The aim of this paper is to bring together the notions of quantum game and game isomorphism. The work is intended as an attempt to introduce a new criterion for quantum game schemes. The generally accepted requirement forces a quantum scheme to generate the classical game in a particular case. Now, given a quantum game scheme and two isomorphic classical games, we additionally require the resulting quantum games to be isomorphic as well. We are concerned with the Eisert-Wilkens-Lewenstein quantum game scheme and the strong isomorphism between games in strategic form.

Sixteen years of research on quantum games have given us many ideas of how quantum games could be described. For example, we have learned from [

Two games in strategic form

It is clear that, given two strategically equivalent games

The strategy equivalence can be extended to take into account different orders of players’ strategies. This type of equivalence is included in the definition of strong isomorphism. Clearly, if, for example, two bimatrix games differ only in the order of a player’s strategies we still have the games that describe the same problem from the game-theoretical viewpoint. Given a quantum scheme, it appears reasonable to assume that the resulting quantum game will not depend on the numbering of players’ strategies in the classical game. As a result, if there is a strong isomorphism between games, we require that the quantum counterparts of these games are also isomorphic.

In order to make our paper self-contained we give the important preliminaries from game theory and quantum game theory.

First we recall the definition of strategic form game [

A game in strategic form is a triple

The notion of strong isomorphism defines classes of games that are the same up to numbering of the players and the order of players’ strategies. The following definitions are taken from [

Given

Let us consider two bimatrix games:

In general case, mapping

The notion of game mapping is a basis for definition of game isomorphism. Depending on how rich structure of the game is to be preserved we can distinguish various types of game isomorphism. One that preserves the players’ payoff functions is called the strong isomorphism. The formal definition is as follows.

Given two strategic games

From the above definition it may be concluded that if there is a strong isomorphism between games

Let

Relabeling players or their strategies does not affect a game with regard to Nash equilibria. If

Let

The proof is based on the following observation. Since

Let us consider a strategic game

Then

(

The numbers

Having specified the notion of strong isomorphism and the generalized Eisert-Wilkens-Lewenstein scheme we will now check if the isomorphism between the classically played games makes the corresponding quantum games isomorphic. We first examine the case when the players’ unitary strategies depend on two parameters. The quantum game

The generalized Prisoner’s Dilemma game and one of its isomorphic counterparts may be given by the following bimatrices:

The example given above shows that the EWL approach with the two-parameter unitary strategies may output different Nash equilibria depending on the order of players’ strategies in the classical game. This appears to be a strange feature since games (

One way to make games (

Let us consider the following three-person games:

Let us now define the EWL quantum extensions

The same conclusion can be drawn for games with arbitrary but finite number

Let

The proof follows by the same method as in Example

As the following example shows, the converse is not true in general.

Let us consider two 2 × 2 bimatrix games that differ only in the order of payoff profiles in the antidiagonal; that is,

The theory of quantum games has no rigorous mathematical structure. There are no formal axioms, definitions that would give clear directions of how a quantum game ought to look like. In fact, only one condition is taken into consideration. It says that a quantum game ought to include the classical way of playing the game. As a result, this allows us to define a quantum game scheme in many different ways. The scheme we have studied in the paper is definitely ingenious. It has made a significant contribution to quantum game theory. However, it leaves the freedom of choice of the players’ strategy sets. Our criterion for quantum strategic game schemes requires the quantum model to preserve strong isomorphism. This specifies the strategy sets to be

The author declares that they have no competing interests.

This work was supported by the Ministry of Science and Higher Education in Poland under the project Iuventus Plus IP2014 010973 in the years 2015–2017.