A Tense Dinner: Quantizing the Prisoner's Dilemma and its Variants
By Tuan, Shehani
Game theory is a field of applied mathematics that models interactions between independent agents, called players, in games of strategy. Game theory analyzes each player’s choices and its impact on both the overall game and every other player’s choices [1]. One of the most studied games in game theory is the Prisoner’s Dilemma [1]. In the classical version, two players, called Alice and Bob, must choose to cooperate or defect. If both cooperate, they both receive one year. If both defect, they both receive five years. If one defects and the other cooperates, the defector will get 0 years and the cooperator 10. Logically, both will defect, because if the other cooperates, defecting will get them no punishment, and if the other defects, getting five years is better than getting 10. This is called a Nash equilibrium, a game state where no player can individually benefit from changing their option [1]. Even though it would be Pareto-optimal, or best overall for both to cooperate, defecting gives players the greatest payoff [2]. Quantum game theory allows players to use quantum strategies, i.e., apply gates to qubits. Often, this allows for communication purely through the player’s choices. This project will test several questions: Can a quantum player achieve a greater payoff than a classical player by using quantum strategies?. Can quantized games achieve Pareto-optimal Nash equilibria where classical variants fall short? This project will use the Eisert-Wilkins-Lewenstein (EWL) scheme to ”quantize”, or make quantum, the Prisoner’s Dilemma. We will also quantize different versions of the Dilemma, including a version where players may choose to abstain and a version with n players. We will then compare by simulating each version using Qiskit, measuring each player’s theoretical and simulated payoff.
This project will leverage the principles of quantum mechanics to analyze variants of the Prisoner’s dilemma for differences in payoffs and shifts in the Nash equilibrium. Viewing game theory through a quantum mechanical lens can give rise to new optimal strategies in these games that are absent when viewing game theory through only a classical lens [3].
Game theory models arise in several disciplines, including economics and network flow. Much of the work that has been done in the field uses a classical model. Applying quantum mechanics to these settings allows us to gain a new framework for understanding them. In addition, the use of quantum strategies in game theory may resemble real-world interactions like those occurring in finance or communication networks [3]. Finally, this project can also provide a precedent and inspiration for future QLab projects on quantum game theory, as it is one area where work is less plentiful.