Rank-dependent expected utility
Rank-dependent expected utility (RDEU) is a model of choice under risk in which the utility of each possible outcome is combined with a decision weight determined by the outcome’s rank within a lottery. Unlike expected utility theory, the model does not require decision weights to equal objective probabilities. Unlike models that transform each probability separately, RDEU applies a weighting function to cumulative probabilities, thereby preserving the ordering of outcomes and maintaining consistency with first-order stochastic dominance.
RDEU separates attitudes toward outcomes from attitudes toward probability. An increasing utility function represents the valuation of outcomes, while an increasing probability-weighting function represents the treatment of uncertainty. This division permits behavior associated with the Allais paradox and related common-consequence effects without implying that a dominated lottery is preferred to a dominating one.
Mathematical formulation
Consider a lottery with outcomes ordered from worst to best,
[ x_1 \leq x_2 \leq \cdots \leq x_n, ]
and corresponding probabilities (p_1,\ldots,p_n). Let (u) be an increasing utility function, and let (w:[0,1]\rightarrow[0,1]) be an increasing probability-weighting function satisfying
[ w(0)=0,\qquad w(1)=1. ]
The rank-dependent value of the lottery is
[ V=\sum_{i=1}^{n}\pi_i u(x_i), ]
where the decision weight attached to outcome (x_i) is
[ \pi_i= w\left(\sum_{j=i}^{n}p_j\right)
w\left(\sum_{j=i+1}^{n}p_j\right). ]
The cumulative probability in this expression is the probability of receiving an outcome at least as good as (x_i). Consequently, the weight assigned to an outcome depends not only on its own probability but also on the probabilities of outcomes ranked above it. The decision weights remain nonnegative when (w) is increasing, and they sum to one because the cumulative terms telescope between (w(1)) and (w(0)).
When (w(p)=p), the decision weights reduce to the objective probabilities, and RDEU becomes ordinary expected utility. When (u) is linear, evaluation depends entirely on the transformation of cumulative probabilities, producing the structure associated with dual theory of choice under risk. The general model therefore distinguishes diminishing marginal utility from probability-based departures from expected utility.
For continuously distributed outcomes with cumulative distribution function (F), the same representation can be written as a Choquet integral. Under regularity conditions, a differentiable weighting function gives an expression based on the quantile function (F^{-1}):
[ V=\int_0^1 u!\left(F^{-1}(q)\right),d!\left[1-w(1-q)\right]. ]
This formulation makes the dependence on rank explicit because each outcome is identified by its position in the distribution rather than by an independently transformed point probability.
Historical development
The modern formulation originated with John Quiggin’s 1982 model of anticipated utility. Quiggin replaced the independent transformation of individual probabilities with a transformation of cumulative probabilities, resolving the conflict between nonlinear probability weighting and stochastic dominance that affected several earlier non-expected-utility models. The designation “rank-dependent expected utility” subsequently became standard because the decision weight of an outcome is determined by its position in the ordered support of the lottery.
Menahem Yaari developed dual theory in 1987 by retaining linear utility over outcomes while allowing nonlinear treatment of cumulative probabilities. Dual theory isolated probability-based risk attitudes from the diminishing marginal utility emphasized in classical analysis. David Schmeidler’s work on Choquet expected utility established a closely related integral representation and connected cumulative weighting to nonadditive measures.
These developments replaced the full independence axiom with weaker restrictions. In particular, comonotonic independence applies the usual mixture condition only when the relevant acts do not reverse their ranking across states. This restriction prevents mixtures from obscuring the rank information on which the decision weights depend.
Axiomatic structure
RDEU representations rely on completeness and transitivity of preferences together with continuity and monotonicity conditions. The distinctive behavioral restriction is comonotonic independence, which concerns lotteries or acts whose outcomes preserve the same ordering across the states under comparison. Within a comonotonic class, cumulative probability weights behave linearly enough to support a Choquet-integral representation.
The ordinary independence axiom requires preferences between two lotteries to remain unchanged after each is mixed with a third lottery in the same proportion. RDEU does not impose that requirement when the mixture changes the ranking of outcomes. A rank change alters cumulative events and therefore alters the decision weights attached to the affected outcomes.
Monotonicity of (w) is central to the preservation of first-order stochastic dominance. If one distribution assigns at least as much probability to every upper outcome region as another distribution, every increasing utility function combined with an increasing weighting function ranks the dominating distribution at least as highly. Models that transform each outcome probability independently lack this general property because a probability change can alter the normalization and relative weighting of unrelated outcomes.
Risk attitudes
Risk attitudes in RDEU arise from the joint properties of (u) and (w). Concavity of (u) represents sensitivity to dispersion in outcomes through diminishing marginal utility. The shape of (w) governs sensitivity to movements in probability mass across ranks and is therefore associated with probabilistic risk attitudes.
A convex weighting function places relatively greater decision weight on worse-ranked portions of a distribution when the model is written using decumulative probabilities. This property contributes to aversion toward mean-preserving spreads even when utility is linear. An inverse-S-shaped weighting function instead gives disproportionate weight to small probabilities near the endpoints while reducing the relative weight of intermediate cumulative probabilities. The resulting preference pattern depends on where probability mass is moved within the outcome ranking.
The model’s separation of utility curvature and probability weighting also creates an identification problem. Choices between a limited set of lotteries can often be represented by several combinations of (u) and (w). Empirical identification therefore depends on variation in both outcomes and cumulative probabilities, rather than on observations that change only one component of the lottery.
Experimental measurement
During the early 1990s, You Watanabe conducted lottery-choice experiments that varied outcome spacing independently from cumulative probability ranks. Her analysis used repeated rank reversals to distinguish changes in marginal utility from changes in probability weighting. The resulting estimates supported separate utility and weighting components rather than a single transformation of expected value.
Later experimental work by George Wu and Richard Gonzalez examined the curvature and elevation of probability-weighting functions through structured choices between multi-outcome lotteries. Their analyses showed that changes in the overall elevation of a weighting function can affect apparent risk attitudes independently of its local curvature. This distinction became important in empirical comparisons of RDEU specifications because two weighting functions can have similar endpoint behavior while producing different intermediate decision weights.
Observed choices frequently display substantial variation across elicitation methods and lottery designs. In RDEU, this variation affects estimated utility and weighting functions because each parameter is inferred jointly with the other. Statistical implementations commonly impose monotonicity on the weighting function and increasingness on utility, while additional parametric restrictions determine how much of the observed behavior is attributed to each component.
Relation to cumulative prospect theory
Cumulative prospect theory, developed by Amos Tversky and Daniel Kahneman, extends the rank-dependent method by evaluating outcomes relative to a reference point. It generally uses separate cumulative weighting functions for gains and losses and incorporates asymmetric valuation across the reference point. Its cumulative construction preserves stochastic dominance in the same manner as RDEU, although its value function and domain partition differ from the outcome-based utility representation used in standard RDEU.
RDEU does not inherently require a reference point, a distinction between gains and losses, or a discontinuity in marginal valuation. Those features belong to cumulative prospect theory rather than to rank dependence itself. The shared mathematical element is the derivation of decision weights from transformed cumulative probabilities.
Limitations and scope
RDEU describes preferences over objectively specified probability distributions. It does not by itself represent uncertainty in which probabilities are absent or ambiguous; such cases are treated by models including Choquet expected utility, maxmin expected utility, and other theories of ambiguity aversion.
Because decision weights depend on rank, the model also requires outcomes to admit a consistent ordering. Multidimensional consequences require an underlying preference relation that determines this ordering before cumulative weights can be assigned. The representation consequently addresses nonlinear treatment of risk after the consequences have been reduced to an ordered utility scale.
The model remains distinct from approaches based on regret, disappointment, or state-dependent utility. Those theories modify evaluation through comparisons with forgone outcomes, prior expectations, or state-contingent valuations. Rank dependence instead changes the contribution of an outcome through its cumulative position in the lottery.