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Indeterminacy of fair infinite lotteries

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Abstract

In ‘Fair Infinite Lotteries’ (FIL), Wenmackers and Horsten use non-standard analysis to construct a family of nicely-behaved hyperrational-valued probability measures on sets of natural numbers. Each probability measure in FIL is determined by a free ultrafilter on the natural numbers: distinct free ultrafilters determine distinct probability measures. The authors reply to a worry about a consequent ‘arbitrariness’ by remarking, “A different choice of free ultrafilter produces a different ... probability function with the same standard part but infinitesimal differences.” They illustrate this remark with the example of the sets of odd and even numbers. Depending on the ultrafilter, either each of these sets has probability 1/2, or the set of odd numbers has a probability infinitesimally higher than 1/2 and the set of even numbers infinitesimally lower. The point of the current paper is simply that the amount of indeterminacy is much greater than acknowledged in FIL: there are sets of natural numbers whose probability is far more indeterminate than that of the set of odd or the set of even numbers.

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Notes

  1. Thanks to an anonymous referee for noting (ii).

  2. In private correspondence, Sylvia Wenmackers noted that, on September 10, 2012, she was sent a draft of Timber Kerkvliet’s Master Thesis (Kervliet 2013), where Kerkvliet observes that the standard part of the probability measure \(Pr_\mathcal {U}\) “is not the same measure for every choice of \(\mathcal {U}\).” Wenmackers subsequently sent me the final version of the thesis (dated June 7, 2013), where Kerkvliet strengthens his earlier observation by noting that the standard part of \(Pr_\mathcal {U}(\{m \in \mathbb {N}: (\exists n \in \mathbb {N})(4^n \le m < 2 \cdot 4^n)\})\) “ranges from \(1/3\) to \(2/3\), completely depending on the (arbitrary) choice of \(\mathcal {U}\).” This is very close to, but slightly weaker than, Theorem 1 of the current paper. Kervliet’s main goal is not to uncover the full extent of the arbitrariness in the FIL approach, but rather to propose a nonarbitrary alternative.

  3. I owe this particular construction of the \(S_n\) to an anonymous referee: it is a huge improvement over my original construction.

References

  • Kervliet, T. (2013). A uniform probability measure on the natural numbers. Master Thesis, Department of Mathematics, University of Amsterdam, Amsterdam.

  • Wenmackers, S., & Horsten, L. (2013). Fair infinite lotteries. Synthese, 190, 37–61.

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  1. Department of Philosophy, University of Toronto, Jackman Humanities Building 170 St. George Street, Toronto, ON, M5R 2M8, Canada

    Philip Kremer

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  1. Philip Kremer
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Correspondence to Philip Kremer.

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Kremer, P. Indeterminacy of fair infinite lotteries. Synthese 191, 1757–1760 (2014). https://doi.org/10.1007/s11229-013-0364-3

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  • DOI: https://doi.org/10.1007/s11229-013-0364-3

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