![]() Let \(U(a)=n\), observe that \(U^g(a)=U(a) 1\) and, as before, let \(V(a)=\phi (U(a))\) for a bounded strictly increasing \(\phi \). The integer n is unique, since otherwise we would have \(g^n b = g^m b\) for distinct n and m. ![]() Given \(a\in \Omega \), let \(b\in E\) be such that \(a=g^n b\) for some integer n. Then \(\sim \) is an equivalence relation, and we can let E contain exactly one element from each equivalence class by Choice. Let \(a\sim b\) if and only if \(a=g^n b\) for some integer n. Let \(G=\Omega =\mathbb \omega = \omega \). Note that Proposition 2 becomes false if “countably additive” is replaced by “finitely additive”, at least given the Axiom of Choice. Our main result also has implications for social choice principles. I shall also argue that someone who accepts Williamson’s famous argument that the probability of an infinite sequence of heads is zero should accept the symmetry conditions, and thus has reason to weaken the strict dominance principle, and I shall propose a restriction of the principle to “implementable” wagers. ![]() In particular, I will show that there is a pair of wagers on the outcomes of a uniform spinner which differ simply in where the zero degrees point of the spinner is defined to be but where one wager dominates the other. I shall show that (given the Axiom of Choice) there is a contradiction between strict dominance and plausible isomorphism or symmetry conditions, by showing how in several natural cases one can construct isomorphic wagers one of which strictly dominates the other. The strict dominance principle that a wager always paying better than another is rationally preferable is one of the least controversial principles in decision theory.
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