On the equivalence between non-factorizable mixed-strategy classical games and quantum games

Abstract

A game-theoretic setting provides a mathematical basis for analysis of strategic interaction among competing agents and provides insights into both classical and quantum decision theory and questions of strategic choice. An outstanding mathematical question is to understand the conditions under which a classical game-theoretic setting can be transformed to a quantum game, and under which conditions there is an equivalence. In this paper, we consider quantum games as those that allow non-factorizable probabilities. We discuss two approaches for obtaining a non-factorizable game and study the outcome of such games. We demonstrate how the standard version of a quantum game can be analysed as a non-factorizable game and determine the limitations of our approach.

A quantum Samaritan’s dilemma cellular automaton

“…Thus, for example, an ad hoc method based in the external parameter 0≤ k ≤1 is given in [26], and shown as follows: $\begin{array}{rlrl}{\pi }_{11}& ={false(2k-1false)}^{2}xy& {\pi }_{12}& =false(1-kfalse)xfalse(1-yfalse)+kfalse(1-xfalse)y\\ \text{and}{\pi }_{21}& =false(1-kfalse)false(1-xfalse)y+kxfalse(1-yfalse)& {\pi }_{22}& =false(1-xfalse)false(1-yfalse)+4kfalse(1-kfalse)xy.\end{array}\}$…”

Spatial correlated games

“…Non-factorizable Π may be generated from independent strategies ( x , y ) as with the ad hoc method based on an external parameter k ∈[0,1] given in [ 13 ], and shown as follows: …”

“…Iqbal et al [ 13 ] give a second method of constructing non-factorizable Π from independent strategies ( x , y ). It departs from the fact that in factorizable Π it is π 11 = xy , π 12 = x − π 11 , π 21 = y − π 11 , π 22 =1+ π 11 −( x + y ).…”

“…Then, it is proposed just to alter the form of π 11 = xy , maintaining those of the other three elements of Π as functions of π 11 . It is concluded in [ 13 ] that π 11 ( x , y )< xy is the only restriction to be imposed on π 11 ( x , y ) in order to make sure that all the elements of Π are in the [0,1] interval and sum to 1.0. The authors propose π 11 =( xy ) 2 and π 11 = x 2 y 3 as examples.…”

“…Collective games on networks have long been studied previously [ 11 , 12 ]. The novelty of this study lies in the consideration of the mechanism for correlating independent strategies given in [ 13 ], and contextualized here in § 2 .…”