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Decision-Making under Combinatorial Risk

This paper introduces an investment-allocation task to study combinatorial risk, revealing that people typically rely on core features like probability increments rather than exact distribution evaluation, but shift toward lottery valuation when the full probability mass function is explicitly displayed, a pattern best explained by a hybrid model combining feature-based heuristics with prospect theory.

Original authors: Yifan Hong, Hongmiao Fan, Chen Wang

Published 2026-06-10

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yifan Hong, Hongmiao Fan, Chen Wang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Idea: The "Mystery Box" vs. The "Menu"

Imagine you are a manager at a company. You have a limited budget for a marketing promotion, and you can only spend it on one of two customers: Customer A or Customer B.

Customer A usually buys 40% of the time. If you promote them, they might buy 60% of the time.
Customer B usually buys 80% of the time. If you promote them, they might buy 90% of the time.

Your goal is to maximize the total number of sales from both customers combined.

This is what the researchers call Combinatorial Risk. It's like a puzzle where the final result isn't given to you directly. Instead, the result is "induced" (created) by combining two separate, uncertain events. You have to figure out the odds of the total outcome based on the odds of the individual parts.

The Experiment: Two Ways to Play

The researchers ran a game with thousands of people to see how they solve this puzzle. They split the players into two groups:

The "Mystery Box" Group (Control): These players were only told the starting odds and how much the promotion would boost them (e.g., "A goes from 40% to 60%"). They had to do the math in their heads to guess which choice would lead to more total sales.
The "Menu" Group (Treatment): These players were given the exact same info, plus a clear "menu" showing the final probabilities. They saw a list like: "If you pick A, there is a 20% chance of 0 sales, a 50% chance of 1 sale, and a 30% chance of 2 sales."

What Did They Find?

1. Humans are Good at "Heuristics" (Shortcuts)

When people didn't have the "Menu" (the final probabilities), they didn't try to calculate the complex math of the whole situation. Instead, they used simple rules of thumb:

Rule A: "Pick the customer who gets the biggest boost." (If A gets a +20% boost and B gets +10%, pick A).
Rule B: "If the boosts are equal, pick the customer who was already more likely to buy." (If both get a +10% boost, pick the one who started at 80% rather than 40%).

The researchers found that people followed these rules very consistently. They were essentially navigating the risk by looking at the core features of the problem (the boost and the starting point) rather than calculating the exact final outcome.

2. The "Menu" Changes How We Think (But Not Necessarily for the Better)

When the researchers gave the "Menu" (the final probabilities) to the second group, something interesting happened:

Behavior Shifted: People stopped relying as much on the simple "boost" rules. They started looking at the final numbers on the menu.
Did they get richer? Surprisingly, no. Even though they had more information, they didn't make significantly better choices in terms of total sales. In fact, their choices became more rigid and less flexible.

It's like giving someone a detailed map of a maze when they were already good at navigating it by feeling the walls. The map didn't help them finish faster; it just made them stare at the paper more and move less intuitively.

The "AI Detective" (Symbolic Regression)

To understand exactly how people were thinking, the researchers didn't just guess theories. They used a special type of AI called Symbolic Regression.

Think of this AI as a detective that looks at thousands of decisions and tries to write a simple math formula that explains them. Instead of the researchers saying, "I think people use Prospect Theory," the AI searched through millions of possible formulas to find the one that fit the data best.

What the AI Discovered:

For the "Mystery Box" group: The best formula was simple. It relied almost entirely on the "boost" and the "starting point." It proved that people were not calculating the complex final odds in their heads.
For the "Menu" group: The best formula was a hybrid. It combined the simple rules with a more complex look at the final numbers (the "Menu"). This confirmed that when people see the final odds, they switch strategies to look at the distribution of outcomes, but they still keep some of their old habits.

The Takeaway

The paper concludes that when we face complex risks made of multiple parts (Combinatorial Risk), we don't act like super-computers calculating every possible outcome.

Without the full picture: We focus on the most obvious, salient features (like "who gets the biggest boost?").
With the full picture: We change our strategy to look at the final distribution, but this doesn't necessarily make us smarter or more profitable.

In short, humans navigate complex risks by focusing on the key ingredients rather than trying to bake the whole cake in their heads. Giving them the recipe (the full probability list) changes how they taste the ingredients, but it doesn't necessarily make the cake taste better.

Technical Summary: Decision-Making under Combinatorial Risk

Problem Definition

The paper addresses a gap in decision-making under risk research, which traditionally focuses on single-shot lottery choices where probability distributions are explicitly given. The authors introduce combinatorial risk, a class of problems where risk arises from the combination of multiple independent risky components, and the outcome distribution is induced rather than presented outright.

The authors formalize this using a two-component investment-allocation task. In this scenario, a decision-maker must allocate a single, indivisible investment to one of two independent Bernoulli components (A or B).

Initial State: Components have initial success probabilities $P^{ini}_A$ and $P^{ini}_B$ .
Action: Investing in a component increases its success probability by a specific increment ( $\Delta P_A$ or $\Delta P_B$ ).
Outcome: The total payoff depends on the sum of successes ( $U = X_A + X_B$ ), inducing a Binomial distribution over $\{0, 1, 2\}$ .
Challenge: The induced probability mass function (PMF) is not directly observable in the "control" condition; decision-makers must infer the risk profile from the component-level features.

The study investigates whether humans represent these problems as explicit induced lotteries or rely on heuristics based on salient combinatorial features. It further examines how providing the induced PMF (the "treatment" condition) alters decision strategies and whether this leads to more advantageous choices.

Methodology

Experimental Design

The authors conducted an online experiment with $N=2,640$ participants using a vendor-marketing framing to avoid moral confounds.

Conditions:
- Control (C): Participants saw only component features ( $P^{ini}$ , $\Delta P$ ).
- Treatment (T): Participants saw the component features plus the induced PMFs for total successes under both options.
- Magnitude: Two payoff magnitudes ($30 and $100) were used to test sensitivity to payoff size.
Stimuli: 1,873 decision problems generated from a discrete grid of probabilities, grouped into batches.
Metrics: Aggregate choice probabilities (proportion choosing Option B) were recorded across conditions.

Modeling Approach

To explain behavioral patterns, the authors employed a multi-stage modeling strategy:

Benchmark Models: They tested standard heuristics (Probability Gain, After-investment Probability, Tail-Probability models) and theory-grounded models (State-Space Utility, Expected Utility, Prospect Theory, Cumulative Prospect Theory) applied to the induced lotteries.
Symbolic Regression (SR): Recognizing that hand-crafted theories were insufficient, the authors developed a hybrid, ontology-guided SR framework.
- Architecture: An island-based evolutionary search (NSGA-II) runs in parallel across conditions.
- LLM Integration: An LLM performs local refinement of Pareto-optimal candidates and synthesizes new expressions based on a shared ontology.
- Ontology: Discovered expressions are parsed into a shared ontology to identify recurring functional forms and behavioral concepts, guiding the exploration of under-explored regions.
- Input Structure: Unlike standard SR, inputs are typed (Scalar/Vector) to preserve the structure of probabilities and outcomes.
Residual Analysis: To isolate the effect of the PMF display, the authors fitted a residual model to the treatment data using the predictions of the best Control model as a baseline, adding a Prospect Theory (PT) component to explain the variance.

Key Results

Behavioral Findings

Dominance of Combinatorial Features: In the absence of PMFs (Control), participants systematically favored options with higher probability increments. When increments were equal, they preferred the option with higher initial (and thus after-investment) success probability, consistent with a preference for lower variance.
Impact of PMF Display: Revealing the induced PMF (Treatment) significantly altered choice patterns:
- Participants became less responsive to the core combinatorial-risk features (increments and initial probabilities).
- Choice variance across problems decreased.
- Crucially, providing the PMF did not improve decision quality; the Treatment group did not achieve significantly higher expected payoffs than the Control group.
Magnitude Sensitivity: In the Control condition, higher payoff magnitudes increased sensitivity to variance differences. In the Treatment condition, this dependency vanished, suggesting a shift in decision strategy.

Modeling Discoveries

Control Condition: The best-performing models discovered via SR relied primarily on after-investment success probabilities ( $P^{aft}_A - P^{aft}_B$ $P_{A}^{a f t} - P_{B}^{a f t}$ ) and probability increments ( $\Delta P_A - \Delta P_B$ $Δ P_{A} - Δ P_{B}$ ).
- These models did not require exact evaluation of the full induced PMF.
- They outperformed standard heuristic and state-space utility benchmarks.
- Exact expected payoff appeared only in the most complex models and had weak influence, suggesting humans do not explicitly calculate expected value in this setting.
Treatment Condition: The best models combined combinatorial features with nonlinear transformations of the expected payoff derived from the induced lottery (e.g., log or power transformations of the expected value difference).
Residual Analysis: The shift from Control to Treatment behavior was effectively captured by layering a Prospect Theory evaluation of the displayed PMF on top of the Control model's predictions. This confirms that the PMF acts as an additional valuation layer rather than replacing the underlying heuristic.

Contributions and Significance

Scientific Contributions

Definition of Combinatorial Risk: The paper formalizes a distinct class of decision problems where risk is induced by component combinations, distinguishing it from standard lottery choice.
Behavioral Insight: It demonstrates that humans navigate combinatorial risk by reasoning over salient probabilistic features (increments, post-investment probabilities) rather than explicitly constructing induced lotteries.
Methodological Framework: The authors present an ontology-guided hybrid symbolic regression framework that combines evolutionary search with LLM-based generation and semantic organization. This approach successfully discovered compact, interpretable models that outperform hand-crafted theories and standard benchmarks.
Information Value: The study provides evidence that providing full distributional information (PMFs) changes decision heuristics but does not necessarily lead to more optimal outcomes, highlighting a potential cognitive cost or shift in attention that does not translate to better performance.

Claims on Significance

The paper claims that its findings challenge the assumption that decision-makers represent complex risks as explicit lotteries. Instead, it suggests a two-layer cognitive process:

Primary Layer: Reasoning over core combinatorial features (increments and success probabilities).
Secondary Layer: Utilizing distributional information (PMFs) as an added valuation layer only when explicitly available, which shifts attention away from the core features but does not improve decision quality.

The authors position this work as a productive domain for the intersection of behavioral theory and interpretable machine learning, demonstrating that symbolic regression can recover psychologically meaningful regularities that standard theories miss, particularly in settings where computational complexity prevents exact evaluation.