The quest for the foundation of utility functions

That question motivated a quest to find assumptions that would justify such functions. In 1944, Von Neumann and Morgenstern posited a set of axioms that together guaranteed the existence of utility functions in the context of choice under uncertainty (understood as risk) (Von Neumann and Morgenstern, 1953 [Reference von Neumann and Morgenstern1944]: 26–27). Two of those axioms were completeness and transitivity.Footnote ¹ Both axioms embody the idea of internal consistency, that is, that one’s preferences ought not ‘contradict’ one another. Together, they guarantee a top-to-bottom ranking of all possible objects of choice. Von Neumann and Morgenstern used the term ‘rational’ for these axioms; to our knowledge, they were the first to do so. However, they did not treat this as a unique definition of rationality, but simply as one that suited their analytical purposes.

Debreu brought the axiomatic approach into choice theory generally. In 1954, he proved that three axioms – completeness, transitivity, and continuity – guarantee the existence of a continuous utility function (Debreu, Reference Debreu1954). Debreu did not use the word ‘rational’ to describe these axioms. Later that same year, however, Arrow and Debreu (Reference Arrow and Debreu1954: 269) employed Debreu’s conclusion as part of their general proof of the existence of a general competitive equilibrium.Footnote ² Uzawa (Reference Uzawa1956) and Arrow (Reference Arrow1959), in the process of merging the axiomatic approach with the theory of revealed preference, reintroduced the word ’rational’ to describe choice functions that satisfy the axioms of transitivity and completeness.

And thus axiomatic rationality was born: from a desire to provide a logical foundation for utility functions in economic models. It was about mathematical tractability. It was about constructing models that (it was hoped) would describe and predict the operation of a market economy. For predictive purposes, and superficially explanatory ones, it did not matter whether anyone really had preferences that satisfied the axioms. All that mattered was whether the resulting theories’ predictions passed empirical muster. And thus, the doctrine of ‘as if’-ism was born at the same time.

Critically, axiomatic rationality was not a normative (i.e., prescriptive) project at this point. Before explaining further, we must clarify a terminological confusion. The word ‘normative’ has multiple meanings. Usually it refers to judgments of value: good/bad, right/wrong, should/should not. In this sense, normative is distinguished from descriptive or positive. However, ‘normative’ is also sometimes used in a quasi-descriptive sense to mean a standard or criterion for the behavior of an idealized agent within a well-specified model. In this sense, it is not directly applicable to changing or correcting the behavior of real-world, non-idealized individuals. For example, Luce and Raiffa (Reference Luce and H.1957: 63) clarify that the axiomatic form of game theory is:

Hereafter, we will use ‘normative’ in its broader prescriptive sense. Our point is that economists of this era had not introduced axiomatic rationality for normative (prescriptive) purposes.Footnote ³

Prominent economists of the period, including Von Neumann and Morgenstern themselves, questioned axiomatic rationality’s descriptive accuracy as well as its normative import and offered persuasive reasons why reasonable people might not satisfy its requirements (Rizzo and Whitman, Reference Rizzo and Whitman2020: 55). Chief among these was that normal people with limited time and cognitive resources would not find it worthwhile to conduct a comprehensive audit of all their preferences for perfect consistency, as doing so would fail a reasonable cost-benefit test.

Nevertheless, the word ‘rational’ itself has a tone of approval. Its opposite, ‘irrational’, is typically an insult. So we should not be surprised that something labeled rational would eventually start to seem like ‘a good thing’.

Furthermore, while the principal goal was positive modeling, it was widely known that general competitive equilibrium had potential normative implications. The famous first and second welfare theorems were already understood by this time: any competitive equilibrium was also Pareto-optimal, and any Pareto-optimal allocation could be the outcome of a competitive equilibrium (Arrow and Debreu, Reference Arrow and Debreu1954: 265). To the extent that we normatively endorse Pareto optimality, therefore, competitive equilibrium would seem like an attractive arrangement. However, and crucially, Pareto optimality’s attractiveness as a norm is contingent on defining ‘betterness’ in terms of the actual well-being of the people involved. Yet we could imagine a world that is Pareto-optimal in terms of as-if utility functions that accurately describe behavior but do not describe what people actually want in a subjective sense. But for the welfare theorems to have normative weight, the utility functions involved also needed to express the actual satisfaction of subjective preferences. What economists did not fully realize is that this required the axioms to be both descriptive and normative. This is the inexorable logic of the theorems. And so, economists who wanted the welfare theorems to matter ended up assuming exactly that, even if they never (or rarely) said so overtly.

Unlike their initial treatment of preferences, economists’ treatment of beliefs was never clearly limited to positive purposes. Classical logic, the rules of probability, and Bayes’ Theorem were assigned normative status as well. As with preferences, this was an axiomatic system embodying the presumption that consistency is necessary and worthwhile, not merely in theory but in practical life. The possibility that rational people with limited time and cognitive resources would not wish to root out all possible inconsistencies in an entire system of beliefs – or that beliefs might serve purposes other than truth-tracking, or that other forms of belief formation and information processing might be viable or even preferable – received little attention.

Thus, in the domain of both preferences and beliefs, the mid-20th century witnessed the elevation of axiomatic rationality from mathematical construct to normative ideal. The same models were used to describe both how people do behave and how they should. And that is where things stood when Kahneman arrived.

Putting axioms to the test

Kahneman and his coauthors took the axioms seriously – not as undeniable truths, but as testable hypotheses about human behavior. Kahneman was not content to test the outputs of theoretical models (e.g., does a higher price lead to a lower quantity demanded?); instead, he tested the inputs (e.g., do people consistently rank two objects of choice in the same way or correctly apply Bayes’ Theorem?). By this method, Kahneman and his coauthors amassed considerable evidence that the axioms were descriptively wrong. Kahneman’s experiments threw a bucket of cold water on the ‘as-if’ approach, because people did not behave as if their preference rankings were complete and transitive, and their inferences did not follow the strict dictates of logic and probability.

Kahneman’s work put the profession at a crossroads. The first path would have involved expanding the notion of rationality beyond the axiomatic straitjacket, remembering that those axioms had never had a strong normative basis to begin with. The second path involved rejecting axiomatic rationality for descriptive purposes but maintaining it for normative ones. The profession largely took the latter (wrong) path.Footnote ⁴ As a result, wide swaths of reasonable and understandable human behaviors were tarred as irrational, problematic, and even pathological.

Among other problems, the maintenance of axiomatic rationality as a normative standard set the stage for the emergence of behavioral paternalism: the use of behavioral findings to justify policy interventions ‘for people’s own good’, even when there is no traditional (i.e., interpersonal) market failure involved. There is a litany of objections here, foremost among them the non sequitur of resolving inconsistent preference rankings (i.e., someone seems to prefer both A to B and B to A, depending on the framing or other supposedly irrelevant factors) in favor of one such ranking, without much basis other than the observer’s judgment for choosing which ranking to treat as someone’s ‘true’ preference (Rizzo and Whitman, Reference Rizzo and Whitman2020: 75–78).

But is Kahneman to blame for this wrong turn in the discipline? Unlike the majority of behavioral economists, Kahneman adopted a broader view of normative rationality than the technical rationality axioms would allow. First, he argued the axioms are too restrictive (Herfeld, Reference Herfeld2014: 3). Since they cannot be satisfied by real human beings, they cannot be normative. Ought implies can, after all. Secondly, Kahneman said the axiomatic requirements are too loose because the pursuit of any ends whatsoever can be rational in the technical, instrumental sense without being genuinely rational. For example, he said addiction to drugs cannot be rational in part because people will come to regret such a choice (Ibid: 3). Instead, Kahneman thought it was better to adopt an evaluative standard of ‘reasonableness’. Although admittedly imprecise, this reasonableness criterion would embody those ends which the individual is unlikely to regret and those means which are most adequate to the task.Footnote ⁵

On the question of the axioms’ restrictiveness, Kahneman was right. On regret, we think not. The problem with a no-regret standard of reasonableness is that regret has no unique meaning. A person may ‘regret’ that a risky choice did not turn out well. A person may willingly accept that he will ‘regret’ some action tomorrow, but think it is worth doing anyway; he would do it again under the same circumstances. A person may ‘regret’ that she did not pay enough attention to unfavorable information that was available about her action. The first two of these are fully consistent with reasonable behavior. Only the last seems to be what Kahneman had in mind. We suspect, however, that even the last type of regret could play an important role in learning processes.Footnote ⁶

Setting aside these concerns about regret, Kahneman evidently had little reverence for the axiomatic approach, rightly regarding it as deficient. Despite his more recent explicit observations on normativity – and his flirtation with a hedonic welfare standard (‘happiness’) in earlier work (Kahneman et al., Reference Kahneman, Wakker and Sarin1997) – Kahneman’s main interest was always the discovery of novel facts about decision-making (Herfeld, Reference Herfeld2014: 6). This was a worthy project that he pursued with alacrity. But what theory did he use to guide his discoveries? The standard neoclassical theory of rational choice. His facts are all identified relative to its axioms; they are essentially amendments to the standard theory.

Paths taken and not taken

One natural approach would have been to first understand why heuristics work – and only then try to explain why and under which conditions they do not. Unfortunately, Kahneman’s laudable descriptive or positive exercise was compromised by calling the new facts ‘errors’ relative to the axiomatic benchmark. To call them such endowed them with negative connotations based simply on their deviation from a theory he said was impossible for human beings to satisfy. The temptation is nigh irresistible; an industry of doctor economists has now emerged to cure people of their rational perversities.

The framing of deviations from axiomatic rationality as ‘errors’ has allowed unjustified and inadvertent prescriptivism to creep into what, in principle, could have been a purely descriptive endeavor. We will offer three brief examples.

First, behavioral economists have typically studied only one bias at a time, a concern Kahneman himself pointed out (Herfeld, Reference Herfeld2014: 9). Analyzing more than one did not seem tractable. However, biases move in various directions and to various degrees. What is the overall effect? Clearly, some biases can offset (or magnify) others (Besharov, Reference Besharov2004). Characterizing one bias in isolation as an ‘error’ is an analytical mistake when biases interact; a system-level approach is called for. Nevertheless, behavioral economists have used lone biases to justify intervention; that is the normative impulse at work. This is what comes of taking axiomatic rationality as a ‘benchmark’: every bias taken in isolation is ipso facto seen as an error.

Second, the axiomatic benchmark privileges the analyst’s or experimenter’s perspective over that of the people involved. The famous Linda problem is an important example of this phenomenon. In this experiment, which has been repeated with variations many times, participants are given a description of ‘Linda’, which suggests that she is a ‘liberal’ with progressive views on many subjects. And then they are asked a question: which is more probable, (1) that Linda is a bank teller or (2) that Linda is a bank teller active in the feminist movement? The original modal result (Linda is a bank teller active in the feminist movement) was deemed an error in probability theory because the probability of a subset can never be greater than the probability of the full set. There have been many criticisms of this experiment, as well as many variations that change (and typically reduce) the proportion of people making the so-called error. The most important objections are these: First, people may understand the term ‘probability’ differently from the experiment designer, which means they are simply answering a different question. Subjects are deemed deficient in understanding the language in a way that does not conform to the experimenter’s usage. Second, the information provided in these experiments is strictly irrelevant, thereby violating standard norms of cooperative communication (Grice, Reference Grice1989). The ‘correct’ answer would have been the same if no information at all had been provided. Subjects are thus deemed deficient for assuming that the experimenters were being cooperative rather than purposely misleading in giving that information. And just how important in daily life is this purported failure to understand probability theory, anyway?

Third, the axiomatic benchmark creates the false impression that there exists a unique normative standard for belief formation. Kahneman and Tversky, following the neoclassical economists before them, suggested that everyone should update their beliefs according to Bayes’s Theorem. That theorem is, of course, entirely correct in its proper place and properly applied. However, it is strictly applicable to decision-making under risk, not uncertainty – and subsequent research has shown that non-Bayesian methods can perform about as well or even better than Bayes in situations of substantial uncertainty (Todd and Goodie, Reference Todd, Goodie, Hallam, Floreano, Hallam, Hayes and Meyer2002; Juslin et al., Reference Juslin, Nilsson and Winman2009). Furthermore, Kahneman and Tversky (among many others) simply assumed that prior probabilities must be equal to the given base rates – which is not required by Bayes’ Theorem. Since subjects were not given full information about problem situations wherein the theorem was supposed to be applied, they were free (implicitly) to make assumptions based on reasonable guesses and their own experience. These assumptions would naturally affect prior probabilities. Lastly, and remarkably, there was no scope for learning over time in the typical experimental setup. People were simply given data and expected to process it via a tautological relation (i.e., Bayes’s Theorem). We know that Bayesian learning is far more likely to be displayed when there is feedback and revision (Gigerenzer, Reference Gigerenzer2023: 64).

Again, these are only examples. The larger moral is this: the treatment of axiomatic rationality as a benchmark, even for purely descriptive purposes, tempts analysts into making unjustified normative conclusions. The axiomatic benchmark further creates a blindspot to the ways that real people can be rational in a more inclusive sense (Rizzo and Whitman, Reference Rizzo and Whitman2020).

Many friends of Daniel Kahneman report that he liked to discuss his ideas with people who did not necessarily agree with him. He enjoyed re-evaluating his own ideas. So we do not think that our criticisms are amiss; we hope he would have welcomed our criticisms and suggestions. Kahneman’s research program has been enormously valuable, opening questions among economists that were long suppressed by conventional wisdom. The challenge now is to purge the program he pioneered of its unjustified elements – especially the remnants of an axiomatic approach that Kahneman himself questioned – so that the enrichment of economics with psychology can move forward with its best parts intact.