On Sat, Feb 13, 1999 at 12:59:53AM +0100, Anders Sandberg wrote:
> Quantum Strategies
> David A. Meyer Physical Review Letters 82:5 1052--1055 1999
> Since computation can be generalized to quantum computation, why not
> generalize game theory? This paper deals with the odd things that can
> happen if you allow players to choose quantum superpositions of
> strategies. As an example Meyer describes a game between captain
> Picard of Star Trek and the superbeing Q, where they take turns of
> either flipping a hidden coin or not. Classically, this is a fair
> zero-sum game with an equilibrium of mixed strategies. But if Q makes
> quantum moves, then he can win against the classical Picard! There is
> no equilibrium if both players play deterministic quantum strategies,
> but Meyer shows that there is an equilibrium if both plays mixed
> quantum strategies (i.e. they select randomly between a number of
> quantum strategies). A fun generalization, and it ties in with quantum
> error correction (where nature is one of the players).
Also available on the web at
http://math.ucsd.edu/~dmeyer/research/publications/qstrat/abs.html. But unfortunately it is not as interesting as it first seem to me. The distinction between quantum and classical strategies is arbitrary (Picard is restricted to a choice of two quantum operations that are called classical even though there is really nothing classical about them, whereas Q is allowed the whole range of possible pure quantum operations).
The PQ Penny Flip game is isomorphic to a game where two players manipulate a point on a plane that starts at (0,1) without being able to see where it is. Each round Q is allowed to rotate the point by an arbitrary amount around the origin, but Picard is only allowed to reflect the point about the line that passes through (0,0) and (1,1). This makes it easy to see why Picard keeps losing. I think all quantum games where no decoherence is possible until the end can similarly be analyzed as equivalent classical games.
I am unable to think of real world applications of these ideas, except perhaps in literal games. Quantum chess anyone?