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# Quantum Decision Making 

Decision-making models are central in economics and psychology. Finding suitable models which can approach human decision-making behavior is, as expected, a very challenging task. Economics has for a long time assumed models which are based on a particular axiomatic skeleton. The axioms are “reasonable” approximations of general decision situations. 
A central starting point in preference modeling in economics consists in proving that there exists an equivalence between the preference relation of an object x over an object $y$, denoted as $x > y$, if and only if there exists a utility function $u(.)$, which maps a set of objects into $R$, such that $u(x) > u(y)$. This is not an easy task. However, as any good microeconomic theory textbook will show for a finite set of objects $X$ (to which $x$ and $y$ belong), the equivalence is relatively easy to show [[D01]](https://doi.org/10.1017/CBO9781139003261). 
In psychology, Decision-Making is regarded as the cognitive process resulting in selecting a belief or a method of action among several possible alternative options. It could be either rational or irrational. Decision-making process is a reasoning process based on assumptions of the decision-maker's values, preferences, and beliefs. Every decision-making process produces a final choice, which may or may not prompt action [[D02]](https://en.wikipedia.org/wiki/Decision-making).
One famous example of decision-making theory is the prison's dilemma. One famous example of decision-making theory is the prison's dilemma. A prisoner's dilemma is a situation where individual decision-makers always have an incentive to choose to create a less than optimal outcome for the individuals as a group.

## Prison's Dilemma
The prisoner's dilemma is a standard example of a game analyzed in game theory that shows why two completely rational individuals might not cooperate, even if it appears that it is in their best interests to do so. It was originally framed by Merrill Flood and Melvin Dresher while working at RAND in 1950. Albert W. Tucker formalized the game with prison sentence rewards and named it "prisoner's dilemma" [[D03]](https://en.wikipedia.org/wiki/Prisoner%27s_dilemma).
The game is presented as:
Two members of a criminal organization are arrested and imprisoned. Each prisoner is in solitary confinement without communicating with the other. The prosecutors lack sufficient evidence to convict the pair on the principal charge, but they have enough to convict both on a lesser charge. Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner can either betray the other by testifying that the other committed the crime or cooperate with the other by remaining silent. The possible outcomes are:
* If A and B betray the other, they serve two years in prison,
* If A betrays B, but B remains silent, A will be set free, and B will serve three years in prison,
* If A remains silent but B betrays A, A will serve three years in prison, and B will be set free,
* If A and B both remain silent, both of them will serve only one year in prison (on the lesser charge).

It is implied that the prisoners will have no opportunity to reward or punish their partner other than the prison sentences they get and that their decision by itself will not affect their reputation in the future. As betraying a partner offers a greater reward than cooperating with them, all purely rational self-interested prisoners will betray the other, meaning the only possible outcome for two purely rational prisoners is for them to betray each other, even though cooperation would yield a greater reward.

This section will show how a Prison's dilemma's game can be modeled based on a quantum walk algorithm.

## Prison's Dilemma as a quantum walk model
Iterated bipartite quantum games can be implemented in the discrete-time quantum walk on the line. This section studies a discrete-time quantum walk on a line with two particles defined as two agents. Classically, random walks with $K$ particles are equivalent to $K$ independent single-particle random walks. In the quantum case, though, a walk with $K$ particles may contain quantum correlation, thus offering a resource unavailable in the classical scenario, introducing exciting features. Also, in the case of identical particles, we have to consider the effects of quantum statistics, giving an additional feature to quantum walks that can also be exploited. 
We model the game based on two rational agents chosen from a restricted set of two-qubit unitary operations. 
In the following, we present a quantum version of the Prisoner’s Dilemma in which both players use mixed strategies as a specific example [[D04]](https://arxiv.org/abs/quant-ph/0607143). 

## Discrete-time quantum walk
The Hilbert space of a quantum walk on a line comprises two parts, $H = H_x \otimes H_c$.
$H_c$ is spanned by two orthonormal states $\ket{0_i}, \ket{1_i}$ as “coin” subspace. The spatial subspace, $H_x$, is traversed by the orthonormal set of position eigenstates, $|\ket{x_i}$, with $x ∈ Z$ labeling discrete sites on a line and we use this symbol to distinguish the position eigenstates from coin space. Repeated application $H_c$ generates the evolution. The evolution is generated by repeated application of a composite unitary operator $U$ which implements a coin operation, followed by a conditional shift in the walker's position.
We model the Prison's dilemma as a game where two walkers play, and two coins can model their decision in each step.
The quantum walk with two walkers A,B takes place in a Hilbert space $H_{AB} = H_A \otimes H_B$, where $H_A$ and $H_B$. After $N$ steps, a pure state characterized by a density operator $\rho_0 = \ket{\psi (0)}\bra{\psi (0)}$ evolves to $\rho_N = U^N \rho_0 U^{\dagger N}$ with [[D05]](http://www.phys-info.org/uploads/3/8/1/3/3813936/physreva.74.042304.pdf)

\begin{equation}
    U = S(I \otimes U_c)
\end{equation}

where $U_c$ is a unitary operation in coin subspace, $I$ is the identity and $S$ is a shift operation in $H_{AB}$.

## Model Prison's dilemma by quantum walk
We can model the game based on the quantum walk algorithm. In this model, two players are considered two walkers, and their decision is made by tossing a coin for each player. We can see a graphic example of Prison's dilemma in the following figure 




We model each condition based on the following codes.




As the figure shows, the player's decision is considered coin states where each decision confessor or being silent is considered getting Heads or Tails. When the prison decides to remain silent or confess, it will be coded as $\ket{0}$ or $\ket{1}$ respectively.
On the other hand, the years of conviction are coded by position states. Since we have four states for each prison, we need two qubits for each one. So, we have two qubits for two coins presenting the decision of each player and four qubits in position space. At first, we consider the decision coins are Hadamard coins where are presented by $U(\pi/2, 0, \pi)$, and each player's state initially is $\ket{00}$. 

Now, we are going to simulate the system:

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2022-01-20T12:42:19.000Z