Divergent Mathematical Treatments in Utility Theory
Abstract
In this paper I study how divergent mathematical treatments affect mathematical modelling, with a special focus on utility theory. In particular I examine recent work on the ranking of information states and the discounting of future utilities, in order to show how, by replacing the standard analytical treatment of the models involved with one based on the framework of Nonstandard Analysis, diametrically opposite results are obtained. In both cases, the choice between the standard and nonstandard treatment amounts to a selection of settheoretical parameters that cannot be made on purely empirical grounds. The analysis of this phenomenon gives rise to a simple logical account of the relativity of impossibility theorems in economic theory, which concludes the paper.
Keywords
Utility Function Utility Theory Mathematical Treatment Normative Constraint Impossibility Result1 Model Theory and Scientific Models
In a 1960 paper, Patrick Suppes claimed that:
This claim was defended against the background thesis that the meaning of the concept of model is the same in mathematics and the empirical sciences (Suppes 1960: 289). Suppes’ view of models is too restrictive in two distinct ways: one of these has become clear through the recent literature on modelling, whereas the other has been neglected and provides the main motivation for the discussion presented in this paper. The first sense in which Suppes’ view is too restrictive is that it wishes to assimilate models tout court to mathematical models. Later developments in philosophy of science have moved away from this perspective, while acknowledging the significance of Suppes’ proposal. For instance, the several contributions included in Morgan and Morrison (1999), a turning point in the philosophy of scientific modelling, work with a notion of model that is much more wideranging than the one proposed by Suppes. This is perhaps most clearly stated in Adrienne van den Bogaard’s contribution to the collection, when she writes:[...] in the exact statement of the theory or in the exact analysis of data the notion of model in the sense of logicians provides the appropriate intellectual tool for making the analysis both precise and clear. (Suppes 1960: 295)
It is clear that, if one is ready to accept the qualification of models for methods, distortions of reality and social devices, Suppes’ more stringent semantic qualification must appear to impose very narrow, perhaps unrealistic, constraints on the study of modelling practices. It does not follow that Suppes’ appeal to notions and techniques from mathematical logic should be deemed irrelevant to the study of modelling practices in general. In this paper, I seek to defend the opposite point of view by applying a modeltheoretic approach to the study of mathematical modelling within utility theory. While doing so, I depart from Suppes’ original aims, which presuppose, in my opinion, too strict a delimitation of the ways in which modeltheoretic considerations may support philosophical investigations of scientific models. The quotation opening this section spells out the delimitation in question by restricting the mobilisation of settheoretical semantics to the purposes of formulating scientific theories (typically as classes of models defined by a settheoretical predicate, an approach whose abstract development has been presented in Da Costa and Chuaqui 1988) or carrying out an exact analysis of data (e.g. by the embedding of a data structure into a representing structure, a strategy adopted in Da Costa and French 2003). These investigations are of clear philosophical interest, but they rely more on the settheoretical representation of structures and mappings between them than on distinctively modeltheoretical constructions and techniques. It is therefore plausible to think that the tasks set by Suppes for a modeltheoretic investigation of the mathematical models used in empirical science cannot be exhaustive. Such an impression is confirmed by a few notable applications of model theory to the theory of measurement, which have been largely neglected by the philosophical literature despite their importance: two of them are P.J. Cameron’s classification of rationalvalued scale types, obtained in Cameron (1989), and the construction of measurement values as modeltheoretic types presented in Niederée (1992). Informally speaking, Cameron’s result shows that, if one thinks of empirical variables as dense orderings that are not continua (isomorphic to the ordered reals), the number of distinct scales they can theoretically give rise to increases from three (for continua) to infinity.^{1} Niederée’s results have shown, among other things, how one can identify measuring numbers with sets of experimental data, as well as the equivalence between certain mathematical assumptions (e.g. the Archimedean property) and properties of experimental procedures. These results shed light on important features of scientific models (especially measurement models) by modeltheoretic means, without pursuing any of the tasks recommended by Suppes, i.e., theory formulation or data analysis. The main objective of this paper is to extend along a further direction the same modeltheoretic style of investigation, which recognises the value of Suppes’ original proposal but transcends its limited scope. The modeltheoretic machinery I shall rely upon comes from Nonstandard Analysis: it is briefly surveyed in Sect. 2 (more details are found in the “Appendix”). I shall apply Nonstandard Analysis to two models from utility theory in order to construct an alternative mathematical treatment for the economic setups they are supposed to describe. This will allow me to show that the fragments of economic theory based on these models are crucially sensitive to a choice of mathematical treatment, more precisely, a selection of settheoretic parameters. What this suggests is that economic theory is, at an abstract level, significantly sensitive to the choice of mathematical resources employed in its articulation. The existence of distinct choices leads to bifurcations in the kind of result one may hope to obtain. In particular, if one wishes to uphold certain normative constraints or introduce certain formal approaches, it is sometimes mandatory to drop traditional mathematical environments based on the real numbers. These remarks will be illustrated in full detail in Sects. 4–7, after a brief semitechnical preliminary.Arguments about the model as a theory, or a method, or a distortion of reality, all focus on the model as a scientific object and how it functions in science. Without denying this dimension of the model at all, this paper wants to broaden the perspective by claiming that the model is also a social and political device. The model will be understood as a practice connecting data, index numbers, national accounts, equations, institutes, trained personnel, laws, and policymaking. (van den Bogaard 1999: 283)
2 Classical and Nonstandard Analysis
A vast amount of work in mathematical social science (especially economics) relies on the availability of the objects of classical analysis in the semantic metatheory. For example, in consumer theory utilities are real numbers, bundles of goods are realvalued vectors and their totality is canonically a subset of some Euclidean space. In many interesting cases there is no particular empirical motivation to select certain specific analytical objects in modelbuilding, either because they (e.g. the metric structure on a set of alternatives ranked by a preference relation) support abstract models without having any empirical interpretation or because, even when they represent some nonmathematical content, they enter a model also as carriers of properties that have no particular connection with this content (e.g. the topological separability of the real numbers representing utilities) and yet influence what can be established about the given model. Because of this, it is of interest to consider what happens if one replaces certain canonically employed analytical objects with alternative objects. In this paper, I consider the objects of Nonstandard Analysis,^{2} which share a number of properties with classical objects but are, at the same time, significantly different. I shall focus on their application to two mathematical models from utility theory. In each case, I study the consequences of using certain extensions of classical numerical sets within a Nonstandard universe as codomains of functions that are canonically selected to be realvalued. While a standard mathematisation based on realvalued functions gives rise to negative results, a Nonstandard mathematisation replaces them by positive results (which may hold under stronger conditions than were sufficient to deduce the negative results by standard means). This divergence highlights the essential relativity (i.e., with respect to a selection of mathematical resources) of negative results in economic theory, since the remarks that hold for the utility models discussed in detail admit of a general reformulation. Such a reformulation will be presented in Sect. 7, after a full discussion of the main examples has taken place, in Sects. 4–6. It is appropriate to note at this point, by way of a concluding remark, that applications of Nonstandard Analysis are not new to the field mathematical economics (see for instance, Skala 1975; Fishburn and Lavalle 1991; Lehmann 2001). However, all those of which I am aware adopt a local point of view, i.e., they construct an ultraproduct of some real structure suitable to specific modelling purposes. Moreover, they are not concerned with showing how canonical and nonstandard resources affect in divergent ways the results of modelling. My approach, on the contrary, is global in the sense that it relies on a Nonstandard universe in which several results involving nonstandard models are simultaneously obtained (this point will be clarified in Sect. 3). Moreover, it is primarily concerned with showing how canonical and nonstandard resources affect the results of modelling.
3 A Formal Preliminary
4 Ranking Information States
The point of the above quotation is that the existence of a utility representation is incompatible with the assumption that an agent should prefer more information to less. This remark is certainly correct, but it hides the following dilemma: is the problem inherent in utility theory as a formal approach to the study of idealised rankings or is it an effect of restricting attention to realvalued utility functions and, thus, of certain properties of \({\mathbb {R}}\)? This dilemma refines the formulation of the problem highlighted by Dubra and Echenique because it does not presuppose that the codomain of a utility function should be \({\mathbb {R}}\). No particular feature of the space of informational states suggests that such a codomain should be selected. It is therefore meaningful to look for alternative numerical codomains, on which utility functions may exist. In other words, it is reasonable to conjecture that a lack of fit exists not between utility functions and spaces of information states, but between these spaces and the ordered reals. An application of Nonstandard Analysis shows that bounded utility functions on \({^*}{\mathbb {N}}\) exhibit a much better fit:Our result is important because it shows that utility theory is not likely to be a useful tool in the analysis of the value of information. This finding should be contrasted with the existing literature on the value of information, where utility representations are used. The use of a utility implies that preferences are not monotone (Dubra and Echenique 2001: 1).
Theorem 4.1
Every complete preorder has an uncountable family of bounded utility functions on \({^*}{\mathbb {N}}\).
The proof that a bounded utility function on \({^*}{\mathbb {N}}\) exists is included in the “Appendix”. Since any strictly increasing, monotonic transformation of a utility function is a utility function, multiplying the values of one by the infinitely large number \(H \in {^*}{\mathbb {N}}\) yields a utility functions. Since there are uncountably many such numbers (see “Appendix”), there are uncountably many utility functions.^{4} In the light of Theorem 4.1, the suggestion that utility theory is not likely to be a useful tool in the analysis of the value of information may be successfully resisted. The applicability of utility theory as a uniform approach to the representation of idealised preferences can be rescued, provided that one does not restrict attention to realvalued utilities, which may be incompatible with the structure of certain preorders. Note that, since \({^*}{\mathbb {N}}\) is compatible with all of them, it is a numerical set and shares with \({\mathbb {N}}\) every formal property statable in \({\mathsf {L}}\), a language powerful enough to express the whole of classical mathematics, there does not seem to be any natural objection to the transition from the codomain \({\mathbb {R}}\) to \({^*}{\mathbb {N}}\). The latter transition may also be regarded as the shift from one mathematical treatment, in which a particular selection of semantic resources has been made, to an alternative treatment, based on an alternative selection. The striking fact is that, on the first treatment, a certain problem (representing monotone rankings of information states) admits of no solution, whereas, on the alternative treatment, it admits of uncountably many. At a certain level of mathematical idealisation, the selection of semantic resources makes a huge difference to the implications of a model. This phenomenon is also responsible for the intimate connection between semantic resources and normative assumptions that some models display, as will be seen in the next two sections.
5 Discounting the Future

Pareto (P): \(s \succ t\) implies \(u(s)\,>\,u(t)\).

Anonymity (A): p(s, t) implies \(u(s) = u(t)\).
Lemma 5.1
Let \({\mathbb {S}}\) be the set of realvalued, infinite utility streams. There is an aggregation u from \({\mathbb {S}}\) to an initial segment of \({^*}{\mathbb {N}}\) that satisfies (P) and (A).
Proof
A fortiori, Lemma 5.1 ensures the existence of an aggregation u when infinite utility streams take values in \({\mathbb {N}}\). Nevertheless, this result does not establish any continuity between the finite and the infinite case: in particular, it does not decide whether it is possible to aggregate utilities by summation in both cases. The latter possibility may be viewed as a desirable requirement because homogeneity in aggregation could be taken as evidence that infinite utility streams are a good generalisation of finite utility streams. It turns out that, for utility streams that take only finitely many values,—a plausible restriction—this type of homogeneity can be established only if one gives up (A), in particular by discounting the future. Such a move is not unfamiliar to economists, since e.g. it is often adopted to solve optimal control problems over an infinite time horizon. In the present context a discounting approach takes the following form:
Remark
Let \({\mathbb {S}}_k\) be the set of all infinite utility streams generated by a finite set of real numbers \(\{r_{1},\ldots ,r_{k}\}\). There are uncountably many discounted utility representations of \({\mathbb {S}}\) that satisfy (P).
Proof
In the light of the last remarks alone, one may conjecture that the only way to obtain a ‘natural’ aggregation (i.e., by summation) for utility streams is to impose a discount rate on the future. If this conjecture were true, then, theoretically, there would be no way of implementing an ethically desirable condition like (A) by way of an aggregation.^{7} It would look as though one were forced to drop (A) and adopt the normatively problematic assumption that utility decays at a certain rate in time, if one wished to compare alternatives by means of aggregations. This standpoint is, however, not reasonable, since it would lead to theoretical scenarios in which e.g. the fasterpaced consumption of an exhaustible resource or a more intensive exploitation of natural resources was deemed preferable to an alternative policy, independently of its environmental effects. One might advocate such an ethically unacceptable perspective only by suggesting that more utility is to be extracted from a certain resource in the present or near future than it could be in the remote future.^{8} Abstractly speaking, such an argument has any force only insofar as one takes the semantic components of the model that licenses it to be fixed. When this constraint is abandoned, it becomes meaningful to search for a mathematical treatment that satisfies (A) and rules out discounting future utilities. There is no particular reason to take the settheoretical parameters of the aggregation problem as fixed and then test the satisfiability of (A) under these parameters. In fact, there are ethical reasons to subordinate the choice of mathematical resources to the satisfiability of a normative constraint like (A). In other words, one may wish to detect the mathematical resources that conflict with (A) in order to replace them with others that do not. In fact, discountfree aggregations that add utilities and satisfy both (P) and (A) can be obtained if one chooses the codomain of the aggregation to be \({^*}{\mathbb {R}}\) instead of \({\mathbb {R}}\) (or \({^*}{\mathbb {N}}\) instead of \({\mathbb {N}}\)). The applicability of Nonstandard Analysis to this problem is adumbrated (but not fully articulated) in Lauwers (2010)^{9} and has been briefly illustrated in Pivato (2014) but none of these authors has emphasised it as an instance of the effects of alternative mathematical treatments. Furthermore, none of them points out that one can easily obtain an uncountable family of aggregations satisfying (A) and (P). This is proved in the next section.
6 DiscountFree Aggregations
Within the Nonstandard universe quickly described in Sect. 3 (see also the “Appendix”), any realvalued, infinite utility stream s has a \({^*}{\mathbb {R}}\)valued extension \({^*}s\): since \(s(n) = r\) is a true \({\mathsf {L}}\)sentence about s, it follows that, for every \(n \in {\mathbb {N}},\,s(n) = {^*}s(n)\). The difference between these two sequences is that \({^*}s\) also has infinitely large arguments, at which it takes uniquely determined values. An additive, discountfree representation of utility streams may be obtained by truncating \({^*}s\) at an arbitrary, infinitely large argument. Since, intuitively, the truncation of \({^*}s\) at any argument behaves like a finite subsequence, its values can be \({^*}\)summed, in the sense that one applies to them a function which is the nonstandard extension of a finite summation, also known as a \({^*}\)finite, or hyperfinite, summation. Taking such a summation preserves all information carried by s (since s and \({^*}s\) agree at every finite argument) in a format that can be handled much like an ordinary, finite summation. This allows a straightforward generalisation of the treatment of finite utility streams described in Sect. 4. In particular both (P) and (A) are satisfied by \({^*}\)finite summations because of the following two lemmas (\({\mathbb {S}}\) below is the set of realvalued, infinite utility streams):
Lemma 6.1
Lemma 6.2
Lemmas 6.1 and 6.2 in turn imply:
Theorem 6.3
Let \({\mathbb {S}}\) be the set of all infinite, realvalued utility streams. There is an uncountable family of \({^*}{\mathbb {R}}\)valued aggregations for \({\mathbb {S}}\) satisfying both (P) and (A).^{10}
The last theorem shows that normative constraints may be deeply connected to a choice of mathematical resources. Under a canonical choice (realvalued aggregations) there is no way of meeting constraint (A). An alternative choice gives rise to uncountably many ways of meeting it. The latter choice is not only preferable on ethical grounds, i.e., because it gives rise to a theoretical model that does not presuppose the viability of unacceptable policies, but may even be supported on technical grounds. This is clarified by two representative quotations from economists working on the problem of intertemporal choice and finding it problematic to work with the reals. The first quotation comes from Koopmans:
The very same issue is raised by Basu and Mitra when they point out that:[...] there is not enough room in the set of real numbers to accommodate and label numerically all the different satisfaction levels that may occur in relation to consumption programs for an infinite future (Koopmans 1960: 288)
These quotations seem to reveal a recognition of the fact that there is an inherent problem with reconciling the structure of the real numbers with that of certain formal preferences. This may be read as a statement against the use of real numbers under these circumstances. The fact that one ‘runs out of real numbers’ is not to be seen as an obstacle to tackling an aggregation problem, but rather as a call for methods that can tackle them.The proof of our result, roughly speaking, involves showing that in trying to represent any social welfare function respecting equity and the Pareto principle [these are (A) and (P) above], one “runs out of real numbers”. (Basu and Mitra 2003: 1558)
7 Impossibility as Unsatisfiability
8 Concluding Remarks
It is easy to think of mathematics as a system of resources whose main effect on modelling is to constrain the setup to which they are applied, by deploying forms of reasoning that conclusively establish results and exclude alternatives. This picture captures one of the advantages of any mathematical treatment, namely control over the object of investigation, but neglects another, distinct, character of mathematical modelling, namely its openendedness. As the examples discussed in the previous sections have shown, mathematical resources offer the means to construct alternative treatments, which may give rise to different modelling trajectories. Mathematics is thus not only to be seen an instrument that stringently fixes outcomes by way of proofs, but also as the matrix of a plurality of formal strategies. As the previous sections have shown, the latter characteristic of mathematics has a nonnegligible impact on modelling practices, in view of its close connection with the viability of certain formal approaches or the satisfiability of certain normative constraints.
Footnotes
 1.
Here scales are distinguished on the basis of the number of reference points, e.g. origins or units, that uniquely determine them.
 2.
As is wellknown, Nonstandard Analysis was created by Robinson (1966). The most widely adopted approaches in this field are the superstructure approach (see e.g. Davis 1977) and the axiomatic approach based on Nelson’s Internal Set Theory (Nelson 1986; Robert 1988, but see also, as a result of many refinements, Hrbacek et al. 2014). Here I make use of a version of the superstructure approach, which is outlined in the “Appendix” and proceeds along the lines of Bell and Machover (1978).
 3.
It proves convenient to have in the language also a twoplace \(\langle , \rangle \) termforming operator that designates ordered pairs. This term does not need to be assumed, since it could be introduced using the language \({\mathsf {L}}\). I have assumed its availability in Sect. 6.
 4.
The existence of a utility function on a suitable ultraproduct of the reals is proved in Skala (1975: 44) and Narens (1985: 258–259). Theorem 4.1 shows that one does not in general need to consider the reals, since a suitable extension of \({\mathbb {N}}\) suffices. Moreover, Narens and Skala are primarily interested in the existence of at least one utility function, and do not emphasise that, in fact, a large family of them can be obtained. It is worth pointing out that Theorem 4.1 goes through for A of arbitrary, infinite cardinality, although any given cardinality will call for an enlargement that satisfies suitable saturation properties.
 5.
Here permutations cannot but be finite. I emphasise finiteness because it will appear also in the discussion of infinite utility streams to follow.
 6.
In the present notation, this condition states that \(s \succ t\) implies \(u(s)\,>\,u(t)\) if strict inequality holds for infinitely many components of s, t (see Crespo et al. 2009, p. 52).
 7.
On the other hand, it would be possible to determine an intrinsic ranking of \({\mathbb {S}}\) that satisfies (P) and (A). This possibility is actually exploited in the proof of Lemma 5.1.
 8.
 9.
Lauwers shows how one may directly order utility streams by means of an ultrafilterbased argument which would determine the same aggregate ranking on a corresponding ultrapower of the reals.
 10.
In fact, these aggregations do not only satisfy (P) and (A) but also a further interesting condition that does not follow from them. This is called ‘Hammond equity for the future’ (H) and has been introduced in Asheim and Tungodden (2005): formally, it states that, if \(s(i)\,>\,t(i)\) for \(i\,\ge\,2\) and \(s(1)\,<\,t(1)\), then \(u(s)\,>\,u(t)\). The intuitive motivation for (H) is the idea that a relative loss in the present is acceptable if it is to be followed by a relative gain for all future generations. The plausibility of this condition depends, among other things, on the rather optimistic assumption that correct estimations of future utility levels can be made for an arbitrary distant future.
 11.
Each set of Uconcurrent formulae is turned into a set of sentences by replacing the free variable in each of these formulae by the same, new (i.e., not in \({\mathsf {L}}\)) constant symbol.
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