A game theoretical perspective on the quantum probabilities associated with a GHZ state

Abstract

In the standard approach to quantum games, players' moves are local unitary transformations on an entangled state that is subsequently measured. Players' payoffs are then obtained as expected values of the entries in the payoff matrix of the classical game on a set of quantum probabilities obtained from the quantum measurement. In this paper, we approach quantum games from a diametrically opposite perspective. We consider a classical three-player symmetric game along with a known expression for a set of quantum probabilities relevant to a tripartite Einstein-Podolsky-Rosen (EPR) experiment that depends on three players' directional choices in the experiment. We define the players' moves as their directional choices in an EPR setting and then express their payoff relations in the resulting quantum game in terms of their directional choices, the entries of the payoff matrix, and the quantum probability distribution relevant to the tripartite EPR experiment.

Two-player quantum games: When player strategies are via directional choices

“…The present paper extends the approach developed in Ref. [10] towards two-player symmetric games. That is, we present a scheme for playing a two-player quantum game in which each player's strategies consist of orientating his/her unit vector along any direction in three dimensions, measurements along which generate dichotomic outcomes ±1.…”

“…More recently, a more direct approach that is suggested in Ref. [38] while considering a quantized version of a classical three-player symmetric game. Building up on an expression reported in Ref.…”

“…A tripartite EPR experiment performed with a GHZ state [10] has been used to develop a quantum version of a three-player non-cooperative game. Each player's strategic choices are three directions a, b, and c along which the dichotomic observables n • σ are measured, where n = a, b, c and σ is a vector whose components are the standard Pauli matrices.…”

“…Using explicit expressions reported by Kaszlikowski and Żukowski [11] for the quantum probabilities involved in such experiments, Ref. [10] develops a three-player quantum game with a tripartite EPR experiment as the underlying physical setup.…”

“…We obtain the set of four quantum probabilities using Eqs. ( 9) that use associations given in (10) i.e. S 1 ∼ (â, m = +1),…”