# Is quantum game theory reducible to classical game theory?

Quantum game theory is an extension of classical game theory to the quantum domain. It differs from classical game theory in three primary ways:

1. Superposed initial states,
2. Quantum entanglement of initial states,
3. Superposition of strategies to be used on the initial states.

This theory is based on the physics of information much like quantum computing.

I wondered if QGT is reducible to classical GT, i.e., whether any quantum game can be transformed to some classical game.

Related issues: To prove the opposite, would we need a space-like separation between players' acts? Would one need Bell's theorem? Should players' acts be outside each other's light-cones? Do we have to appeal to physics (e.g., QM itself and/or GR)? Would we need counterfactual definiteness? Would we need to dismiss superdeterminism? Are, some or all, such issues already covered (by hidden assumptions) in classical game theory or even economics?

Can anyone perhaps point to relevant literature that specifically deals with this (the title) question?

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Does a mathematical definition of quantum games exist? –  Michael Greinecker Jul 4 '13 at 21:16

There are several versions of quantum game theory. First, there are Eisert-Wilkens type models in which players submit strategies that are quantum superpositions of classical strategies. In these models, a classical game {\bf G} is replaced by a quantum game {\bf G}, which is itself a classical game with a much larger strategy space than {\bf G}. So here the answerr to your question is: Yes, the "quantum game" is itself in fact a classical game.

Next, there are models in which players use quantum technology to randomize among (classical) strategies, where these technologies might include observations of entangled particles. As long as players have no private information, this is equivalent to allowing players to choose their strategies contingent on classical correlated random variables. (E.g. perhaps you and I each choose our strategies contingent on whether we see the sun shining; whether or not the sun shines near my house is correlated with whether or not it shines near yours.) So here all we've done by adding quantum strategies is imbed our classical game {\bf G} into a larger (classical) game. Every equilibrium in the expanded game is a correlated equilibrium in the original game (but not, in general, vice versa). Here again, the answer to your question is that a "quantum game" is nothing but a particular classical game.

However, genuinely new phenomena occur when players have access to private information, e.g. when they have only partial information about the payoff matrix (and one player's information might differ from the other's). Here each player has a set of basic strategies (say "Confess (C)" and "Not Confess" (N)), and a piece of information $x$ chosen from some information set $P$. We can then model a (classical) mixed strategy as either

a) A map from $P$ to probability distributions over $\lbrace N,C\rbrace$ or

b) A probability distribution over maps from $P$ to $\lbrace N, C\rbrace$

It turns out not to matter which model you choose; the two models (usually called "the game of behavioral strategies" and "the game of mixed strategies") are equivalent in an appropriate sense and make all the same predictions about payoffs. Whichever model you choose, you've got a new (classical) game to study.

But in the quantum environment, the equivalence between mixed and behavioral strategies breaks down, and genuinely new phenomena emerge. In this case, the quantum version of the behavioral-strategy game has equilibria that cannot be recognized as correlated equilibria in any naturally defined classical game. So in this case, the answer to your question becomes no.

In particular: If we allow players to choose a map from the information set $P$ to the set of quantum strategies over $\lbrace N,C\rbrace$ --- where a quantum strategy'' is a strategy contingent on the realization of a quantum observable --- and if the observables available to the two players are entangled --- then one gets equilibria that are not in any sense equivalent to correlated equilibria in the original classical game.

This is all spelled out in careful detail in Dahl and Landsburg, to which Carlo has already referred. But since the key diagrams are missing from that arxiv'd version (due to insane policies on the part of the arxiv), you might prefer to read it here.

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just out of curiosity: why couldn't you include this diagram as a pdf figure in your arXiv source? –  Carlo Beenakker Jul 5 '13 at 8:18
Many thanks for your answer! What is "naturally defined" here? Why may unnatural isomorphisms not be thought of as "the same" (related to footnote 4)? What would an "unreasonable analogue" be (see p. 20), and what would be unreasonable about it? –  Glen The Udderboat Jul 5 '13 at 11:15
Gugg: Once one has computed the equilibrium outcome of a quantum game, one can, of course, trivially write down some totally unrelated classical game which happens to have the same expected payoffs for both players. That's what I want to rule out when I talk about a "naturally occurring" classical game that mimics the quantum game. This is captured in the Dahl/Landsburg paper with the notion of a stochastic extension. –  Steven Landsburg Jul 5 '13 at 15:09
Carlo: The story of why the arxiv wouldn't let me include an image is much too long for a comment, but I'll be glad to send you an email. –  Steven Landsburg Jul 5 '13 at 15:10

A careful analysis was provided by Brandenburger, The relationship between quantum and classical correlation in games (2010). The answer to your question is "no", provided that the game does not allow for direct communication between the players. A simple example ("the game of cats and dogs") of the difference between a classical and a quantum game is given by Dahl and Landsburg: Quantum strategies (2011), and Quantum game theory (2011).

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