Nonmeasurable Invariant Sets

Rosenthal, John

doi:10.2307/2319743

Cited by 6 publications

(4 citation statements)

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“…The tail sets of

form a sigma-algebra, denoted by

. The sigma-algebras

and

are generally not related: neither of them includes the other [Rosenthal ( 16 ) and Blackwell and Diaconis ( 17 )]. A function from

is called tail-measurable if it is measurable with respect to

.…”

Section: Model and Main Resultsmentioning

confidence: 99%

Existence of equilibria in repeated games with long-run payoffs

Ashkenazi-Golan

Flesch

Predtetchinski

et al. 2022

Proc. Natl. Acad. Sci. U.S.A.

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Significance Nash equilibrium, of central importance in strategic game theory, exists in all finite games. Here we prove that it exists also in all infinitely repeated games, with a finite or countably infinite set of players, in which the payoff function is bounded and measurable and the payoff depends only on what is played in the long run, i.e., not on what is played in any fixed finite number of stages. To this end we combine techniques from stochastic games with techniques from alternating-move games with Borel-measurable payoffs.

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“…The tail sets of

form a sigma-algebra, denoted by

. The sigma-algebras

and

are generally not related: neither of them includes the other [Rosenthal ( 16 ) and Blackwell and Diaconis ( 17 )]. A function from

is called tail-measurable if it is measurable with respect to

.…”

Section: Model and Main Resultsmentioning

confidence: 99%

Existence of equilibria in repeated games with long-run payoffs

Ashkenazi-Golan

Flesch

Predtetchinski

et al. 2022

Proc. Natl. Acad. Sci. U.S.A.

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“…We note that the tail sigma-algebra T and the Borel sigma-algebra B are not nested. For constructions of tail sets that are not Borel, see Rosenthal [41] and Blackwell and Diaconis [8].…”

Section: Blackwell Games With Tail-measurable Payoffsmentioning

confidence: 99%

Regularity of the minmax value and equilibria in multiplayer Blackwell games

Ashkenazi-Golan¹,

Flesch²,

Predtetchinski³

et al. 2022

Preprint

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A real-valued function ϕ that is defined over all Borel sets of a topological space is regular if for every Borel set W, ϕ(W) is the supremum of ϕ(C), over all closed sets C that are contained in W, and the infimum of ϕ (O), over all open sets O that contain W.We study Blackwell games with finitely many players. We show that when each player has a countable set of actions and the objective of a certain player is represented by a Borel winning set, that player's minmax value is regular.We then use the regularity of the minmax value to establish the existence of ε-equilibria in two distinct classes of Blackwell games. One is the class of n-player Blackwell games where each player has a finite action space and an analytic winning set, and the sum of the minmax values over the players exceeds n − 1. The other class is that of Blackwell games with bounded upper semi-analytic payoff functions, history-independent finite action spaces, and history-independent minmax values. For the latter class, we obtain a characterization of the set of equilibrium payoffs.

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“…Equivalently, f i is shift-invariant if whenever two runs have the form hr and h ′ r, i.e., they only differ in the prefixes h and h ′ , then f i (hr) = f i (h ′ r). The set of shift-invariant functions is not included by, neither does it include, the set of Borel-measurable functions, see Rosenthal [25] and Blackwell and Diaconis [3].…”

Section: Expected Payoffs and Equilibriummentioning

confidence: 99%