Non-Classical Rules in Quantum Games

. 2021 May 13;23(5):604. doi: 10.3390/e23050604

Mixed extension

Γ = (N, {(Σ_{i})}_{i \in N}, {(u_{i}^{'})}_{i \in N})

of a

2 \times 2 \times \dots \times 2

game

N = {1, 2, \dots, n}

Σ_{i} = \{σ_{i} : \{s_{0}^{i}, s_{1}^{i}\} \to [0, 1] : σ_{i} (s_{0}^{i}) + σ_{i} (s_{1}^{i}) = 1\}

u_{i}^{'} (σ_{1}, σ_{2}, \dots, σ_{n}) = \sum_{j_{1}, \dots, j_{n} \in {0, 1}} u_{i} (s_{j_{1}}^{1}, s_{j_{2}}^{2}, \dots, s_{j_{n}}^{n}) σ_{1} (s_{j_{1}}^{1}) σ_{2} (s_{j_{2}}^{2}) \dots σ_{n} (s_{j_{n}}^{n})

The EWL extension

Γ_{E W L} = (N, {(D_{i})}_{i \in N}, {(v_{i})}_{i \in N})

of a

2 \times 2 \times \dots \times 2

game

N = {1, 2, \dots, n}

D_{i} \subset SU (2) = \{U (θ, α, β) : θ \in [0, π], α, β \in [0, 2 π)\}

v_{i} (U_{1}, U_{2}, \dots, U_{n}) = \sum_{j_{1}, \dots, j_{n} \in {0, 1}} u_{i} (s_{j_{1}}^{1}, s_{j_{2}}^{2}, \dots, s_{j_{n}}^{n}) {| 〈 Ψ | j_{1}, \dots, j_{n} 〉 |}^{2}