q is at least as high as in p. The main cryptographic tool used in the construction is the quantum weak coin flipping protocol of Mochon.

💡 Research Summary

The paper addresses a fundamental limitation of correlated equilibria (CE) in game theory: the need for a trusted mediator to sample from the joint distribution that defines the equilibrium. While Dodis, Halevi, and Rabin (DHR, 2000) showed that a computationally bounded pair of players can achieve any CE by adding a pre‑game communication phase that relies on classical cryptographic primitives (essentially Oblivious Transfer), their solution is conditional on computational hardness assumptions and does not provide an unconditional guarantee.

The authors propose a completely unconditional protocol that replaces the classical communication channel with a quantum one. The key quantum primitive they exploit is Mochon’s quantum weak‑coin‑flipping protocol, which allows two parties with opposite preferences (one prefers “heads”, the other “tails”) to generate a coin toss whose bias can be made arbitrarily small (ε) regardless of how a cheating party deviates. This primitive is information‑theoretically secure and does not require any computational assumptions.

The construction proceeds in three stages, forming an extended game Q that incorporates the original strategic game G:

1. Quantum Communication Stage – The players repeatedly invoke the weak‑coin‑flipping protocol to sample a joint strategy s from the distribution p that defines a desired CE of G. Unlike DHR, privacy of each player’s individual move is not preserved; after this stage both players know the entire sampled pair (s₁, s₂). The authors argue that this relaxation is harmless because the weak‑coin‑flipping protocol already limits any cheating advantage to at most ε.

2. Game Stage – The players play the original game G, ideally following the suggested moves s₁ and s₂. If a player deviates, the deviation will be detected in the next stage.

3. Checking Stage – Each player announces either “Accept” or “Reject”. If both announce Accept, the payoffs from G are awarded; otherwise both receive zero. The checking stage need not be simultaneous; it simply enforces a punishment for deviation.

The main theorem states that for any CE p of G there exists such an extended game Q with a Nash equilibrium σ whose expected payoffs for both players are at least those of p. Moreover, σ can be computed efficiently (the protocol is polynomial‑time) and the equilibrium is an ε‑approximate Nash equilibrium, where ε can be made arbitrarily small by choosing a sufficiently precise weak‑coin‑flipping sub‑routine.

The authors compare their approach with DHR. DHR’s protocol achieves privacy of each player’s move by using a blindable encryption scheme, which is essentially as strong as Oblivious Transfer. This requires heavy cryptographic assumptions and only yields a computational equilibrium. By abandoning privacy and using the weaker primitive of weak‑coin‑flipping, the quantum protocol eliminates all computational assumptions while still deterring cheating: any attempt to bias the coin yields at most an ε gain, and any deviation in the game stage is punished by the zero payoff in the checking stage.

The paper also discusses when the checking stage may be unnecessary. If the optimal CE of G places its support only on pure Nash equilibria, then knowing the opponent’s suggested move does not create an incentive to deviate, because the suggested profile is already a Nash equilibrium. In many practical settings—such as sports where breaking a pre‑agreed rule automatically results in a loss—the checking stage is implicitly enforced.

In conclusion, the work demonstrates that quantum communication alone suffices to implement any correlated equilibrium for two‑player strategic games without a mediator and without any computational assumptions. This provides a concrete illustration of a quantum advantage in mechanism design and opens avenues for extending the technique to multi‑player games, extensive‑form games, and quantum‑enhanced economic protocols.