### Quantum Game Theory and EWL Scheme

In a previous QETEL blog post we talked about the **Braess Paradox** and some related ideas such as **Game Theory** or **Nash Equilibrium**. Furthermore, we claimed that **Quantum Game Theory** could offer answers to the problems derived from this paradox. However, we did not explain what Quantum Game Theory is and why it can achieve better results. To do so, we will fix ideas by reasoning with a concrete game and a particular quantization scheme. The game we have chosen is the **Prisoner's Dilemma** (see Fig. 1), which to some extent can be considered as a simplified version of the Braess Paradox game. Furthermore, the selected quantization scheme is the **EWL Scheme** (see Fig. 2), named after the initials of its authors Eisert, Wilkens and Lewenstein.

To understand the existence of the dilemma, it is necessary to understand two concepts:

**Nash equilibrium**: a situation in which none of the players can improve their reward by unilaterally changing their strategy. In this game it corresponds to situation 11, because if one of the players unilaterally varies their strategy, they will go from a 1-year penalty reduction to no penalty reduction.**Pareto Optimal**: a situation in which neither player can improve their reward without worsening the reward of the other. In this game it corresponds to situation 00.

Nash equilibrium is the situation that is reached when rational players act to achieve their greatest individual benefit, as changing their strategy unilaterally will not improve their payoff. However, in the proposed game, the payoff for both players in the Nash equilibrium is lower than in the Pareto optimum. This is the paradox: that thinking selfishly about individual benefit does not achieve the best outcome for both players. This is clearly reminiscent of the traffic problem and Braess Paradox.

According to Fig. 2 we can see that the quantum circuit allows for a (quantum) computational process prior to classical decision making. The entanglement between the qubits links the decisions of each of the players, giving rise to essentially two different consequences. On the one hand, it offers the possibility of generating new Nash equilibria that take into account the general situation of the game. To illustrate this first point and for purely pedagogical interest, we can limit ourselves to the particular case of two-parameter U strategies with the general form of Fig. 5, as is done in the paper [1].

According to this article, the quantum strategy ϕ=π/2, θ =0 now becomes a new Nash Equilibrium which is also Pareto Optimal, as we can easily check in Fig. 6 by limiting ourselves to that subset of three strategies or by considering all strategies by means of the theoretical calculation. In this simple and non-general case, the Prisoner's Dilemma is thus solved.

However, on the other hand, each of the players loses their intuition about the consequences of their strategy, because now the decision strategy that results after the circuit (0 or 1) will also depend on the quantum strategies of the other players. Therefore, the decision capacity of each of the players now seems more debatable. In this regard, literature has addressed people’s ability to play quantum games [2].

Therefore, much work remains to be done to show whether quantum game theory can, provide an advantage over traditional game theory in solving real problems such as traffic and the Braess Paradox.