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Nash equilibria in quantum games

Steven E. Landsburg

2011
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Proceedings of the American Mathematical Society
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When the players in a game G can communicate with a referee via quantum technology (e.g. by sending emails composed on a quantum computer), their strategy sets naturally expand to include quantum superpositions of pure strategies. These superpositions lead to probability distributions among payoffs that would be impossible if players were restricted to classical mixed strategies. Thus the game G is replaced by a much larger "quantum game" G Q . When G is a 2 x 2 game, the strategy spaces of G Q
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... are copies of the threedimensional sphere S 3 ; therefore a mixed strategy is an arbitrary probability distribution on S 3 . These strategy spaces are so large that Nash equilibria can be difficult to compute or even to describe. The present paper largely overcomes this difficulty by classifying all mixed-strategy Nash equilibria in games of the form G Q . One result is that we can confine our attention to probability distributions supported on at most four points of S 3 ; another is that these points must lie in one of several very restrictive geometric configurations. A stand-alone Appendix summarizes the relevant background from quantum mechanics and quantum game theory. 5/5/05 In classical (i.e. "ordinary") game theory, an n-person game is characterized by n strategy spaces S i and n payoff functions Nothing in this formulation attempts to model the process by which the payoffs are actually computed, though in applications there is usually some story to be told about, say, a market mechanism or a referee who observes the strategies and calculates the payoffs. When the real world imposes limits on what referees can observe and calculate, we can incorporate those limits in the model by restricting the allowable strategy spaces and payoff functions. To take an entirely trivial example, consider a game where each player is required to play one of two pure strategies, say "cooperate" (C) and "defect" (D). 1 No mixed strategies are allowed. Although such games make perfect sense in the abstract, it's hard to see how they could ever be implemented. Player One announces "I cooperate!" How is the referee to know whether Player One arrived at this strategy through a legal deterministic process or an illegal random one? So to bring our model more in line with reality, we replace the game with a larger game, 1 I am using the words "cooperate" and "defect" as generic names for alternative strategies. I do not mean to imply that the strategy "cooperate" has anything to do with cooperation. 1 abandoning the two-point strategy space {C, D} for the space of all convex combinations of C and D, while extending the payoff function in the obvious way. Quantum game theory 2 begins with the observation that the technology of the near future is likely to dictate that much communication will occur through quantum channels, that is, through the interactions of very small particles. For example, players might communicate their strategies to the referee via email composed on quantum computers. Such communication automatically expands the player's strategy spaces in ways that cannot be prohibited. Instead of declaring either "I cooperate" or "I defect", a prisoner can send a message that is in some quantum superposition of the states "I cooperate" and "I defect". (In Section One, I will be entirely explicit about what this means; for now, I will merely note that a superposition is not in general equivalent to playing a mixed strategy.) In the quantum context, there is no way for the referee to detect this kind of "cheating" and hence no way to rule it out. We can deal with the possibility of quantum strategies just as we deal with the possibility of mixed strategies-by imbedding the original game in a larger one. So for each game we have an associated quantum game-the game that results when players' strategy spaces are expanded to include quantum superpositions, and the payoff function is extended accordingly. (Eventually, we will want to allow for mixed quantum strategies, which will require us to expand the strategy spaces still further.) There are in fact several different ways to convert a classical game to a quantum game, depending on exactly how one models the communication between players and referees. In Section One, I will 2 The idea of using quantum strategies in game theory was introduced by the physicist David Meyer in [M].

doi:10.1090/s0002-9939-2011-10838-4
fatcat:g3vvckwcp5a27jdkl7gw2elozq