The Intrinsic Quantum Nature of Nash Equilibrium Mixtures

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

Teoria dos jogos: conceitos, formalização matemática e aplicação à distribuição de custo conjunto Game theory: concepts, mathematical formalization and applications to the assignment of joint cost

O presente trabalho tem por objetivo a sistematização e a apresentação dos conceitos fundamentais da Teoria dos Jogos, a formalização matemática dos principais conceitos clássicos de solução de um jogo e ilustrar sua aplicação ao problema da distribuição de custo conjunto. Como produto de uma pesquisa de alguns anos, conseguimos reunir sistematicamente o conjunto de conceitos de solução de jogos donde se derivam todos conceitos encontrados nos textos atuais. Apresentamos os conceitos de solução para os chamados jogos cooperativos através do conceito de objeção e contra-objeção de forma a mostrar que os mesmos podem ser vistos como um processo de negociação.<br>This paper presents the basic concepts of Game Theory and the mathematical formalizations of classic concepts of game solutions. We surveyed the fundamental concepts from where we derive the concepts of game solutions that can be found in the modern papers. We feature the concepts to the game named cooperative-games by concepts of objection and counterobjection to show that they can be understood as a negotiation process. In addition, we show how the Game Theory can be applied to the problem of the assignment of joint cost