# Duality, phase structures, and dilemmas in symmetric quantum games

## Abstract

Symmetric quantum games for 2-player, 2-qubit strategies are analyzed in detail by using a scheme in which all pure states in the 2-qubit Hilbert space are utilized for strategies. We consider two different types of symmetric games exemplified by the familiar games, the Battle of the Sexes (BoS) and the Prisoners' Dilemma (PD). These two types of symmetric games are shown to be related by a duality map, which ensures that they share common phase structures with respect to the equilibria of the strategies. We find eight distinct phase structures possible for the symmetric games, which are determined by the classical payoff matrices from which the quantum games are defined. We also discuss the possibility of resolving the dilemmas in the classical BoS, PD, and the Stag Hunt (SH) game based on the phase structures obtained in the quantum games. It is observed that quantization cannot resolve the dilemma fully for the BoS, while it generically can for the PD and SH if appropriate correlations for the strategies of the players are provided.

- Authors:

- High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801 (Japan). E-mail: tsubasa@post.kek.jp
- High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801 (Japan)

- Publication Date:

- OSTI Identifier:
- 20976734

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Annals of Physics (New York); Journal Volume: 322; Journal Issue: 3; Other Information: DOI: 10.1016/j.aop.2006.05.001; PII: S0003-4916(06)00110-2; Copyright (c) 2006 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DUALITY; GAME THEORY; HILBERT SPACE; MATRICES; QUANTIZATION; QUANTUM ENTANGLEMENT; QUANTUM MECHANICS; QUBITS

### Citation Formats

```
Ichikawa, Tsubasa, and Tsutsui, Izumi.
```*Duality, phase structures, and dilemmas in symmetric quantum games*. United States: N. p., 2007.
Web. doi:10.1016/j.aop.2006.05.001.

```
Ichikawa, Tsubasa, & Tsutsui, Izumi.
```*Duality, phase structures, and dilemmas in symmetric quantum games*. United States. doi:10.1016/j.aop.2006.05.001.

```
Ichikawa, Tsubasa, and Tsutsui, Izumi. Thu .
"Duality, phase structures, and dilemmas in symmetric quantum games". United States.
doi:10.1016/j.aop.2006.05.001.
```

```
@article{osti_20976734,
```

title = {Duality, phase structures, and dilemmas in symmetric quantum games},

author = {Ichikawa, Tsubasa and Tsutsui, Izumi},

abstractNote = {Symmetric quantum games for 2-player, 2-qubit strategies are analyzed in detail by using a scheme in which all pure states in the 2-qubit Hilbert space are utilized for strategies. We consider two different types of symmetric games exemplified by the familiar games, the Battle of the Sexes (BoS) and the Prisoners' Dilemma (PD). These two types of symmetric games are shown to be related by a duality map, which ensures that they share common phase structures with respect to the equilibria of the strategies. We find eight distinct phase structures possible for the symmetric games, which are determined by the classical payoff matrices from which the quantum games are defined. We also discuss the possibility of resolving the dilemmas in the classical BoS, PD, and the Stag Hunt (SH) game based on the phase structures obtained in the quantum games. It is observed that quantization cannot resolve the dilemma fully for the BoS, while it generically can for the PD and SH if appropriate correlations for the strategies of the players are provided.},

doi = {10.1016/j.aop.2006.05.001},

journal = {Annals of Physics (New York)},

number = 3,

volume = 322,

place = {United States},

year = {Thu Mar 15 00:00:00 EDT 2007},

month = {Thu Mar 15 00:00:00 EDT 2007}

}