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aAeq ajaq aqeqs aqq fo Armbruster and Boege assigns positive probability attributes probability Aumann axdmexa belief hierarchies belief system chooses Colin and Matt Colin knows common knowledge common prior commonly known conditions for Nash conjectures p1 denote epistemic conditions event example expected payoff finite fo adgaxMnok formal game being played game g Harsanyi implies inba individual conjectures irrational japjo jaq^o John Harsanyi K2(GflF knovn knowledge of rationality Lemma lyonmmod matching pennies maximizes g middle box mixed strategies mutual knowledge mutually known n-person n-tuple naAa naaq seq naqM Nash equilibrium onmmod payoff functions player knows players are mutually players are rational prior in Figure probability assessments probability distribution Proof pure strategy q.eqq qDea qeqq qseN Rowena and Colin Rowena knows Rowena's conjecture Rowena's type saur^DaronD Section stochastic independence strategy choices strategy profiles Theorem theory two-person universal belief space utility function xetunm yaqq