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Permissible four-strategy quantum extensions of classical games

Piotr Frąckiewicz, Anna Gorczyca-Goraj, Marek Szopa

Abstract

The study focuses on strategic-form games extended in the Eisert-Wilkens-Lewenstein scheme by two unitary operations. Conditions are determined under which the pair of unitary operators, along with classical strategies, form a game invariant under isomorphic transformations of the input classical game. These conditions are then applied to determine these operators, resulting in five main classes of games satisfying the isomorphism criterion, and a theorem is proved providing a practical criterion for this isomorphism. The interdependencies between different classes of extensions are identified, including limit cases in which one class transforms into another.

Permissible four-strategy quantum extensions of classical games

Abstract

The study focuses on strategic-form games extended in the Eisert-Wilkens-Lewenstein scheme by two unitary operations. Conditions are determined under which the pair of unitary operators, along with classical strategies, form a game invariant under isomorphic transformations of the input classical game. These conditions are then applied to determine these operators, resulting in five main classes of games satisfying the isomorphism criterion, and a theorem is proved providing a practical criterion for this isomorphism. The interdependencies between different classes of extensions are identified, including limit cases in which one class transforms into another.
Paper Structure (12 sections, 5 theorems, 64 equations, 2 figures, 1 table)

This paper contains 12 sections, 5 theorems, 64 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classical game theory
  4. Eisert-Wilkens-Lewenstein quantum game scheme
  5. Determining the criteria for valid unitary matrices $U_1$ and $U_2$
  6. Permissible extensions of classical $2\times 2$ games combining four strategies
  7. Extension of the A class
  8. Extension of the B class
  9. Extension of the C class
  10. Extension of the D class
  11. Extension of the E class
  12. Conclusions

Key Result

Lemma 1

Let $(u_{1}, u_{2})$ be a payoff vector given by (generalEWLpayoff). Then do not change the payoff vector.

Figures (2)

  • Figure 1: The EWL scheme
  • Figure 2: The convergence of extensions D and E to extension A at $\theta_{1} \to 0$ and $\theta_{1} \to \pi$.

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Example 1
  • Definition 3
  • Definition 4
  • Example 2
  • Example 3
  • Lemma 1
  • Corollary 1
  • Proposition 1
  • ...and 2 more