# Quantum Conditional Strategies and Automata for Prisoners’ Dilemmata under the EWL Scheme

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

_{W}, the GPD

_{F}, and the GPD

_{N}, which are based on the weak, the full, and the normalized prisoner’s dilemma game, respectively [48].

## 3. Definitions and Background

**Definition**

**1.**

- $N=\{1,2,\cdots ,n\}$ denotes a finite set of players;
- ${S}_{i}$ denotes the set of strategies of player i, for each player $i\in N$;
- ${u}_{i}:{S}_{1}\times {S}_{2}\times \cdots \times {S}_{n}\to \mathbb{R}$ is a function that associates each vector of strategies $s={\left({s}_{i}\right)}_{i\in N}$ with the payoff ${u}_{i}\left(s\right)$ to player i, for every player $i\in N$.

#### 3.1. Classical PD

**A set of players**N, where $N=\{1,2\}$;- for each player i, there is a set of actions ${S}_{i}=\{C,D\}$, where C stands for “cooperate” and D stands for “defect”;
- the notion of
**strategy profiles**captures the combinations of players’ strategies such as (DC), (CC), which represent all the combinations of actions that can arise based on the rules of the game. In turn, the actions of the players lead to the payoff utilities assigned to each one of these. Each player’s aim is to maximize their own payoff.

- $(D,C)$, meaning that player 1 defects and player 2 cooperates (resulting in $(t,s)$ payoff);
- $(C,C)$, where both 1 and 2 cooperate (resulting in $(r,r)$ payoff);
- $(D,D)$, where both 1 and 2 defect (resulting in $(p,p)$ payoff), and;
- $(C,D)$, where 1 cooperates and 2 defects (resulting in $(s,t)$ payoff).

#### 3.2. Quantum Computing Background

#### 3.3. PD under Quantum Rules

#### How It Is “Quantumly” Played

#### 3.4. Repeated PD and Conditional Strategies

#### 3.4.1. Tit for Tat Conditional Strategy

#### 3.4.2. Pavlov Conditional Strategy

#### 3.5. Repeated Quantum PD

- Each player has a choice to either cooperate |0⟩ $\to C$, or defect |1⟩ $\to D$ at each round of the game.
- A unitary operator (or an action), denoted by ${U}_{1}$ or ${U}_{2}$, respectively, is applied by each player to her qubits. Players are unable to communicate with each other.
- “Gates” are introduced to entangle the qubits. The actions of the players in the quantum game will result in a final state that will be a superposition of the basis kets. When the final state is measured, the payoff is determined from the table.

## 4. Quantum Automata

**Definition**

**2.**

## 5. Quantum Automata for the Classical Repeated PD

#### 5.1. Classical Pavlov in PD and Automata

#### 5.1.1. Player 1 Plays Pavlov

#### 5.1.2. Player 1 Plays Tit for Tat

#### 5.1.3. Player 1 Plays Reversed Tit for Tat

## 6. Quantum Version of Conditional Strategies

#### Strategies for the Quantum PD

## 7. Defining the Quantum “Disruptive” Conditional Strategy

**Definition**

**3.**

#### 7.1. Quantum Disruptive Strategy Against Others

#### Player 1 Plays Disruptively

#### 7.2. Changing the Action to C when the Other Played M

#### 7.3. Remarks on the “Disruptive” Conditional Strategy and the Use of Automata

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

EWL scheme | Eisert–Wilkens–Lewenstein scheme (from [2]) |

PD | Prisoner’s dilemma |

RPD | Repeated prisoner’s dilemma |

C | Cooperate |

D | Defect |

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**Figure 1.**An extensive form of the classical prisoner’s dilemma game. Two stages of the game are shown, but one can easily see how it evolves.

**Figure 2.**A depiction of the the circuit model of the quantum prisoner’s dilemma as proposed by Eisert et al. in [2].

**Figure 3.**A simple quantum periodic automaton inspired from the work in [19]. The measurement period is $\mu =5$ (the period is equal to the sum 4 + 1 of the exponents), and it accepts with a probability of 1 the language ${\left({a}^{4}b\right)}^{\mu}$ (or in case the input and the measurement mode is infinite, it accepts the ${\left({a}^{4}b\right)}^{\omega}$).

Player 2 | |||
---|---|---|---|

Cooperate (C) | Defect (D) | ||

Player 1 | Cooperate (C) | $(r,r)$ | $(s,t)$ |

Defect (D) | $(t,s)$ | $(p,p)$ |

Player 2 | |||
---|---|---|---|

Cooperate | Defect | ||

Player 1 | Cooperate | $(3,3)$ | $(0,5)$ |

Defect | $(5,0)$ | $(1,1)$ |

Action | Representation |
---|---|

Cooperation | C |

Defect | D |

Pair of Actions | Letter |
---|---|

CC | a |

CD | b |

DC | c |

DD | d |

**Table 5.**The payoff matrix for the quantum PD game, where players can choose to apply four different actions. C stands for cooperate, D for defect, M for the miracle move, and Q for the Q move.

Player 2 | |||||
---|---|---|---|---|---|

C | D | M | Q | ||

Player 1 | C | $r,r$ | $s,t$ | $(s,t)$ or $(p,p)$ | $p,p$ |

D | $t,s$ | $p,p$ | $(s,t)$ or $(p,p)$ | $s,t$ | |

M | $(t,s)$ or $(p,p)$ | $(t,s)$ or $(p,p)$ | All | $(r,r)$ or $(s,t)$ | |

Q | $p,p$ | $t,s$ | $(r,r)$ or $(t,s)$ | $r,r$ |

Player 2 | |||||
---|---|---|---|---|---|

C | D | M | Q | ||

Player 1 | C | $3,3$ | $0,5$ | $(0,5)$ or $(1,1)$ | $1,1$ |

D | $5,0$ | $1,1$ | $(0,5)$ or $(1,1)$ | $0,5$ | |

M | $(5,0)$ or $(1,1)$ | $(5,0)$ or $(1,1)$ | All | $(3,3)$ or $(0,5)$ | |

Q | $1,1$ | $5,0$ | $(3,3)$ or $(5,0)$ | $3,3$ |

Action | Representation |
---|---|

Cooperation | C |

Defect | D |

Miracle Move | M |

Q move | Q |

Pair of Actions | Letter |
---|---|

CC or QQ | a |

CD or DQ | b |

CM or DM | c |

DC or QD | d |

DD or CQ or QC | e |

QM | f |

MQ | g |

MC or MD | h |

MM | i |

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**MDPI and ACS Style**

Giannakis, K.; Theocharopoulou, G.; Papalitsas, C.; Fanarioti, S.; Andronikos, T.
Quantum Conditional Strategies and Automata for Prisoners’ Dilemmata under the EWL Scheme. *Appl. Sci.* **2019**, *9*, 2635.
https://doi.org/10.3390/app9132635

**AMA Style**

Giannakis K, Theocharopoulou G, Papalitsas C, Fanarioti S, Andronikos T.
Quantum Conditional Strategies and Automata for Prisoners’ Dilemmata under the EWL Scheme. *Applied Sciences*. 2019; 9(13):2635.
https://doi.org/10.3390/app9132635

**Chicago/Turabian Style**

Giannakis, Konstantinos, Georgia Theocharopoulou, Christos Papalitsas, Sofia Fanarioti, and Theodore Andronikos.
2019. "Quantum Conditional Strategies and Automata for Prisoners’ Dilemmata under the EWL Scheme" *Applied Sciences* 9, no. 13: 2635.
https://doi.org/10.3390/app9132635