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The Intrinsic Quantum Nature of Nash Equilibrium Mixtures / Yohan Pelosse

Journal of Philosophical Logic

Swansea University Author: Yohan Pelosse

Abstract

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...

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Published in: Journal of Philosophical Logic
ISSN: 1573-0433
Published: 2015
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URI: https://cronfa.swan.ac.uk/Record/cronfa21659
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first_indexed 2015-05-23T02:05:04Z
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spelling 2021-01-18T17:10:25.9526263 v2 21659 2015-05-22 The Intrinsic Quantum Nature of Nash Equilibrium Mixtures 455a04210e95a07e6fbea54f2cc4d6be 0000-0001-8546-918X Yohan Pelosse Yohan Pelosse true false 2015-05-22 ECON 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. Journal Article Journal of Philosophical Logic 1573-0433 4 3 2015 2015-03-04 10.1007/s10992-015-9349-7 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. COLLEGE NANME Economics COLLEGE CODE ECON Swansea University 2021-01-18T17:10:25.9526263 2015-05-22T14:23:28.5693839 School of Management Economics Yohan Pelosse 0000-0001-8546-918X 1 0021659-19012016150736.pdf PelosseIntrinsicQuantumNaturePostprint.pdf 2016-01-19T15:07:36.1130000 Output 558317 application/pdf Accepted Manuscript true 2016-03-04T00:00:00.0000000 false
title The Intrinsic Quantum Nature of Nash Equilibrium Mixtures
spellingShingle The Intrinsic Quantum Nature of Nash Equilibrium Mixtures
Yohan, Pelosse
title_short The Intrinsic Quantum Nature of Nash Equilibrium Mixtures
title_full The Intrinsic Quantum Nature of Nash Equilibrium Mixtures
title_fullStr The Intrinsic Quantum Nature of Nash Equilibrium Mixtures
title_full_unstemmed The Intrinsic Quantum Nature of Nash Equilibrium Mixtures
title_sort The Intrinsic Quantum Nature of Nash Equilibrium Mixtures
author_id_str_mv 455a04210e95a07e6fbea54f2cc4d6be
author_id_fullname_str_mv 455a04210e95a07e6fbea54f2cc4d6be_***_Yohan, Pelosse
author Yohan, Pelosse
author2 Yohan Pelosse
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publishDate 2015
institution Swansea University
issn 1573-0433
doi_str_mv 10.1007/s10992-015-9349-7
college_str School of Management
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hierarchy_top_title School of Management
hierarchy_parent_id schoolofmanagement
hierarchy_parent_title School of Management
department_str Economics{{{_:::_}}}School of Management{{{_:::_}}}Economics
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description 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.
published_date 2015-03-04T03:35:29Z
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