Quantum Game Theory and EWL Scheme

Quantum game theory

Jul

01

2024

0

Quantum Game Theory and EWL Scheme

By QETEL

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.

Figure 1. Prisoner's Dilemma reward table for two players P1 and P2. The possible strategies for each of the prisoners are two, 0 (cooperate with the other prisoner and do not betray him) or 1 (betray the other prisoner). Depending on all possible outcomes of the game the judge will apply a certain sentence reduction to each player: 00 (both cooperate and get a sentence reduction of 3 years), 01 or 10 (one cooperates and gets no sentence reduction, the other betrays and gets a sentence reduction of 5 years) and 11 (both betray and get a sentence reduction of 1 year).

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.

Figure 2. Multiplayer EWL scheme for decision making (0 or 1) in a game such as Prisoner's Dilemma. The operation of the quantum circuit in the figure defining the EWL scheme is described as follows: each player {P1, ..., PN} is assigned a qubit, then the game judge applies a gate J to create an entangled state (the judge can set the degree of entanglement by adjusting a parameter according to Fig. 3), each player applies their strategy U (which depends on some parameters that can be set according to Fig. 4) on their qubit, the judge undoes the entanglement operation J and each qubit is measured, resulting in one of the classical possible strategies.

Figure 3. Parameter γ of entanglement and the range of values it can take.

Figure 4. Parameters that can be adjusted by the player to set their strategy.

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].

Figure 5. Here the player can adjust the parameters θ and ϕ.

Figure 6. Payoff table considering now also the quantum Q strategy.

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.

[1] Eisert, J., Wilkens, M., & Lewenstein, M. (1999). Quantum games and quantum strategies. Physical Review Letters, 83(15), 3077.

[2] Chen, K. Y., & Hogg, T. (2006). How well do people play a quantum prisoner’s dilemma?. Quantum Information Processing, 5, 43-67.