Quantum Game Theory
Classical game theory can be extended to the quantum regime to form “quantum games”. In such games, players are replaced with sites (particles) and strategies are replaced with their spin (a property of the particle). One significant difference when considering quantum games is the existence of a new strategy, the quantum strategy. The quantum strategy stems from the quantum principle of entanglement. Entanglement is a phenomenon in which certain particles have correlated quantities that cannot be described separate from one another. One such property is spin, the quantity used analogously to strategies in quantum games. Entanglement thus creates links between players’ strategies where they didn’t exist before. This can change the strategies in games where the players are isolated and are classically unable to confer strategically.
For example, the Prisoner’s Dilemma, a well-studied game in the classical realm, has a quantum counterpart which has also been studied extensively. The classical prisoner’s dilemma has a known Nash Equilibrium strategy: defection by both players (that is, each player implicates the other in the crime). However, the result is more complicated in the quantum case. If the two “players” are entangled, then they are not forced to choose a strategy independently, as in the classical case. In the prisoner’s dilemma, not allowing the prisoners to speak to each other is essential, as it drives both players to defect. However, in the quantum prisoner’s dilemma, the mutual “quantum strategy” (the one involving entanglement) is the Nash Equilibrium, not defection. The game is thus quite different in the quantum realm.
In both the classical realm and the quantum realm, games like the prisoner’s dilemma can be extended to the “thermodynamic limit”. The thermodynamic limit is the extension of the game to infinite players, rather than just 2. In the thermodynamic limit of the classical prisoner’s dilemma, the majority defect, which is logical, as this is the dominant strategy. However, the result is different in the quantum realm. The ratio of defection to quantum strategy depends on the amount of entanglement in the system, but if entanglement is maximum, then the defectors are always in the minority. This is interesting in that is completely contradictory to the classical solution to the dilemma.
Sources: https://arxiv.org/abs/1806.07343
https://jila.colorado.edu/news-events/articles/playing-games-quantum-entanglement