A Quantum-Like View to a Generalized Two Players Game

Abstract

This paper consider the possibility of using some quantum tools in decision making strategies. In particular, we consider here a dynamical open quantum system helping two players, $\mathcal {G}_{1}$ and $\mathcal {G}_{2}$, to take their decisions in a specific context. We see that, within our approach, the final choices of the players do not depend in general on their initial mental states, but they are driven essentially by the environment which interacts with them. The model proposed here also considers interactions of different nature between the two players, and it is simple enough to allow for an analytical solution of the equations of motion.

Appendix: Few Results on the Number Representation

To keep the paper self-contained, we discuss here few important facts in quantum mechanics and in the so–called number representation. More details can be found, for instance, in [19, 20], as well as in[4].

Let $\mathcal {H}$ be an Hilbert space, and $B(\mathcal {H})$ the set of all the (bounded) operators on $\mathcal {H}$. Let $\mathcal {S}$ be our physical system, and $\mathfrak {A}$ the set of all the operators useful for a complete description of $\mathcal {S}$, which includes the observables of $\mathcal {S}$. For simplicity, it is convenient (but not really necessary) to assume that $\mathfrak {A}$ coincides with $B(\mathcal {H})$ itself. The description of the time evolution of $\mathcal {S}$ is related to a self–adjoint operator H = H ^† which is called the Hamiltonian of $\mathcal {S}$, and which in standard quantum mechanics represents the energy of $\mathcal {S}$. In this paper we have adopted the so–called Heisenberg representation, in which the time evolution of an observable $X\in \mathfrak {A}$ is given by

$$ X(t)=\exp(iHt)X\exp(-iHt), $$

(A.1)

or, equivalently, by the solution of the differential equation

$$ \frac{dX(t)}{dt}=i\exp(iHt)[H,X]\exp(-iHt)=i[H,X(t)], $$

(A.2)

where [A,B]:=A B−B A is the commutator between A and B. The time evolution defined in this way is a one–parameter group of automorphisms of $\mathfrak {A}$.

An operator $Z\in \mathfrak {A}$ is a constant of motion if it commutes with H. Indeed, in this case, equation (A.2) implies that $\dot Z(t)=0$, so that Z(t)=Z for all t.

In some previous applications, [4], a special role was played by the so–called canonical commutation relations. Here, these are replaced by the so–called canonical anti–commutation relations (CAR): we say that a set of operators $\{a_{\ell },~a_{\ell }^{\dagger }, \ell =1,2,\ldots ,L\}$ satisfy the CAR if the conditions

$$ \{a_{\ell},a_{n}^{\dagger}\}=\delta_{\ell n}\mathbb{1},\quad \{a_{\ell},a_{n}\}=\{a_{\ell}^{\dagger},a_{n}^{\dagger}\}=0 $$

(A.3)

hold true for all ℓ,n = 1,2,…,L. Here, $\mathbb {1}$ is the identity operator and {x,y}:=x y+y x is the anticommutator of x and y. These operators, which are widely analyzed in any textbook about quantum mechanics (see, for instance, [19, 20]) are those which are used to describe L different modes of fermions. From these operators we can construct $\hat {n}_{\ell }=a_{\ell }^{\dagger } a_{\ell }$ and $\hat {N}={\sum }_{\ell =1}^{L} \hat {n}_{\ell }$, which are both self–adjoint. In particular, $\hat {n}_{\ell }$ is the number operator for the ℓ–th mode, while $\hat {N}$ is the number operator of $\mathcal {S}$. Compared with bosonic operators, the operators introduced here satisfy a very important feature: if we try to square them (or to rise to higher powers), we simply get zero: for instance, from (A.3), we have $a_{\ell }^{2}=0$. This is related to the fact that fermions satisfy the Fermi exclusion principle [19].

The Hilbert space of our system is constructed as follows: we introduce the vacuum of the theory, that is a vector φ ₀ which is annihilated by all the operators a _ℓ : a _ℓ φ ₀ =0 for all ℓ = 1,2,…,L. Such a non zero vector surely exists. Then we act on φ ₀ with the operators $a_{\ell }^{\dagger }$ (but not with higher powers, since these powers are simply zero!):

$$ \varphi_{n_{1},n_{2},\ldots,n_{L}}:={(a_{1}^{\dagger})}^{n_{1}}{(a_{2}^{\dagger})}^{n_{2}}\cdots {(a_{L}^{\dagger})}^{n_{L}}\varphi_{\mathbf{0}}, $$

(A.4)

n _ℓ =0,1 for all ℓ. These vectors form an orthonormal set and are eigenstates of both $\hat {n}_{\ell }$ and $\hat {N}$: $\hat {n}_{\ell }\varphi _{n_{1},n_{2},\ldots ,n_{L}}=n_{\ell }\varphi _{n_{1},n_{2},\ldots ,n_{L}}$ and $\hat {N}\varphi _{n_{1},n_{2},\ldots ,n_{L}}=N\varphi _{n_{1},n_{2},\ldots ,n_{L}},$ where $N={\sum }_{\ell =1}^{L}n_{\ell }$. Moreover, using the CAR, we deduce that

$$\hat{n}_{\ell}\left(a_{\ell}\varphi_{n_{1},n_{2},\ldots,n_{L}}\right)=(n_{\ell}-1)(a_{\ell}\varphi_{n_{1},n_{2},\ldots,n_{L}}) $$

and

$$\hat{n}_{\ell}\left(a_{\ell}^{\dagger}\varphi_{n_{1},n_{2},\ldots,n_{L}}\right)=(n_{\ell}+1)(a_{l}^{\dagger}\varphi_{n_{1},n_{2},\ldots,n_{L}}), $$

for all ℓ. Then a _ℓ and $a_{\ell }^{\dagger }$ are called the annihilation and the creation operators. Notice that, in some sense, $a_{\ell }^{\dagger }$ is also an annihilation operator since, acting on a state with n _ℓ =1, we destroy that state.

The Hilbert space $\mathcal {H}$ is obtained by taking the linear span of all these vectors. Of course, $\mathcal {H}$ has a finite dimension. In particular, for just one mode of fermions, $\dim (\mathcal {H})=2$. This also implies that, contrarily to what happens for bosons, all the fermionic operators are bounded.

The vector $\varphi _{n_{1},n_{2},\ldots ,n_{L}}$ in (A.4) defines a vector (or number) state over the algebra $\mathfrak {A}$ as

$$ \omega_{n_{1},n_{2},\ldots,n_{L}}(X)= \langle\varphi_{n_{1},n_{2},\ldots,n_{L}},X\varphi_{n_{1},n_{2},\ldots,n_{L}}\rangle, $$

(A.5)

where 〈 , 〉 is the scalar product in $\mathcal {H}$. As we have discussed in [4], these states are useful to project from quantum to classical dynamics and to fix the initial conditions of the considered system.