Bookmaking over infinite-valued events
type: Item
instance of: scholarly article
title: Bookmaking over infinite-valued events
zbMATH Open document ID: 1123.03011
published in: International Journal of Approximate Reasoning
publication date: 2007-02-19T00:00:00Z
review text: A valuation is a function \(V\) from the set of formulas to the unit interval of real numbers such that \(V(\neg\phi)= 1- V(\phi)\), \(V(\phi\oplus\psi)= \min(1, V(\phi)+ V(\psi))\), \(V(\phi\odot\psi)= \max(0, V(\phi)+ V(\psi)- 1)\). Two formulas \(\phi\), \(\psi\) are equivalent if \(V(\phi)= V(\psi)\) for any valuation \(V\). A mapping \(s\) defined on the MV-algebra \({\mathcal L}_k\) of all equivalent classes is a state if \(s(1)= 1\) and \(s(f\oplus g)= s(f)+ s(g)\) whenever \(f\oplus g= 0\). Given a finite set of formulas \(\psi_1,\dots,\psi_n\) and real numbers \(\beta_1,\dots,\beta_n\in [0,1]\), the numbers arise from a state \(s\) (i.e. \(\beta_i= s(f_{\psi_i})\), where \(f_{\psi_i}\) is the class of \({\mathcal L}_k\) obtaining \(\psi_i\)) if and only if there are no real numbers \(\sigma_i\) such that \(\sum^n_{j=1} \sigma_j(\beta_j- V(\psi_j))< 0\) for every valuation \(V\). The result solves a problem of \textit{J. Paris} [``A note on the Dutch book method'', in: G. de Cooman et al. (eds.), Proc. 2nd Int. Symp. on Imprecise Probabilities and their Applications, ISIPTA 2001. Ithaka, NY, USA: Shaker, 301--306 (2001)] and generalizes a result by \textit{B. Gerla} [Int. J. Approx. Reasoning 25, No. 1, 1--13 (2000; Zbl 0958.06007)]. The author also extends the result for infinitely many formulas and he deals with the problem of deciding if a book is Dutch.
Mathematics Subject Classification ID: 03B48
Mathematics Subject Classification ID: 03B50
Mathematics Subject Classification ID: 06D35
Mathematics Subject Classification ID: 60A05
zbMATH DE Number: 5128175
zbMATH Keywords: Dutch book
zbMATH Keywords: De Finetti coherence criterion
zbMATH Keywords: many-valued logic
zbMATH Keywords: Łukasiewicz logic
zbMATH Keywords: MV-algebra
zbMATH Keywords: state
zbMATH Keywords: finitely additive measure
zbMATH Keywords: subjective probability
author: Daniele Mundici
reviewed by: Beloslav Riečan
MaRDI profile type: Publication
full work available at URL: j.ijar.2006.04.004
OpenAlex ID: W1984784892
cites work: Algebraic foundations of many-valued reasoning
cites work: Q4094173
cites work: Q5687941
cites work: MV-algebras, multiple bets and subjective states
cites work: Q3717102
cites work: Measures in Boolean Algebras
cites work: Q5595275
cites work: Interpretation of AF \(C^*\)-algebras in Łukasiewicz sentential calculus
cites work: Farey stellar subdivisions, ultrasimplicial groups, and \(K_ 0\) of AF \(C^*\)-algebras
cites work: Averaging the truth-value in Łukasiewicz logic
cites work: Q4788820
cites work: Q5667781
cites work: A new decision method for elementary algebra
cites work: Q5807665
cites work: Lectures on Polytopes
DOI: 10.1016/J.IJAR.2006.04.004
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description: scientific article; zbMATH DE number 5128175