Millionaires' Problem with Rational Players: a Unified Approach in Classical and Quantum Paradigms

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. This paper introduces another novel concept. Both our classical and quantum protocols follow a unified approach; both use a rational third party rather than a trusted or untrusted third party, to mediate between the players. We exploit 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).