Beyond the Odds: A Theory of How Paired Outcomes Shape Choice

Why the pairing matters

An uncertain alternative is more than a list of possible prizes. It also specifies which outcome occurs in each possible state of the world. Comparing two alternatives therefore creates a second layer of information: how their outcomes are paired across those states.

In Correlation Probabilistic Sophistication, Yi-Chun Chen, Soo Hong Chew and Xinhan Zhang make that pairing central. Their November 2026 arXiv preprint develops a mathematical framework, not an experimental finding about how frequently people choose this way.

The starting distinction is between an alternative’s individual outcome distribution and the joint distribution generated by comparing two alternatives. Individual distributions describe each option separately. The joint distribution describes the probabilities of outcome pairs: what one option delivers alongside what the other would deliver. The authors connect this perspective to theories of regret and salience, where the comparison with a forgone outcome can matter.

Keeping probabilities without discarding comparisons

The proposed principle, correlation probabilistic sophistication, or CPS, says that two comparisons producing the same joint distribution must receive the same preference judgment. Other details of their implementation become irrelevant once that distribution is fixed.

This remains a restriction on choice. It does not say that anything can affect a decision. But it allows more information to matter than ordinary probabilistic sophistication, which reduces each alternative to its own outcome distribution. Under CPS, unchanged individual distributions do not necessarily imply an unchanged comparison if the pairing changes.

The framework also does not initially require every pair of alternatives to be comparable or preferences to be transitive. Transitivity means, roughly, that preferring one option to a second and the second to a third cannot lead to preferring the third over the first.

Where subjective probabilities come from

Rather than simply assigning probabilities, the authors provide conditions under which probabilities can be derived from comparisons.

Their central device is “simultaneous event exchangeability.” Two events count as exchangeable if swapping their assigned outcomes in both alternatives never changes the comparison, whatever the background outcomes. The simultaneous swap preserves the structure of the outcome pairs being moved.

Together with three assumptions about comparative likelihood—Event Completeness, an Event Archimedean Property and Event Nonsatiation—the authors show that this structure yields a unique agreeing subjective probability and the CPS reduction.

That probability is finitely additive: probabilities add across finitely many disjoint events. It is also solvable, meaning that an event can contain a subevent matching the probability of an existing event with no greater probability. These properties do not automatically allow every imaginable probability mixture.

For that stronger flexibility, later results assume “convex range”: every event can be divided into any desired probability proportion. The distinction matters because the initial probability theorem alone does not establish the richness required for all subsequent representations.

A score for receiving one outcome and forgoing another

With convex range, every finite joint outcome distribution can be implemented by a pair of alternatives. The authors can then transfer representation results from choices with specified probabilities to subjective uncertainty.

Under continuity, they obtain a continuous numerical representation of binary comparisons. Adding completeness and Strong Independence yields the more restrictive correlation expected utility form.

Here, each outcome pair receives a comparison score. Reversing the pair reverses the score’s sign, and receiving the same outcome under both alternatives contributes zero. The overall comparison averages these scores using subjective probabilities. Unlike ordinary expected utility, this can retain sensitivity to how outcomes are paired and can permit nontransitive preferences.

When correlation sensitivity disappears

A central result also establishes a boundary. Within the paper’s single-source setting, CPS becomes ordinary transitive probabilistic sophistication when preferences are complete and transitive, the probability has convex range, and the induced joint-distribution preference satisfies the stated closedness condition.

Under those assumptions, only the individual outcome distributions matter. This is not a claim that transitivity alone eliminates every possible concern about correlation.

The final extension keeps different sources of uncertainty distinct through “small worlds,” each with its own subjective probability and potentially different comparison rule. Cross-source choices can be specified using source-specific certainty equivalents when those exist. An urn illustration shows how different source-specific utility functions can generate aversion to a mixed source even with symmetric probabilities.

The contribution is a conditional theoretical account: it clarifies how subjective beliefs and sensitivity to outcome pairing can coexist, and which additional commitments remove that sensitivity. It does not establish how well the framework predicts observed choices.