The history of expected utility theory (EUT) started with Bernoulli’s work resolving the St. Petersburg paradox (Stearns, 2000). Several axiomatic schemes for EUT are known (Luce and Raiffa, 1989, Savage, 1972). Currently, EUT has applications in a wide range of fields, including economics (Hey, 1997), psychology (Baron, 2000), evolutionary game theory (Hofbauer, Sigmund, et al., 1998), and general artificial intelligence (Everitt, Leike, & Hutter, 2015).

EUT shows how to choose between two lotteries (Hey, 1997, Luce and Raiffa, 1989, Savage, 1972): (x,p)=x1x2...xnp1p2...pn,(y,q)=y1y2...ynq1q2...qn,∑k=1npk=∑k=1nqk=1, where (p1,…,pn) and (q1,…,qn) are (resp.) the probabilities of monetary outcomes (x1,…,xn) and (y1,…,yn) within each lottery. The probabilities reflect the objective and actual chances of different outcomes happening in a given situation. These probabilities are considered accurate and truthful representations of reality (veridical probabilities). EUT proposes the following functional for each lottery (Hey, 1997, Luce and Raiffa, 1989, Savage, 1972): V(x,p)=∑i=1nu(xi)pi,where u(xi) is the utility of the monetary value xi. EUT recommends choosing in (1) the first lottery, if V(x,p)>V(y,q).

Experiments revealed problems with EUT and its axiomatic foundations. In particular, several classic experiments cannot be explained by EU for any choice of the utility function u(.) in (3) (Allais and Hagen, 2013, Machina, 1992). People generally choose in contradiction to EUT, violating the independence axiom, one of the four axioms of the von Neumann–Morgenstern formulation of EUT (Luce & Raiffa, 1989). The most prominent example of this is Allais’s paradox (Allais & Hagen, 2013), where each human subject chooses between two lotteries. The prospect theory (Kahneman and Tversky, 2013, Tversky and Kahneman, 1992), and rank-dependent utility theory (Diecidue and Wakker, 2001, Quiggin, 2012) discarded the independence axiom, and proposed functionals similar to V(x,p) in (3), where instead of probabilities pi one employs weights πi that generally depend both on (p1,…,pn) and (x1,…,xn). Baron, 2000, Hey, 1997, Machina, 1992 discuss these and other alternatives to EUT.

There are also other situations where EUT does not apply. EUT cannot be used directly when the lottery outcome remains uncertain even after the lottery choice has been made. A good example of this situation is the decision problem known as Savage’s omelet (Savage, 1972). To our knowledge, this problem was never studied from the perspective of EUT’s inapplicability. The problem is not in questioning a specific axiom of EUT. Rather, EUT does not work for deeper, conceptual reasons.

We confine ourselves to these two problems because they amount to important (but different) limitations of EUT, which are resolved the regret theory. Other violations of EUT including cases with unknown probabilities (e.g. Ellsberg paradox), and known examples of transitivity violations such as framing effects and preference reversal phenomena are not addressed.

As we show below, both Allais’ paradox and Savage’s omelet can be resolved by the regret theory (RT), which is one of the alternatives of EUT. The main difference of RT compared to EUT is that RT does not operate with a value functional for a single lottery. Instead it counter-factually compares two lotteries. RT has an intuitive emotional appeal, and it is also related to cognitive aspects of decision making (Bourgeois-Gironde, 2010). RT was first proposed by Savage in minimax form (Savage, 1972) [see Acker (1997) for an update of this approach], and later brought to its current form in Bell, 1982, Loomes and Sugden, 1982; see Bleichrodt and Wakker, 2015, Machina, 1992 for a review. Loomes and Sugden (1982) extended the regret to independent lotteries and noted its potential in explaining Allais’ paradox.1 Loomes and Sugden (1982) also analyzed transitivity, common ratio effect, and preference reversals. Functional forms involving two lotteries were given axiomatic foundation in Fishburn (1981). An axiomatic formulation of regret was attempted in Sugden (1993).

This work has three purposes. First, we want to show how Allais’ paradox is solved by a transitive and super-additive RT. People mentioned regret in the context of Allais’ paradox [see e.g. Baron, 2000, Bourgeois-Gironde, 2010, Loomes and Sugden, 1982], but so far no systematic and complete solution of this paradox was provided. Our solution is rather complete, because it also predicts conditions under which the paradox does not hold. Both transitivity and super-additivity have transparent meaning for regret theories in general. We do clarify their applicability range. This is especially important for transitivity, because generally regret theories do not lead to transitive predictions (Starmer, 2000). While in context of Allais’ paradox lotteries are compared in pairs, in more general frameworks, with more than two lotteries involved, the violation of transitivity can lead to predictive indecisiveness (Bar-Hillel & Margalit, 1988). So transitivity condition provides applicability of regret theory to general decision problems. Transitivity is one of cornerstones of classic rationality, and it may intrinsically be present even in some of those situations, which look intransitive at an aggregated level (Regenwetter, Dana, & Davis-Stober, 2011).2

Second, we prove that the transitive and super-additive regret theory is consistent with the stochastic dominance criterion (Luce & Raiffa, 1989). Stochastic dominance is a useful tool, but it does not apply to comparing any pair of lotteries. The previous literature in this direction is mostly negative showing that regret-based approaches violate first order stochastic dominance (Levy, 2017, Quiggin, 1990).3 Third, we demonstrate—using as an example Savage’s omlet problem—that RT can recommend choosing between lotteries with not resolved outcomes, a task which cannot be consistently addressed by EUT.

The paper is organized as follows. Section 2 is devoted to regret functional for independent lotteries and some of its properties related to the expected utility. In decision making theory the functional form is frequently derived from axiomatic foundation. In contrast, here we first introduce the functional considered, then derive its properties. Section 3 is devoted to Allais’ paradox and its relations to other concepts. Stochastic dominance abidance is considered in Section 4. Section 5 analyzes Savage’s omelet problem, identifies an aspect that prevents the applicability of the expected utility theory, and solves this problem via the regret. We summarize in the last section.