Abstract
A bookmaker is said to be non-reversible if she does not accept negative stakes on the events. A rational criterion to choose stakes on a book Γ arranged by a non-reversible bookmaker, avoids bad bets. A bad bet is a stake δ on an event φ, that gives the bettor a strictly better payoff independently of the truth values of the events involved. In this paper we study the computational complexity for the problem of deciding whether a book arranged on many-valued events, admits bad bets, or it does not. In this short paper we show our problem to be NP-complete.
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References
Bova, S., Flaminio, T.: The coherence of Łukasiewicz assessments is NP-complete. International Journal of Approximate Reasoning 51, 294–301 (2010)
Chang, C.C.: Algebraic Analysis of Many-valued Logics. Trans. Am. Math. Soc. 88, 467–490 (1958)
Cignoli, R., D’Ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of Many-valued Reasoning. Kluwer, Dordrecht (2000)
Fedel, M., Keimel, K., Montagna, F., Roth, W.: Imprecise probabilities, bets and functional analytic methods in Łukasiewicz logic (submitted)
Fedel, M.: Uncertainty, Indeterminacy and Fuzziness: A Probabilistic Approach. In: Hosni, H., Montagna, F. (eds.) Probability, Uncertainty and Rationality, Publications of the Scuola Normale Superiore. CRM series. Springer, Heidelberg (In print)
Flaminio, T., Montagna, F.: Models for Many-valued Probabilistic Reasoning. Journal of Logic and Computation, doi:10.1093/logcom/exp013
Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998)
Hájek, P.: Complexity of Fuzzy Probabilistic Logics II. Fuzzy Sets and Systems 158(23), 2605–2611 (2007)
Kühr, J., Mundici, D.: De Finetti Theorem and Borel states in [0,1]-valued Algebraic Logic. International Journal of Approximate Reasoning 46(3), 605–616 (2007)
McNaughton, R.: A Theorem about Infinite-valued Sentential Logic. J. Symb. Log. 16, 1–13 (1951)
Mundici, D.: Averaging the Truth-value in Łukasiewicz Logic. Studia Logica 55(1), 113–127 (1995)
Mundici, D.: Bookmaking over Infinite-valued Events. International Journal of Approximate Reasoning 43, 223–240 (2006)
Paris, J.B.: The uncertain reasoner’s companion, - a mathematical perspective. Cambridge University Press, Cambridge (1994)
Schrijver, A.: Theory of Linear and Integer Programming. Wiley, Chichester (1998)
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Fedel, M., Flaminio, T. (2010). On the Complexity of Non-reversible Betting Games on Many-Valued Events. In: Hüllermeier, E., Kruse, R., Hoffmann, F. (eds) Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Methods. IPMU 2010. Communications in Computer and Information Science, vol 80. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14055-6_10
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DOI: https://doi.org/10.1007/978-3-642-14055-6_10
Publisher Name: Springer, Berlin, Heidelberg
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