Inherent individual differences in utility.

Opinion Article published: 15 November 2011 doi: 10.3389/fpsyg.2011.00297 Inherent individual differences in utility R. Duncan Luce* Institute for Mathematical Behavioral Sciences, University of California, Irvine, CA, USA *Correspondence: rdluce@uci.edu 1 Who knows a utility representation Repeatedly it is alleged that this or that theory concerns a decision maker maxi- mizing some utility function. But, as I have been at pains to discuss, the repre- sentation is not a creature of the deci- sion maker but of the scientist studying the decision maker’s behavior “The representation theorems go in only one direction: from behavioral and structural properties to numerical representations. The behavioral properties embody the scientific information that we, as scien- tists, have about the decision maker. It is the scientist, not the decision maker, who formulates both the properties (axi- oms) and the representation. The decision maker exhibits the behavior which, pre- sumably, arises from some fairly complex neuronal processes…” (Luce, 2000, p. 25). This observation holds far more broadly than just in decision making. 2 Uniqueness of the representation Uniqueness of representations is, of course, a very well trodden topic. The gist is that in addition to proving a rep- resentation theorem, one must also for- mulate a uniqueness theorem – how do the several representations relate? In most of the popular measurement theories, the uniqueness falls into one of Stevens’ (1946, 1951) well known classification: nominal, ordinal, interval, log-interval, ratio, and absolute scales (see Krantz et al., 1971; Luce et al., 1990). As I recently came to realize, when we are dealing with a preference weak order ≥ and a binary operation ⊕, such as the con- catenation of two sums of money, for which 0 money is an identity of ⊕ and ⊕ is com- mutative, associative, and monotonic in ≥, things are not really quite as simple as has been long believed. Recall that Holder (1901) proved in this context the existence of an additive representation www.frontiersin.org c ( x ⊕ y ) = c ( x ⊕ 0 ) + c ( 0 ⊕ y ) = c ( x ) + c ( y ) , which is unique up to a ratio scale. A century later it was recognized that, in the context of utility theory, we also have the interplay between value and risk, which means that multiplication as well as addition is in play. See, e.g., (3) below. So the mapping should be into 〈R, ≥, +, ×〉, in which case the full set of polynomial (p-additive) represen- tations are (see Luce, 2000, p.151, § 4.4.6) c ( x ⊕ y ) = c ( x ) + c ( y ) + dc ( x ) c ( y ) d = − 1 , 0 , 1 . Of course, d = 0 is Holder’s original solution. The other two are quite different in that they are absolute scales (see Luce, 2010a). Here is the reason: when d ≠ 0, (2) is equivalent to 1 + dc ( x ⊕ y ) = ( 1 + dc ( x ) ) ( 1 + dc ( y ) ) . But this expression is not invariant under any transformation of c other than the identity. The fact they are absolute scales has several important implications. 3.1 Three types of people There are three inherently different types of people corresponding to their values of d. The following simple behavioral criterion to decide a particular person’s type was reported by Luce (2010a). Let (x, p; y, 1 − p) denote the gamble where x occurs with probability p and y with probability 1 − p and suppose, as is true in a great many theories of utility, that + U ( y ) W ( 1 − p ) , where ψ has been replaced in this context by U for utility, W is a strictly increasing sub- jective probability function, and W(0) = 0. Further, assume that Find the probability p 1/2 with subjective probability 1/2, i.e., such that for the repre- sentation (3) and for all x > y x , p 1 ; y , 1 − p 1      y , p 1 ; x , 1 − p 1   ⇔ W ( p 1 ) I assume that W(p) = p is satisfied for p = 0, p = p 1/2 , possibly for p = 1, but not elsewhere. Note that it follows that relative to the line f(p) = p, the weighting func- tion W must be either S-shaped or inverse S-shaped. See Section 3.4 of Luce (2000) for a summary of empirical estimates which seem to agree with this prediction. Then the criterion for d is that for all x > x′ > y > y′ d =  0 x ⊕ x ′, p 1 ; y ⊕ y ′, 1 − p 1    ∼ 3 Some implications of the p-additive representation U ( x , p ; y , 1 − p ) = U ( x ) W ( p ) W ( p ) + W ( 1 − p ) = 1 . x ⊕ y,, p 1 ; x ′ ⊕ y ′, 1 − p 1   . Observe that under the ordering con- straint on the consequences, the gamble on the left has a greater variance than the one on the right. For that reason, the three d types are called, respectively, risk-seeking (d = 1), risk-neutral (d = 0), and risk-averse (d = −1). It would not surprise me that in the academic population, which is the source of many experimental respondents, risk-averse types will predominate except, perhaps, in schools of business. It has yet to be shown empirically that this classification holds up in the population. But should it, one simply should not aver- age data over people without, at the least, dealing with the three classes separately. Indeed, really only the d = 0 types should ever be averaged. Individual ­differences are November 2011 | Volume 2 | Article 297 | 1