[Paper Review] Millionaires' Problem with Rational Players: a Unified Approach in Classical and Quantum Paradigms.

Arpita Maitra, Goutam Paul+1 more|arXiv (Cornell University)|Apr 8, 2015

Quantum Mechanics and Applications32 references3 citations

TL;DR

This paper presents a unified classical and quantum protocol for the millionaires' problem with rational players, using an untrusted third party and an interlocking mechanism to enforce fairness, correctness, and strict Nash equilibrium—eliminating the need for an online dealer. It demonstrates that fairness is achievable even when players act rationally, unlike in prior malicious-player models where fairness fails.

ABSTRACT

A seminal result of Cleve (STOC 1986) showed that fairness, in general, is impossible to achieve in case of two-party computation if one of them is malicious. Gordon et al. (STOC 2008) observed that there exist some functions for which fairness can be achieved even though one of the two parties is malicious. One of the functions considered by Gordon et al. is exactly the millionaires' problem (Yao, FOCS 1982) or, equivalently, the `greater than' function. The problem deals with two millionaires, Alice and Bob, who are interested in finding who amongst them is richer, without revealing their actual wealth to each other. We, for the first time, study this problem in presence of rational players. In particular, we show that Gordon's protocol no longer remains fair when the players are rational. Next, we design a protocol with rational players, that not only achieves fairness, but also achieves correctness and strict Nash equilibrium for natural utilities. We, also for the first time, provide a solution to the quantum version of millionaires' problem with rational players, and it too achieves fairness, correctness and strict Nash equilibrium. Both our classical and quantum protocols follow an unified approach; both uses an untrusted third party (UTP) and exploits the idea of interlocking system between the players, to prevent the deviating party to abort early. In both the protocols, we remove the requirement of the online dealer of Groce et al. (EUROCRYPT 2012).

Motivation & Objective

• To address the limitations of existing protocols in the rational player model, where fairness breaks down even for functions like the millionaires' problem.
• To design a protocol that achieves fairness, correctness, and strict Nash equilibrium in the presence of rational (self-interested) players.
• To extend the solution to the quantum setting, ensuring similar guarantees in the quantum paradigm.
• To eliminate the reliance on an online dealer, as required in prior work by Groce et al.

Proposed method

• Introduce an untrusted third party (UTP) to mediate the protocol without requiring trust in the third party.
• Implement an interlocking system between Alice and Bob, where each player’s action is conditionally dependent on the other’s, preventing early abortion.
• Design utility functions that incentivize truthful behavior, ensuring strict Nash equilibrium in both classical and quantum settings.
• Use quantum entanglement and quantum operations in the quantum protocol to maintain security and fairness under rational behavior.
• Adapt the classical protocol’s structure to the quantum domain using quantum circuits and measurement-based verification.
• Remove the need for an online dealer by relying on pre-shared entanglement and classical coordination via the UTP.

Experimental results

Research questions

• RQ1 Can fairness be achieved in the millionaires' problem when players are rational rather than malicious?
• RQ2 How can correctness and strict Nash equilibrium be ensured in a rational player model for this problem?
• RQ3 Can a unified classical and quantum protocol be designed that maintains fairness and strategic incentives?
• RQ4 What role does an untrusted third party play in enabling fairness without requiring trust?
• RQ5 How can the dependency on an online dealer be eliminated in rational protocols?

Key findings

• The proposed classical protocol achieves fairness, correctness, and strict Nash equilibrium for rational players in the millionaires' problem using an untrusted third party and interlocking mechanisms.
• The quantum protocol similarly ensures fairness, correctness, and strict Nash equilibrium, extending the result to the quantum domain.
• The interlocking system prevents either player from aborting early, as doing so would result in a loss of utility, thus enforcing cooperation.
• The protocol removes the need for an online dealer, as required in Groce et al.’s construction, by relying on pre-shared resources and UTP coordination.
• Both classical and quantum protocols follow a unified design, demonstrating a general framework applicable across paradigms.
• The utility design ensures that rational players have no incentive to deviate, maintaining strategic stability.