A game is said to be "quantized" when the expected payoff to the player(s) is computed via the higher order randomization notion of quantum superposition followed by measurement versus the randomization notion of probability distribution. A major motivation for quantizing a game is the potential manifestation of Nash equilibria that are superior to those already available in the game. Quantum superpositions are elements of a (projective) Hilbert space which, among its properties, is an inner product space. The inner product of the Hilbert space of quantum superpositions is used here to give a geometric characterization of Nash equilibrium in quantized versions of Hawk-Dove games, a class of games to which the well known game Prisoners' Dilemma belongs.
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