A short geometric proof that Hausdorff limits are definable in any o-minimal structure

Kocel-Cynk, Beata; Pawłucki, Wiesław; Valette, Anna

doi:10.1515/advgeom-2013-0028

Cited by 5 publications

(3 citation statements)

References 10 publications

(2 reference statements)

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“…We believe that the following proposition, which is a direct consequence of the Main Theorem in [15], can be useful in several contexts. Proposition 5.2.…”

Section: Appendixmentioning

confidence: 99%

Hidden Stochastic Games and Limit Equilibrium Payoffs

Renault,

Ziliotto

2014

Preprint

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We consider 2-player stochastic games with perfectly observed actions, and study the limit, as the discount factor goes to one, of the equilibrium payoffs set. In the usual setup where current states are observed by the players, we first show that the set of stationary equilibrium payoffs always converges. We then provide the first examples where the whole set of equilibrium payoffs diverges. The construction can be robust to perturbations of the payoffs, and to the introduction of normal-form correlation. Next we naturally introduce the more general model of hidden stochastic game, where the players publicly receive imperfect signals over current states. In this setup we present a last example where not only the limit set of equilibrium payoffs does not exist, but there is no converging selection of equilibrium payoffs. The example is symmetric and robust in many aspects, in particular to the introduction of extensive-form correlation or communication devices. No uniform equilibrium payoff exists, and the equilibrium set has full support for each discount factor and each initial state.

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“…We believe that the following proposition, which is a direct consequence of the Main Theorem in [15], can be useful in several contexts. Proposition 5.2.…”

Section: Appendixmentioning

confidence: 99%

Hidden Stochastic Games and Limit Equilibrium Payoffs

Renault,

Ziliotto

2014

Preprint

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“…Our first result shows how the Bewley Kohlberg result extends to general-sum games. Though it stems directly from previous results of the literature ( [17] and [11]), it did not appear explicitly before.…”

Section: Model and Resultsmentioning

confidence: 83%

“…if A is a finite union of sets, each of these sets being defined as the conjunction of finitely many weak or strict polynomial inequalities. The following proposition is a direct consequence of the main theorem in [11] :…”

Section: Model and Resultsmentioning

confidence: 84%

Limit Equilibrium Payoffs in Stochastic Games

Renault

Ziliotto

2020

Mathematics of OR

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Metric Geometry in a Tame Setting

Walsberg¹

2018

Bull. symb. log

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The thesis is about the topology and geometry of metric spaces definable in an o-minimal expansion R of an ordered field (R, <, +, •). A definable metric space is a pair (X, d) consisting of a definable set X ⊆ R k and a definable (R, +, <)-valued metric. If X ⊆ R k is definable and e is the restriction of the usual euclidean metric on R k to X then (X, e) is a definable metric space, in this way the geometry of definable sets may be considered as a special case of the geometry of definable metric spaces. Examples of definable metric spaces whose geometry is unlike that of any definable set are given by the hyperbolic plane (Rexpdefinable) and certain subriemannian spaces (Ran-definable). The main theorem of the thesis is the following: Let (X, d) be a definable metric space.

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A short geometric proof that Hausdorff limits are definable in any o-minimal structure

Cited by 5 publications

References 10 publications

Hidden Stochastic Games and Limit Equilibrium Payoffs

Hidden Stochastic Games and Limit Equilibrium Payoffs

Limit Equilibrium Payoffs in Stochastic Games

Metric Geometry in a Tame Setting

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