Linear Core-Based Criterion for Testing Extreme Exact Games

Milan Studený, Václav Kratochvı́l
Proceedings of the Tenth International Symposium on Imprecise Probability: Theories and Applications, PMLR 62:313-324, 2017.

Abstract

The notion of a (discrete) coherent lower probability corresponds to a game-theoretical concept of an exact (cooperative) game. The collection of (standardized) exact games forms a pointed polyhedral cone and the paper is devoted to the extreme rays of that cone, known as extreme exact games. A criterion is introduced for testing whether an exact game is extreme. The criterion leads to solving simple linear equation systems determined by (the vertices of) the core polytope (of the game), which concept corresponds to the notion of an induced credal set in the context of imprecise probabilities. The criterion extends and modifies a former necessary and sufficient condition for the extremity of a supermodular game, which concept corresponds to the notion of a 2-monotone lower probability. The linear condition we give in this paper is shown to be necessary for an exact game to be extreme. We also know that the condition is sufficient for the extremity of an exact game in an important special case. The criterion has been implemented on a computer and we have made a few observations on basis of our computational experiments.

Cite this Paper

BibTeX
@InProceedings{pmlr-v62-studený17a, title = {Linear Core-Based Criterion for Testing Extreme Exact Games}, author = {Studený, Milan and Kratochvı́l, Václav}, booktitle = {Proceedings of the Tenth International Symposium on Imprecise Probability: Theories and Applications}, pages = {313--324}, year = {2017}, editor = {Antonucci, Alessandro and Corani, Giorgio and Couso, Inés and Destercke, Sébastien}, volume = {62}, series = {Proceedings of Machine Learning Research}, month = {10--14 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v62/studený17a/studený17a.pdf}, url = {https://proceedings.mlr.press/v62/studen%C3%BD17a.html}, abstract = {The notion of a (discrete) coherent lower probability corresponds to a game-theoretical concept of an exact (cooperative) game. The collection of (standardized) exact games forms a pointed polyhedral cone and the paper is devoted to the extreme rays of that cone, known as extreme exact games. A criterion is introduced for testing whether an exact game is extreme. The criterion leads to solving simple linear equation systems determined by (the vertices of) the core polytope (of the game), which concept corresponds to the notion of an induced credal set in the context of imprecise probabilities. The criterion extends and modifies a former necessary and sufficient condition for the extremity of a supermodular game, which concept corresponds to the notion of a 2-monotone lower probability. The linear condition we give in this paper is shown to be necessary for an exact game to be extreme. We also know that the condition is sufficient for the extremity of an exact game in an important special case. The criterion has been implemented on a computer and we have made a few observations on basis of our computational experiments.} }
Endnote
%0 Conference Paper %T Linear Core-Based Criterion for Testing Extreme Exact Games %A Milan Studený %A Václav Kratochvı́l %B Proceedings of the Tenth International Symposium on Imprecise Probability: Theories and Applications %C Proceedings of Machine Learning Research %D 2017 %E Alessandro Antonucci %E Giorgio Corani %E Inés Couso %E Sébastien Destercke %F pmlr-v62-studený17a %I PMLR %P 313--324 %U https://proceedings.mlr.press/v62/studen%C3%BD17a.html %V 62 %X The notion of a (discrete) coherent lower probability corresponds to a game-theoretical concept of an exact (cooperative) game. The collection of (standardized) exact games forms a pointed polyhedral cone and the paper is devoted to the extreme rays of that cone, known as extreme exact games. A criterion is introduced for testing whether an exact game is extreme. The criterion leads to solving simple linear equation systems determined by (the vertices of) the core polytope (of the game), which concept corresponds to the notion of an induced credal set in the context of imprecise probabilities. The criterion extends and modifies a former necessary and sufficient condition for the extremity of a supermodular game, which concept corresponds to the notion of a 2-monotone lower probability. The linear condition we give in this paper is shown to be necessary for an exact game to be extreme. We also know that the condition is sufficient for the extremity of an exact game in an important special case. The criterion has been implemented on a computer and we have made a few observations on basis of our computational experiments.
APA
Studený, M. & Kratochvı́l, V.. (2017). Linear Core-Based Criterion for Testing Extreme Exact Games. Proceedings of the Tenth International Symposium on Imprecise Probability: Theories and Applications, in Proceedings of Machine Learning Research 62:313-324 Available from https://proceedings.mlr.press/v62/studen%C3%BD17a.html.

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