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Lexicographic probability, conditional probability, and nonstandard probability

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  • Halpern, Joseph Y.

Abstract

The relationship between Popper spaces (conditional probability spaces that satisfy some regularity conditions), lexicographic probability systems (LPS's), and nonstandard probability spaces (NPS's) is considered. If countable additivity is assumed, Popper spaces and a subclass of LPS's are equivalent; without the assumption of countable additivity, the equivalence no longer holds. If the state space is finite, LPS's are equivalent to NPS's. However, if the state space is infinite, NPS's are shown to be more general than LPS's.

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  • Halpern, Joseph Y., 2010. "Lexicographic probability, conditional probability, and nonstandard probability," Games and Economic Behavior, Elsevier, vol. 68(1), pages 155-179, January.
    • Handle: RePEc:eee:gamebe:v:68:y:2010:i:1:p:155-179
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    Cited by:

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    3. Di Tillio, Alfredo & Halpern, Joseph Y. & Samet, Dov, 2014. "Conditional belief types," Games and Economic Behavior, Elsevier, vol. 87(C), pages 253-268.
    4. Christian W. Bach & Jérémie Cabessa, 2023. "Lexicographic agreeing to disagree and perfect equilibrium," Post-Print hal-04271274, HAL.
    5. Marcus Pivato, 2014. "Additive representation of separable preferences over infinite products," Theory and Decision, Springer, vol. 77(1), pages 31-83, June.
    6. Asheim, Geir B. & Brunnschweiler, Thomas, 2023. "Epistemic foundation of the backward induction paradox," Games and Economic Behavior, Elsevier, vol. 141(C), pages 503-514.
    7. David McCarthy & Kalle Mikkola & Teruji Thomas, 2019. "Aggregation for potentially infinite populations without continuity or completeness," Papers 1911.00872, arXiv.org.
    8. Tsakas, E., 2010. "Belief hierarchies in standard state space models and epistemic equivalence of belief spaces," Research Memorandum 048, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
    9. Lee, Byung Soo, 2013. "Conditional Beliefs and Higher-Order Preferences," MPRA Paper 48551, University Library of Munich, Germany.
    10. Shuige Liu, 2018. "Ordered Kripke Model, Permissibility, and Convergence of Probabilistic Kripke Model," Papers 1801.08767, arXiv.org.
    11. Guarino, Pierfrancesco, 2020. "An epistemic analysis of dynamic games with unawareness," Games and Economic Behavior, Elsevier, vol. 120(C), pages 257-288.
    12. Asheim, Geir B. & Perea, Andres, 2005. "Sequential and quasi-perfect rationalizability in extensive games," Games and Economic Behavior, Elsevier, vol. 53(1), pages 15-42, October.
    13. Basu, Pathikrit, 2019. "Bayesian updating rules and AGM belief revision," Journal of Economic Theory, Elsevier, vol. 179(C), pages 455-475.
    14. Tsakas, Elias, 2014. "Epistemic equivalence of extended belief hierarchies," Games and Economic Behavior, Elsevier, vol. 86(C), pages 126-144.
    15. Streufert, Peter A., 2015. "An elementary proof that additive i-likelihood characterizes the supports of consistent assessments," Journal of Mathematical Economics, Elsevier, vol. 59(C), pages 37-46.
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    17. Nehring, Klaus & Pivato, Marcus, 2019. "Majority rule in the absence of a majority," Journal of Economic Theory, Elsevier, vol. 183(C), pages 213-257.
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    19. Tsakas Elias, 2018. "Agreeing to Disagree with Conditional Probability Systems," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 18(2), pages 1-7, July.
    20. Peter A. Streufert, 2012. "Additive Plausibility Characterizes the Supports of Consistent Assessments," University of Western Ontario, Departmental Research Report Series 20123, University of Western Ontario, Department of Economics.
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    22. Joseph Y. Halpern & Yoram Moses, 2017. "Characterizing solution concepts in terms of common knowledge of rationality," International Journal of Game Theory, Springer;Game Theory Society, vol. 46(2), pages 457-473, May.

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