Abstract
Savage denied that Bayesian decision theory applies in large worlds. This paper proposes a minimal extension of Bayesian decision theory to a large-world context that evaluates an event \(E\) by assigning it a number \(\pi (E)\) that reduces to an orthodox probability for a class of measurable events. The Hurwicz criterion evaluates \(\pi (E)\) as a weighted arithmetic mean of its upper and lower probabilities, which we derive from the measurable supersets and subsets of \(E\). The ambiguity aversion reported in experiments on the Ellsberg paradox is then explained by assigning a larger weight to the lower probability of winning than to the upper probability. However, arguments are given here that would make anything but equal weights irrational when using the Hurwicz criterion. The paper continues by embedding the Hurwicz criterion in an extension of expected utility theory that we call expectant utility.
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Notes
European roulette has no 00. Bets on low or high remain on the table when 0 occurs.
Vitali’s argument needs the Axiom of Choice, without which all sets on the circle can be taken to be Lebesgue measurable (assuming “inaccessible cardinals” exist—Solovay 1970). But to deny the Axiom of Choice would seem to deny the ethos that led Savage to insist on the relevance of large worlds by assuming that our current formalism is adequate to describe anything that nature might throw at us.
His theorem has a countable set of exceptional points, but the later work of Banach and Tarski perfected his result on the way to proving their even more spectacular paradox.
All limits in Postulate 3 are taken from within \(D\). If \((0,0)\) were excluded from the set \(D\), the theorem would fail as \(v(x,y)=x^{1-\alpha }y^{\alpha }\) would remain a possibility.
\(f(x+y)=f(x)+f(y)\) implies \(f'(x+y)=f'(x)\). So \(f'(y)=f'(0)\) and \(f(y)=f'(0)y\).
The identity does not hold if \(E\) is unmeasured and \({\alpha }\not ={1 \over 2}\), because implies that \(1=(1-{\alpha })p+{\alpha }P + (1-{\alpha })(1-P)+{\alpha }(1-p) = 1+(2\alpha -1)(P-p)\), where \(p\) and \(P\) are the lower and upper probabilities of \(E\).
Postulate 9 does not even imply that a gamble can be identified with \({\mathcal{P}}\) when all the prizes are \({\mathcal{P}}\). But when \(\mathcal{M}\not =\emptyset \) and so \(B\) is a measured event, this conclusion follows from requiring that Bayesian decision theory holds for gambles constructed only from measured events.
Note that we are asking more than that \(\pi \) be a measure on \(\mathcal{M}\). In particular, although our standard normalization of VN&M utility functions ensures that \(\pi (\emptyset )=0\) and \(\pi (B)=1\) and so \(\pi \) is always a measure on \(\{\emptyset , B\}\), it need not be true that \(\{\emptyset , B\}\subseteq \mathcal{M}\). Example 5 is a case in which \(\mathcal{M}\) is taken to be empty (Sect. 2.1).
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Acknowledgments
I am grateful for funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC grant 295449. I am also grateful to David Kelsey and Karl Schlag for commenting on a previous version of the paper.
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Binmore, K. A minimal extension of Bayesian decision theory. Theory Decis 80, 341–362 (2016). https://doi.org/10.1007/s11238-015-9505-0
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DOI: https://doi.org/10.1007/s11238-015-9505-0