Separating the Value of a Gamble From the Cost of Its Complexity

What makes a gamble complicated?

Standard expected utility theory evaluates a gamble by weighting the utility of each possible prize by its probability. The probability structure has no separate value. But experiments discussed in the preprint suggest that splitting an outcome into nearby, distinct prizes can change how people value a gamble beyond its expected-utility consequences.

In Decomposing the Cost of Complexity Under Risk, Xiaoge Dong and Lasse Mononen investigate which behavioral rules justify separating these components. Their contribution is mathematical: they derive representations of preferences from axioms, rather than running a new experiment or estimating how much complexity costs people.

A crucial distinction concerns what counts as a branch. The model combines events that pay exactly the same prize. Its “probability tree” describes probabilities attached to distinct final outcomes, not an arbitrary diagram or the number of lines used to present a gamble. Merely drawing an identical payoff twice does not create additional complexity in this framework.

Separating prizes from structure

The central representation evaluates a lottery as expected utility of its prizes minus an adjustment determined only by its probability tree. That adjustment is zero for a certain outcome. Initially, however, it need not be positive: the general model allows probability structure to be attractive as well as costly.

One key assumption is branch cancellation. Suppose two gambles share a branch with the same probability and prize. Replacing that shared prize with another common prize should not change their ranking, provided it remains distinct from the other outcomes. This restricts how prize values can interact with probability structure.

Another assumption requires respect for first-order stochastic dominance when gambles have the same probability tree. Roughly, when the structural component is unchanged, an unambiguously better distribution of prizes must be preferred. The authors impose this within-tree requirement rather than demanding dominance across all gambles.

Together with ordering and continuity assumptions, these restrictions deliver expected-utility weighting for branches whose probabilities are at most one-half. Larger-probability branches require another condition: adding a common outside prize to two gambles with the same tree must not reverse their ranking. With that restriction, the expected-utility-plus-adjustment representation extends to all lotteries.

Different meanings of complexity

The general adjustment allows interactions between branches: the structural effect of splitting one branch can depend on the rest of the probability tree. Strengthening cancellation from a common prize to a common sublottery eliminates those interactions. The adjustment then becomes a sum of contributions associated with individual branch probabilities, with a normalization for certainty.

A separate characterization asks when only the number of distinct prizes matters. Its key additional rule is same-support independence: if two lotteries offer exactly the same possible prizes, mixing each with a common lottery preserves their ranking.

This route permits discontinuities when outcomes appear or disappear, and uses strong unboundedness assumptions instead of mixture continuity. The resulting representation subtracts an outcome-count adjustment from expected utility for risky lotteries. Certain prizes may still be evaluated differently. A further consistency rule involving certainty equivalents—the guaranteed prizes considered equally good as particular gambles—aligns the evaluation of certainty and risk.

When the adjustment becomes a cost

A separate axiom says that mixing a preferred gamble with a weakly worse one cannot produce something better than the preferred gamble. Under the general representation, and assuming utility ranges over all real values, the authors show that this condition is equivalent to concavity of the structural adjustment.

That mathematical property makes the adjustment nonnegative and ensures that splitting a branch into distinct outcomes cannot reduce it. In the outcome-count model, the same behavioral condition makes costs weakly increase with the number of distinct prizes. These are conditional results, not evidence that everyone dislikes complexity.

Implications for experiments and insurance

For event-splitting experiments, the model separates changing payoffs from changing structure. As two distinct prizes approach one another, their expected-utility difference can shrink continuously while the structural adjustment remains unchanged. When they become exactly identical, their branches merge, potentially producing a discrete valuation change. The authors relate this prediction to previously published experimental findings, not new observations.

Deductible insurance supplies another illustration. Losses above a deductible produce the same final wealth because insurance covers the excess. Moving the deductible between adjacent loss levels leaves the probability tree unchanged. Reaching a loss threshold merges additional outcomes. With continuous premiums and utility, any resulting valuation jump comes from the structural adjustment. Different cost specifications predict different gains from that merger.

What remains unestablished

The preprint provides a framework for distinguishing models, not a measurement of cognitive effort or a universal account of insurance decisions. Its conclusions depend on behavioral axioms, finite monetary lotteries and particular continuity or unboundedness requirements. Whether actual choices satisfy those conditions—and which structural cost best describes them—requires empirical testing.