## 1. Introduction

## 2. Preliminaries on Games with Unawareness

#### 2.1. Introductory Example

#### 2.2. Strategic-Form Games with Unawareness

**Definition**

**1.**

- 1.
- For every $v\in \mathcal{V}$,$$v^\mathrm{v}\in \mathcal{V}\phantom{\rule{3.33333pt}{0ex}}ifandonlyif\phantom{\rule{3.33333pt}{0ex}}\mathrm{v}\in {N}_{v}.$$
- 2.
- For every $v^\tilde{v}\in \mathcal{V}$,$$v\in \mathcal{V},\phantom{\rule{1.em}{0ex}}\varnothing \ne {N}_{v^\tilde{v}}\subset {N}_{v},\phantom{\rule{1.em}{0ex}}\varnothing \ne {({S}_{i})}_{v^\tilde{v}}\subset {({S}_{i})}_{v}\phantom{\rule{3.33333pt}{0ex}}forall\phantom{\rule{3.33333pt}{0ex}}i\in {N}_{v^\tilde{v}}.$$
- 3.
- If $v^\mathrm{v}^\overline{v}\in \mathcal{V}$, then$$v^\mathrm{v}^\mathrm{v}^\overline{v}\in \mathcal{V}\phantom{\rule{3.33333pt}{0ex}}and\phantom{\rule{3.33333pt}{0ex}}{G}_{v^\mathrm{v}^\overline{v}}={G}_{v^\mathrm{v}^\mathrm{v}^\overline{v}}.$$
- 4.
- For every strategy profile ${(s)}_{v^\tilde{v}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{\left\{{s}_{j}\right\}}_{j\in {N}_{v^\tilde{v}}}$, there exists a completion to a strategy profile ${(s)}_{v}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{\{{s}_{j},{s}_{k}\}}_{j\in {N}_{v^\tilde{v}},k\in {N}_{v}\backslash {N}_{v^\tilde{v}}}$ such that$${({u}_{i})}_{v^\tilde{v}}({(s)}_{v^\tilde{v}})={({u}_{i})}_{v}({(s)}_{v}).$$

#### 2.3. Extended Nash Equilibrium

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Proposition**

**1.**

- 1.
- σ is rationalizable for G if and only if ${(\sigma )}_{v}=\sigma $ is part of an extended rationalizable profile in ${\left\{{G}_{v}\right\}}_{v\in \mathcal{V}}$.
- 2.
- σ is a Nash equilibrium for G if and only if ${(\sigma )}_{v}=\sigma $ is part of on an ENE for ${\left\{{G}_{v}\right\}}_{v\in \mathcal{V}}$ and this ENE also satisfies ${(\sigma )}_{v}={(\sigma )}_{v^\overline{v}}$.

**Remark**

**1.**

#### 2.4. The Role of the Notion of Games with Unawareness in Quantum Game Theory

## 3. Eisert–Wilkens–Lewenstein Scheme

#### 3.1. Construction

- ${D}_{i}$ is a set of unitary operators, ${D}_{i}\subset \mathsf{SU}(2)$. The commonly used parameterization for $U\in SU(2)$ is given by$$U(\theta ,\alpha ,\beta )=\left(\begin{array}{cc}{e}^{i\alpha}cos\frac{\theta}{2}& i{e}^{i\beta}sin\frac{\theta}{2}\\ i{e}^{-i\beta}sin\frac{\theta}{2}& {e}^{-i\alpha}cos\frac{\theta}{2}\end{array}\right),\theta \in [0,\pi ],\alpha ,\beta \in [0,2\pi ).$$Each set ${D}_{i}$ is assumed to include $\{U(\theta ,0,0)\mid \theta \in [0,\pi \left]\right\}$. Elements ${U}_{i}\in {D}_{i}$ play the role of strategies of Player i. Each Player i, by choosing ${U}_{i}\in {D}_{i}$, determines the final state $|\Psi \rangle $ according to the following formula:$$|\Psi \rangle ={J}^{\u2020}\left(\underset{i=1}{\overset{n}{\u2a02}}{U}_{i}({\theta}_{i},{\alpha}_{i},{\beta}_{i})\right)J{|0\rangle}^{\otimes n},\phantom{\rule{3.33333pt}{0ex}}\mathrm{where}\phantom{\rule{3.33333pt}{0ex}}J=\frac{1}{\sqrt{2}}({\mathrm{\U0001d7d9}}^{\otimes n}+i{\sigma}_{x}^{\otimes n}).$$
- ${u}_{i}^{\ast}$ is Player i’s payoff function. It is defined as the average value of the observable ${M}_{i}$,$${M}_{i}=\sum _{{j}_{1},\dots ,{j}_{n}\in \{0,1\}}{a}_{{j}_{1},\dots ,{j}_{n}}^{i}|{j}_{1},\dots ,{j}_{n}\rangle \langle {j}_{1},\dots ,{j}_{n}|.$$The numbers ${a}_{{j}_{1},\dots ,{j}_{n}}^{i}$ are Player i’s payoffs in G such that ${a}_{{j}_{1},\dots ,{j}_{n}}^{i}={u}_{i}({s}_{{j}_{1}}^{1},\dots ,{s}_{{j}_{n}}^{n})$. The function ${u}_{i}^{\ast}$ may be written as$${u}_{i}^{\ast}\left(\underset{i=1}{\overset{n}{\u2a02}}{U}_{i}({\theta}_{i},{\alpha}_{i},{\beta}_{i})\right)=\mathrm{tr}(|\Psi \rangle \langle \Psi |{M}_{i}).$$

#### 3.2. Quantum Counterparts of Classical Strategies

#### 3.3. Nash Equilibria in Eisert–Wilkens–Lewenstein-Type Game

**Lemma**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Definition**

**6.**

**Proposition**

**3.**

**Corollary**

**1.**

**Proof.**

## 4. Quantum Games with Unawareness

- the set of relevant views $\mathcal{V}$ is equal to the set of all potential views, i.e.,$$\mathcal{V}=\bigcup _{n=0}^{\infty}{N}^{(n)},\phantom{\rule{3.33333pt}{0ex}}\mathrm{where}\phantom{\rule{3.33333pt}{0ex}}{N}^{(n)}=\prod _{j=1}^{n}N,$$
- for all $v\in \mathcal{V}$$${({D}_{i})}_{v}\in \{\{\mathrm{\U0001d7d9},i{\sigma}_{x}\},\mathsf{SU}(2)\},$$$$if\phantom{\rule{3.33333pt}{0ex}}{({D}_{i})}_{v}=\{\mathrm{\U0001d7d9},i{\sigma}_{x}\}\phantom{\rule{3.33333pt}{0ex}}\mathrm{then}\phantom{\rule{3.33333pt}{0ex}}{({D}_{i})}_{v^\tilde{v}}={({D}_{i})}_{v},$$
- for $v^\mathrm{v}^\mathrm{v}^\tilde{v}\in \mathcal{V}$,$${G}_{v^\mathrm{v}^\mathrm{v}^\tilde{v}}={G}_{v^\mathrm{v}^\tilde{v}}$$
- for $i\in N$, $v\in \mathcal{V}$ and $\tau \in {\u2a02}_{i=1}^{n}{({D}_{i})}_{v}$,$${({u}_{i}^{\ast})}_{v}(\tau )={({u}_{i}^{\ast})}_{\varnothing}(\tau ).$$

**Proposition**

**4.**

**Proof.**

**Example**

**1.**

**Example**

**2.**

## 5. Conclusions

## Funding

## Conflicts of Interest

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