Unknown odds leave room for different kinds of rational choice

Choosing a gamble with known odds is different from choosing one whose odds are unclear. Economists call the first situation risk and the second uncertainty. Daniel Ellsberg’s famous paradox challenged the idea that people always handle uncertainty by assigning probabilities and then treating it like risk: they may instead prefer to avoid ambiguous odds.

In Ever since Ellsberg, Aluma Dembo, Shachar Kariv, Matthew Polisson and John K.-H. Quah revisit that distinction using detailed laboratory choices. Their central finding has two parts. Decisions under uncertainty were about as consistent with expected utility theory as decisions under risk. Nevertheless, uncertainty left substantially more room for alternative models that accommodate attitudes toward ambiguity.

Three possible outcomes, fifty decisions

The researchers reanalyzed an earlier experiment involving 154 participants, each making 50 portfolio choices. Participants divided an experimental budget among three assets. Each asset paid tokens in one possible outcome and nothing in the other two.

One outcome had a known probability of one-third. The other two together accounted for the remaining two-thirds, but their individual probabilities were unknown. Prices changed between rounds, allowing the researchers to examine how each person adjusted allocations across many different opportunities.

For comparison, they used a separate experiment with 168 participants and the same portfolio-choice structure, except that all three outcomes were explicitly equally likely. One randomly selected round determined payment.

The design also offered a way to avoid ambiguity: holding equal token payoffs in the two uncertain outcomes meant that their unknown probability split no longer affected the payoff distribution.

Testing consistency without prescribing a personality

Rather than assume a particular mathematical shape for someone’s preferences, the authors used revealed preference analysis. This asks whether observed choices could come from a consistent preference system, given what was affordable each time.

They applied progressively stronger requirements. The broadest asked whether choices could maximize a continuous utility function that prefers more to less. Further tests required respecting clear improvements in payoff distributions while allowing multiple possible probability assessments, then using one fixed probability distribution. The strongest required expected utility: evaluating outcomes by averaging their utility using probabilities.

The main single-probability benchmark assigned each outcome one-third, reflecting the experiment’s symmetry. Allowing unequal probabilities for the uncertain outcomes did not materially improve the reported results.

Because exact consistency is demanding across 50 choices, the researchers calculated cost-efficiency scores between zero and one. A score of one means full compatibility. Lower scores indicate how much the comparison budgets must shrink before the choices become compatible with a model. This is a money-based measure of inconsistency, not a measurement of actual cash lost.

Similar overall scores, different explanations

Only 20 of the 154 uncertainty participants were perfectly consistent with basic utility maximization. None were perfectly consistent with expected utility, although many scores were close to one.

Crucially, the distributions of basic-consistency and expected-utility scores looked similar across the risk and uncertainty experiments. Statistical tests did not detect differences: the reported p-values were approximately 0.434 and 0.226, respectively. That is evidence against an obvious deterioration in this setting, not proof that the two conditions are equivalent.

Alternative models nevertheless gained additional explanatory room under uncertainty. Models permitting multiple probability assessments could accommodate patterns that a fixed-probability approach could not. Comparisons with simulated choices supported the interpretation that this improvement reflected structured behavior rather than merely extra flexibility under the simulation’s assumptions.

The authors also directly tested a “kinked” model that can give extra weight to the worse-paying uncertain outcome. Its best-fitting ambiguity parameters clustered near the expected-utility benchmark, with a tilt toward mild ambiguity aversion.

No alternative won universally. The kinked model scored better than a broader fixed-probability alternative for 42.9 percent of participants; the reverse held for 44.8 percent. Choosing the better of those two approaches separately for each person raised the share scoring above 0.9 from 28.6 percent under expected utility to 50 percent.

What the findings leave open

These are results from an arXiv preprint, based on particular laboratory tasks and separate participant samples—not a demonstration of how everyone makes consequential real-world decisions. Better compatibility with observed choices also does not establish better prediction of future choices.

The authors did not directly test a prominent smooth ambiguity model because its unrestricted mathematical structure made a comparable test difficult. Their broader multiple-probability score instead bounds what models satisfying those restrictions could achieve. They also acknowledge debate over whether randomly paying one round can influence measured ambiguity attitudes.

The takeaway is therefore narrower than declaring people rational or irrational: unknown odds did not clearly worsen overall consistency here, but they changed which accounts of choice could explain the details. Individual differences remained central.

Source: Ever since Ellsberg, arXiv preprint, version 2.