If all lotteries are indifferent to each other, any utility function representing is trivially a von Neumann–Morgenstern utility function. If a binary relation satisfies Order, Continuity and Independence, then it can be represented by a von Neumann-Morgenstern utility function U. If in addition it satisfies non-divergence, then the functional form of a von Neumann-Morgenstern utility function representing is unique to an increasing affine transformation.7.
In the version of the expected utility theorem presented in Weymark (1991), I neglected to explicitly rule out the possibility of universal indifference when considering the uniqueness properties of von Neumann–Morgenstern utility functions. For example, calculations are simpler when a von Neumann–Morgenstern utility function is used rather than a utility function of the form described by Arrow. Note that if Independent Prospects are satisfied and i can be represented by a von Neumann–Morgenstern utility function, then i must satisfy non-generation.
The Harsanyi–Sen Debate
These sign restrictions can be satisfied if stronger forms of the Pareto condition are invoked. To focus on the essence of the argument, let me further assume that Independent Prospects is satisfied, so that it is the only weight vector for which (5) holds for this profile. As Sen noted, if we subject the individual utility functions in U to non-affine transformations, then we can represent the social preference relation by a non-linear function of the individual utilities.
Therefore, the social objective function appears to be weighted utilitarian with one representation of individual preferences and non-utilitarian with another. 16. To illustrate this objection to Harsanyi's utilitarian interpretation of his aggregation theorem, I use a variant of the example used in Weymark (1991, Section 4). Let be the profile obtained by subjecting each of the individual utility functions to an exponential transformation.
Recall that each of the functions Ui are von Neumann-Morgenstern auxiliary functions whose images are nondegenerate intervals. The need for difference comparability can be seen most clearly by rewriting (5) as. 8) The sum in (8) is generally not invariant for independent increasing transformations of the individual utility functions, although these transformations are constrained to be affine. 17In the absence of the non-degeneracy assumption, the set of profiles of utility functions in Uc is a strict subset of the set of profiles of utility functions preserving utility difference comparisons exhibited by U .
Morgenstern utility representations of the impartial observer's preferences are admissible, and it is therefore illegitimate to use this argument to conclude that interpersonal comparisons of utility differences are meaningful. Harsanyi has indeed made fundamental contributions to our understanding of the logical basis for making interpersonal utility comparisons.
Measurement Theory
The early analyzes of the nature of measurements, such as Campbell's (1920), were about measurement in the physical sciences. A measurement involves constructing a mapping h, called a homomorphy, between an empirical relational structure E and a numerical relational structure N that preserves all the relations in E. The concatenation x ◦y corresponds to putting both x and y in the same pan of the scale, say x on top of y.
For example, in the case of measuring weight or length, the required homomorphism exists if W is what is known as a closed extensive structure when the numerical relational structure is N1.25. The uniqueness theorem identifies the scale type of h (ie, the structure of the class group of transformations that can be applied to h).26 For example, in the case of weight or length, a homomorphism is unique up to a similarity transformation. . Furthermore, N1 and N2 are only two of an infinite number of possible choices for a numeric relational structure.
The essential fact about the uniqueness of the representation is not the particular group of admissible transformations, but that all groups are isomorphic and, in the case of extensive measurement, all are one-parameter groups. The possible appeal of a non-additive representation for a closed comprehensive structure such as weight or length can be seen by considering an alternative empirical interpretation of the concatenation operation used in length measurement to the one described above. 28This interpretation of the empirical structure used to measure length was originally proposed by Ellis (1966).
In practice, it has been pragmatic considerations that have dictated the choice of the numerical relational structure used in a measurement exercise. Moreover, the mathematical form of the physical laws using weight or length may be simpler with an additive representation than with an alternative form.30 Nevertheless.
Expected Utility Theory Reconsidered
In order to determine whether the von Neumann–Morgenstern utility functions are cardinal in the sense required by Harsanyi's theorems, it is necessary to reconsider the von Neumann–Morgenstern utility theory in terms of the representative measurement theory developed in the previous section. 32 In their informal discussion, von Neumann and Morgenstern also consider the lottery formulation of expected utility theory used here. For a very useful exegesis of the sections on expected utility theory in von Neumann and Morgenstern (1944), see Fishburn (1989).
Nevertheless, this finding does not allow us to conclude that we must use the von Neumann–Morgenstern representation of preference. As with weight and length, the choice of N4 over N5 for the numerical relational structure in expected utility theory was dictated by pragmatic considerations, as described in Section 2 for choosing von Neumann–Morgenstern representations from among all possible representations. tationsof. This is perfectly acceptable in describing choice behavior, which is the goal that von Neumann and Morgenstern (1944, p. 20) offer for their theory of measurable utility.
As noted above, von Neumann and Morgenstern (1944) did not use M as their empirical relational structure in their formal theory. The structures M and U have similar formal properties, so it is possible to reformulate the previous argument using von Neumann and Morgenstern's version of expected utility theory. Here von Neumann and Morgenstern refer to the operators that appear in their empirical relational structure; that is, for the convex combination operators.
For von Neumann and Morgenstern, and for most researchers who have followed in their footsteps, it seems clear that the convex combination operator ¯αk for the real numbers must be chosen to correspond to the empirical operator α∗k (or αk). According to von Neumann and Morgenstern (1944, p. 24), these empirical and numerical operators are "synonymous". The choice of the numerical operator ¯αk does not have significant appeal because it shares many formal properties with αk∗ and αk.
Broome and Risse on the Cardinality of Utility
It is my contention that Broome (1991) provided an account of utility in which intrapersonal comparisons of utility differences are meaningful, but he did so only by introducing an additional quaternary operator that is not part of the von Neumann-Morgenstern- approach to expected utility theory. Of course, in order for Harsanyi's propositions to provide support for utilitarianism, differences in utility must be comparable interpersonally, not just intrapersonally. These two differences - all respectability to the good - must be weighed against each other. my emphasis).
In the case where these differences are exactly balanced, the utility presentation should equalize the difference in utility in going from £20 to. The fact that the two utility differences are equal tells us from the definition of utility that the consideration in favor of [a1] exactly balances the consideration in favor of [a2]. Utility tells us how much difference in goodness counts when determining the overall goodness of alternative options for you.
Since the two differences between good are exactly balanced in determining the overall goodness of the prospects, it would be very natural to express this fact by saying that the differences are actually the same. 37 Note that Broome's use of "natural" in this quote is different in meaning from the use of "natural" in von Neumann and Morgenstern (1944). 38 See also Broome (1993, section 2), where it is said that "better" is a cardinal term, and cardinality is defined in terms of the meaning of difference comparisons.
However, as I will explain, it seems to me that Risse's argument is fundamentally different from Broom's. However, Broome has an independent basis for measuring differences in utility (degree of goodness or well-being). Risse infers utility differences by assuming that only the von Neumann-Morgenstern utility functions represent his better-off ratio (moving from (23) to (24)), while Broome starts with an independent quantitative measure of the level of well-being. .
I am not suggesting that Risse has drawn the erroneous conclusion about the importance of intrapersonal utility differences that I discussed in section 4.
Concluding Remarks
The use of the weight analogy in this quote is not particularly appropriate, as the concatenation operator used to measure weight is formally quite different from the convex combination operator used in expected utility theory, but this is a minor quibble. What is more important about this quotation is that it appears that, unlike in Broome (1991), this argument does not use an independent quantitative measure of degree of preference. Expressed in the language of measure theory, Broome clearly recognizes that some justification for the choice of numerical relational structure N4 in expected utility theory is required.
But, as I have argued above, this is an unsatisfactory basis for determining the choice of the numerical relational structure of a normative theory. Given that the main elements of the representational theory of measurement were not systematized until the late 1950s, it is perhaps not surprising that Harsanyi, writing in the early 1950s, does not seem to have grasped the significance for his theory of von Neumann and Morgenstern's theories. notes that they had "discovered" a natural operation that allowed them to quantitatively measure the utility. As far as I know, in his published writings Harsanyi has never explicitly referred to the way in which utility differences are quantified in measure theory.
However, in the paper cited in footnote 18 above, Harsanyi draws an analogy between measuring differences in utility and measuring differences in weight. It is unfortunate that he never presented these ideas in print, as they provide valuable insight into Harsanyi's views on the issues I have examined in this article.
Acknowlegements
