Searching for Nash Equilibrium in Quantum Game Theory

Quantum game theory
Oct
23
2024

As we saw in previous blog posts where we introduced Quantum Game Theory (QGT) and the EWL Scheme, quantum computation allows to reach new Nash equilibria without classical counterpart. In some cases, these new equilibria can provide better results and eventually reach the Pareto Optimum. Generally, however, discovering the quantum play that allows to reach this equilibrium remains a challenge, due to the entanglement of strategies resulting from the quantum circuit. Although in some simple cases with few players and few strategies the quantum play of the equilibrium can be found analytically, for more complex situations this technique is impractical. 

In this blog we show the results of a method that consists of introducing the possibility that in an iterative game each player can adjust the parameters of their quantum plays iteratively to optimise their own payoff through a cost function using, for example, gradient descent techniques. In other words, it consists of setting up a trainable Parametrized Quantum Circuit (PQC), like those commonly used in Quantum Machine Learning (QML), but with as many cost functions as there are players and where each player can only modify their own parameters.

The results shown in the figures below correspond to the game model, which is explained in this blog post, the Prisoner's Dilemma game with two players and two classical strategies, with a two-parameter quantum play θ and Φ (for which Nash equilibrium does exist).  The evolution of the payoff over the iterations is shown in Figure 1.

Figure 1. Evolution of the payoff over 110 iterations. We can see that the final equilibrium situation corresponds to the Quantum Nash Equilibrium (which coincides in this case with the Pareto Optimum) instead of the classical Nash equilibrium situation.

The evolution of the parameters, i.e. the strategies, is shown in Figure 2. As can be seen from the previous blog post, both players start by collaborating, which fits with the initial payoff of each player, which is 3. Player 2 initially does it faster, as it increases its payoff over player 1. However, finally both end up in a strategy very close to Q (θ=0 y Φ=π/2), thus reaching the situation of Quantum Nash Equilibrium that we were looking for, coinciding with the Pareto Optimum of the Game.

Figure 2. Evolution of the parameters θ and Φ of each player over the 110 iterations. Both end up reaching a move close to quantum play Q (θ=0 and Φ=π/2).

We can see that this method is easily generalisable to situations with more players and more strategies, conversely to analytical calculations which hardly cope with such situations. Perhaps it can help in the search for a real application of Quantum Game Theory. This is what future research will focus on, along with quantum games with more parameters for cases where there is no Nash equilibrium.

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