Quantum version of the Monty Hall problem

Abstract

A version of the Monty Hall problem is presented where the players are permitted to select quantum strategies. If the initial state involves no entanglement the Nash equilibrium in the quantum game offers the players nothing more than can be obtained with a classical mixed strategy. However, if the initial state involves entanglement of the qutrits of the two players, it is advantageous for one player to have access to a quantum strategy while the other does not. Where both players have access to quantum strategies there is no Nash equilibrium in pure strategies, however, there is a Nash equilibrium in quantum mixed strategies that gives the same average payoff as the classical game.

Maximum Entropy and Bayesian Inference for the Monty Hall Problem

“…Preceding investigations [1]- [11] into the Monty Hall problem focused predominantly on the winning chances of sticking and switching, which arises from differences between free choice (unconditional probability) and restricted choice (conditional probability). Depicted in Figure 1 are results from the so-called free choice and restricted choice scenarios.…”

“…In contrast, the "coin-flipping" procedure [1] [2] is shown to be unjustified for an uneven variant of the Monty Hall problem. Our findings not only completely resolve the Monty Hall problem and its variants, but also have applications in the decision making, information theory [14], bioinformatics [15]- [17], quantum game theory [11] [12] and beyond.…”

“…What is the rational response in the case where the host adopts a strategy that minimizes the winning chance of the contestant [11]- [13]? To answer this question, we need to consider the connection between the strategies chosen by both the host and the contestant.…”

“…What if, given a choice, Monty prefers to open one door over another? Given that the frequentist approach is used exclusively in the literature [1]- [11], it becomes necessary to develop Bayesian approaches for the Monty Hall problem.…”

Coalitions in the quantum Minority game: Classical cheats and quantum bullies

“…Quantum game theory was initially applied to classical game situations [1,2,3,4,5,6,7,8] and to markets [9,10], the traditional motivator of game theory [11]. In addition, it has been used to shed light on the role of entanglement [12,13,14,15,16], decoherence and noise [17,18,19,20,21], and quantum correlations [22,23,24,25], among other phenomena [26,27,28].…”