Von Neumann and Morgenstern (1944) defined rational preferences in the presence of risk by a series of simple postulates supporting the Expected Utility (EU) representation. They also beautifully demonstrated that EU guarantees the existence of equilibrium in zero-sum games (the minmax theorem), a result famously generalized by Nash (1950) to all finite games. Its normative appeal, elegance and simplicity made EU theory a golden standard in applications involving decisions under risk.

Since then, EU theory applied to monetary risks with wealth levels being the consequence has been shown to be systematically violated in numerous experiments. Many attempts have been made to propose better descriptive theories, with cumulative prospect theory (CPT) standing out as the most popular contender (Kahneman and Tversky, 1979, Tversky and Kahneman, 1992). However, as Barberis (2013, p. 173) notes “It is curious, then, that so many years after the publication of the 1979 paper, there are relatively few well-known and broadly accepted applications of prospect theory in economics”.

Prospect theory builds on two ideas: reference dependence (Markowitz, 1952) and nonlinear probability weighting (Edwards, 1954). While the former reinterprets the domain over which the EU axioms are defined—with outcomes representing changes relative to a reference point rather than levels of wealth—the latter departs from EU by introducing rank-ordering of outcomes and probability weighting.

We claim that by denying the essence of the EU this latter feature makes prospect theory less attractive in applications. For example nonlinear probability weighting implies violations of the reduction of compounded lotteries. To illustrate consider the implementation of a mixed strategy equilibrium, assuming one exists (under CPT, equilibrium may fail to exist). The ex-ante optimal mixed strategy is not optimal after I flip a coin to determine my own strategy. Moreover, CPT requires rank-ordering the outcomes, which introduces complexity and hinders tractability (Baucells et al., 2023).

Instead of entirely replacing EU as prospect theory does, what we propose in this paper is a model that extends EU by allowing for context dependence. While EU is preserved within each decision context, its violations are allowed when a given alternative, or a comparison between two alternatives, is set in a different context. Thus our model retains the elegance and tractability of EU implied by linear probability weighting, and at the same time is capable of accommodating systematic violations of EU. This makes it suitable for applications.

In our model, probability weighting is replaced by a range distortion function that deforms the relative utility of the outcomes compared to the range of outcomes. When eliciting certainty equivalents of binary prospects, and only then, CPT and our model become mathematically identical, with the range distortion function being the inverse of the probability weighting function. To elicit certainty equivalents of prospect with three or more outcomes, or compare any two prospects each having two or more outcomes, the two models are different. The notion of an S-shaped range distortion function has a strong support in psychophysics and neuroscience.

In psychophysical judgment, Parducci (1965) demonstrated that a particular stimulus is judged relative to the range of the stimuli in the context. Such range adaptation effects are well documented in neuroeconomics (Rustichini et al., 2017, Stauffer et al., 2014, Zimmermann et al., 2018) and multi-attribute choice problems (Bushong et al., 2021, Kőszegi and Szeidl, 2013, Soltani et al., 2012, Somerville, 2022). We think of range distortion as ‘continuous’ way to model the process that the brain follows to encode inputs of different width into a fixed neuronal output range. This is in contrast with ‘discrete’ process models such as the priority heuristic (Brandstätter et al., 2006, Katsikopoulos and Gigerenzer, 2008), where the minimum gain and the maximum gain across the prospects being compared also feature prominently, and that can successfully account for some well-documented EU paradoxes. In the same way that Drechsler et al. (2014) axiomatizes the priority heuristic, our main contribution is to axiomatize range utility theory.

Range effects also appear in large empirical studies. In the context of human reinforcement learning, Bavard et al. (2018) find that the algorithm that best matches subject’s behavior is based on range-adaptation and reference dependence (see also Palminteri & Lebreton, 2021). Peterson et al. (2021) conducted largest experiment on risky choice to date and used the resulting dataset to power machine-learning algorithms that were constrained to produce psychological theories of risky choice. They tested a long list of well-known models and heuristics. They found that context-dependence is a key feature of a new, more accurate model, and identified minimum outcome and maximum outcome in the context – i.e. range – as well as outcome variability as the most important context effects.

In line with these findings, we introduce a formal context-dependent decision model that accounts for range adaptation and reference dependence, called range utility theory (RUT). As a key theoretical advantage, RUT retains a vNM linear structure when the range is fixed, and reproduces the utility shapes advanced by Markowitz (1952). On the descriptive side, RUT successfully explains the robust but elusive preference reversal phenomenon (Grether and Plott, 1979, Lichtenstein and Slovic, 1971); and the model is quite compatible with observed risk attitudes (e.g., Gonzalez and Wu (1999)), without the need of invoking nonlinear probability weighting.

Finally, we discuss the theoretical and practical advantages of RUT over rank-dependent models when it comes to elementary game theory applications. Indeed, range utility theory is linear in probabilities, hence allows a standard treatment of mixed strategies, provided the range is set by the payoffs of the game. If a player faces a new game with a different range, then her vNM utility for such game needs to be recalculated.

Our model builds on range dependent utility of Kontek and Lewandowski (2017). In this earlier model, each outcome in a prospect is normalized relative to the highest and lowest outcome of that prospect. Here, such outcome would be normalized relative to the highest and lowest outcome in the context, which may be wider. Thus, range dependent utility cannot in principle address preference reversals. Moreover, range dependent utility leans on scale and shift invariance, a rather simplistic axiom (if all the prizes of a risky prospect are shifted by some amount, then its certainty equivalents must also shift by this same amount, even if the shift turns some gains into losses or losses into gains, or substantially changes the decision maker’s wealth).