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I have benefited from comments and suggestions to ancestors of this paper from many people. I am particularly grateful to Gustaf Arrhenius, Wlodek Rabinowicz, David Donaldson, Walter Bossert, Ben Bradley, Iwao Hirose, Martin Peterson and Nicolas Espinoza. I owe a special debt to an anonymous referee and the editor for several very helpful suggestions which have improved the exposition substantially. I should like to thank The Swedish Rescue Services Agency for financial support.
1.
INTRODUCTION
Consider a sequence of outcomes of descending value, A > B > C > . . . > Z. The continuity axiom of Expected Utility Theory implies, roughly, that for any certain outcome, there will be a lottery with a probability p of a gain and a probability (1-p) of a loss, which is at least as good, however small the gain and however great the loss, if p is sufficiently large. According to Larry Temkin (2001), many people - including Temkin himself - are inclined to deny the continuity axiom in certain 'extreme' cases, i.e. cases of triplets of outcomes A, B and Z, where A and B differ little in value, but B and Z differ greatly. But, Temkin argues, contrary to what many people think, rejection of the axiom of continuity in such cases implies 'deep and irresolvable difficulties for expected utility theory' (2001, p. 95). More precisely, he attempts to show that 'if one rejects the principle of continuity in some 'extreme' cases, then one must reject continuity even in the 'easy' cases for which it seems most plausible [i.e. the cases where the loss is small], reject the axiom of transitivity, or reject the principle of substitution of equivalence' (2001, p. 107, Temkin's emphasis). For if we assume continuity for 'easy' cases, we can derive continuity for the 'extreme' case by applying the axiom of substitution and the axiom of transitivity: the rejection of continuity for 'extreme' cases therefore renders the triad of continuity in 'easy' cases, the axiom of substitution and the axiom of transitivity inconsistent.
There are several problems with Temkin's argument for this alleged inconsistency. First, as Gustaf Arrhenius and Wlodek Rabinowicz (2005) have pointed out, Temkin's proof of the inconsistency is itself...