Edited by: Faisal Shah Khan, Khalifa University, United Arab Emirates
Reviewed by: Adrian Paul Flitney, Melbourne University, Australia; Haozhen Situ, South China Agricultural University, China
Specialty section: This article was submitted to Quantum Computing, a section of the journal Frontiers in ICT
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Our purpose is to focus attention on a new criterion for quantum schemes by bringing together the notions of quantum game and game isomorphism. A quantum game scheme is required to generate the classical game as a special 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 show how this isomorphism condition influences the players’ strategy sets. We are concerned with the Marinatto–Weber type quantum game scheme and the strong isomorphism between games in strategic form.
The Marinatto–Weber (MW) scheme introduced in Marinatto and Weber (
Though it is possible to extend both the MW scheme and our refinement to consider more complex games than 2 × 2, possible generalizations can be defined in many different ways. A result concerning 3 × 3 games can be found in Iqbal and Toor (
Certainly, one can find yet other ways to generalize the MW scheme. Hence, it would be interesting to place additional restrictions on a quantum game scheme and examine how they refine the quantum model. In this paper, we formulate a criterion in terms of isomorphic games. Given two isomorphic games, we require the corresponding quantum games to be isomorphic as well. If, for example, two bimatrix games differ only in the order of players’ strategies, they describe the same problem from a game-theoretical point of view. 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.
In Frąckiewicz (
The scheme proceeds in the similar way as the MW scheme – the players determine the final state by choosing their strategies and acting on operator
Next, the payoffs for player 1 and 2 are
As it was shown in Frąckiewicz (
The notion of strong isomorphism defines classes of games that are the same up to the numbering of the players and the order of players’ strategies. The following definitions are taken from Gabarró et al. (
In general case, the mapping
The notion of game mapping is a basis for the 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 a strong isomorphism. The formal definition is as follows:
From the above definition, it may be concluded that if there is a strong isomorphism between games Γ and Γ′, they may differ merely by the numbering of players and the order of their strategies.
The following lemma shows that relabeling players and their strategies do not affect the game with regard to Nash equilibria. If
It is not hard to see that we can define various schemes based on the MW approach. We can modify operator (3) and the players’ strategies to construct another scheme still satisfying the requirement about generalization of the input game. The following example of such a scheme is particularly interesting. Let us consider a triple
It is immediate that the resulting final state
As a result, the players’ payoff functions
Since
Hence
The values
It follows easily that matrix game (17) is a genuine extension of equation (
To sum up, scheme (11) might seem to be acceptable as long as scheme (2) is acceptable. Matrix game (17) includes equation (
The corresponding isomorphism
Set
It is fairly easy to see that games (21) and (22) differ in the order of the first two strategies and the second two strategies of player 2. Thus, the games are strongly isomorphic. More formally, one can check that a game mapping
In the next section, we prove a more general result about scheme (2).
Let us now consider scheme (11). Matrix (17) in terms of input games (19) implies
With Lemma 1, we can show that games (24) and (25) are not isomorphic. Comparing the sets of pure Nash equilibria in both games, we find the equilibrium profiles
Additional criteria for a quantum game scheme may have a significant impact on the way how we generalize these schemes. It can be easily seen in the case of the MW scheme (Marinatto and Weber,
This construction can be found in Iqbal and Toor (
Both ways to generalize the MW scheme enable us to obtain the classical game. So at this level, neither equation (
Consider the MW-type approaches Γ
Then,
Set the initial state |Ψ⟩ = (1∕2)|00⟩ + (
On the other hand, replacing equation (
Then, we have
There is no pure Nash equilibrium in the first game of equations (
Example 2 shows that players’ strategy sets defined by equations (
Each
The games determine the isomorphism
Using permutation matrices leads us to formulate another generalized MW scheme. For simplicity, we confine attention to (
Let
We let
Before stating the main result of this section, we start with the observation that the MW scheme remains invariant to numbering of the players. Consider two isomorphic bimatrix games:
Clearly, the isomorphism is defined by a game mapping
Games determined by equations (
On account of Definition 3, we have
As a result,
By a similar argument, we can show that
Hence, f˜ maps
This finishes the proof.
Note that operators (42) come down to 𝟙 and
Now, we have
For fixed
By reasoning similar to equation (
It is worth noting that the converse may not be true. Given isomorphic games Γ
The theory of quantum games does not provide us with clear definitions of how a quantum game should look like. In fact, only one condition is taken into consideration. A quantum game scheme is merely required to generalize the classical game. As a result, this allows us to define a quantum game scheme in many different ways. However, various techniques to describe a game in the quantum domain can imply different quantum game results. Therefore, it would be convenient to specify that some quantum schemes work under some further restrictions. We have been working under the assumption that a quantum scheme is invariant with respect to isomorphic transformations of an input game. We have shown that this requirement may be essential tool in defining a quantum scheme. The protocol that replicates classical correlated equilibria is an example that does not satisfy our criterion. The refined definition for a quantum game scheme may also be useful to generalize protocols. Our work has shown that dependence of local unitary operators in the MW scheme on the number of strategies in a classical game is not linear. In fact, the generalized approach to
The author confirms being the sole contributor of this work and approved it for publication.
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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.