Resolving the small improvement argument: a defense of the axiom of completeness

This paper defends an assumption in utility theory, specifically the assumption that agents either hold a strict preference over two options, or that they are indifferent between them. I will call this sort of comparison 'trimodal'. This assumption of comparability will be defended against a particular sort of objection, the small improvement argument (SIA), perhaps most famously presented by Ronald De Sousa as the problem of the 'fairly virtuous wife' (1974, 544).1 The fairly virtuous wife appears to be indifferent "between keeping her virtue for nothing and losing it in Cayucos for $1,000" (1974, 545). The fairly virtuous wife, however, also appears to be indifferent between keeping her virtue and losing it for $1,500, which presents a problem for utility theorists. For them "indifference, like preference, in terms of which it is defined, is a transitive relation" (De Sousa 1974, 545), and the wife's rankings are a case of preference-indifference intransitivity.2 While it is a failure of the assumption of transitivity that brings the problem into focus, De Sousa holds that the options presented to the fairly virtuous wife are actually incomparable. In general, philosophers have shared this interpretation of the choice problem. Joseph Raz, for example, refers to failures of transitivity as "the mark of incommensurability" (1985, 120).The structure of the SIA exposes an inconsistency between the assumption of completeness and the assumption of transitivity. If the wife's deliberative stances-by which I mean her attitudes about the two options (absent any particular theoretical account of choice being applied to those attitudes)-are understood as preference rankings, then those rankings are intransitive and leave the wife vulnerable to a money pump. The response to that apparent inconsistency between completeness and transitivity advanced in this paper involves the claim that agents may consider an option in more or less detail depending on what that option is being compared to. I will argue that the objection to comparability illustrated by the SIA can be answered without abandoning a trimodal approach to explaining choice, provided that the approach also assumes that agents are able to resolve3 choice options at finer or coarser grains-which is to say, that the number of details considered when assessing a choice option and, importantly, the precision with which agents consider those details may vary.4 I will also argue that the costs to agents of making comparisons will vary depending on the resolution at which the comparisons are made. For example, the representation of an outcome as "I receive a bag of oranges" is less finely resolved than the representation of an outcome as "I receive a bag containing 11 oranges"; a fortiori, generating that less finely resolved representation is less costly as I do not have to count the oranges.I will explain the process of resolving in the context of an axiomatic, subjective, folk psychological theory of rational choice, and will provide an account of that utility theory below. However, this paper is not meant to provide a tout court defense of comparability but, rather, a response to a very specific sort of objection particular to the SIA. And, as many examples of the SIA, like De Sousa's, conflate that specific sort of objection with various other objections to comparability, I first want to isolate the particular problem I mean to solve.THE PARTICULAR PROBLEM POSED BY THE SIAIn examples such as De Sousa's, at least part of the reason for focusing the objection on the assumption of comparability-rather than on the assumption of transitivity-is the idea that the two options are "qualitatively different" (De Sousa 1974, 545). Sinnot-Armstrong illustrates preference-indifference intransitivity using choices between death and amounts of pain, and the problem is often illustrated via choice situations between various sorts of careers, such as the choice between becoming a lawyer or a clarinetist (Raz 1985, 126). …