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The Intrinsic Quantum Nature of Nash Equilibrium Mixtures
Journal of Philosophical Logic
Swansea University Author: Yohan Pelosse
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Every undergraduate textbook in game theory has a chapter discussing the difficulty to interpret the mixed Nash equilibrium strategies. Unlike the usual suggested interpretations made in those textbooks, here we prove that these randomised strategies neither imply that players use some coin flips to...
|Published in:||Journal of Philosophical Logic|
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Every undergraduate textbook in game theory has a chapter discussing the difficulty to interpret the mixed Nash equilibrium strategies. Unlike the usual suggested interpretations made in those textbooks, here we prove that these randomised strategies neither imply that players use some coin flips to make their decisions, nor that the mixtures represent the uncertainty of each player about the others' actions.Instead, the paper demonstrates a fundamental connection between the Nash equilibrium 'randomised' or 'mixed' strategies of classical game theory and the pure quantum states of quantum theory in physics. This link has some key consequences for the meaning of randomised strategies:In the main theorem, I prove that in every mixed Nash equilibrium, each player state of knowledge about his/her own future rational choices is represented by a pure quantum state. This indicates that prior making his/her actual choice, each player must be in a quantum superposition over her/his possible rational choices (in the support of his probability measure). This result notably permits to show that the famous 'indifference condition' that must be satisfied by each player in an equilibrium is actually the condition that ensures each player is in a 'rational epistemic state of ignorance' about her/his own future choice of an action.
This paper uses techniques from epistemic game theory in logic as well as key notions and results in the foundation of quantum mechanics in physics. Knowledge of results in classical game theory, basic understanding of the logical Kripke frames and some basic understanding of some foundational results in quantum theory (e.g, Gleason's theorem) are essential to fully understand the paper.
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