Decision Trees & Expected Value

Decision trees map out your choices and their possible consequences like a branching path, with each branch ending in an outcome and its chance of happening. Expected value is the long-run average result you'd get if you could repeat the exact same decision many times under identical conditions—it's calculated by multiplying each outcome by its probability and adding them all together. For example, a coin flip with a 50% chance of $1,200 and 50% chance of nothing has an expected value of $600—because (0.5 × $1,200) + (0.5 × $0) = $600. Try this: The next time you face an uncertain choice, sketch out the branches and calculate the expected value of each option.

How Decision Trees Work

A decision tree starts with your initial choice, then branches outward showing each possible outcome. At each branch point, you list what could happen and the probability of each outcome. For example, deciding whether to launch a product might branch into "high demand" and "low demand," with probabilities estimated from market research. The tree continues branching until you reach end points with concrete outcomes—profits, losses, or other results you care about.

Calculating Expected Value

Once your tree is built, you work backward from the outcomes. Multiply each final outcome by its probability to get its expected value. When multiple branches lead to the same decision point, add those expected values together. This process, called "folding back," lets you compare any set of choices by their average long-term result. For investment decisions, you'd compare the expected value of stocks versus bonds. In healthcare, doctors use this to compare treatments by their expected quality-adjusted life years.

Real-World Applications

Businesses use decision trees for capital investments—comparing expansion projects, new product launches, or strategic partnerships. A pharmaceutical company might map clinical trial outcomes: success probability, revenue if approved, costs if it fails. Insurance companies set premiums using expected value calculations of accident risks across millions of drivers. Even oil companies use decision trees to weigh drilling prospects: the probability of striking oil, the revenue if successful, and the drilling costs either way.

Limitations and Nuances

Expected value works best for repeated decisions where you can "play the averages." For one-time, high-stakes choices, people often care about risk tolerance—losing $10,000 hurts more than gaining $10,000 helps. Additionally, your probabilities are only as good as your estimates, and overconfidence can lead to inflated expectations. Decision trees also struggle with complex situations that have too many branches or unknown unknowns.

Expected value theory dates to the 17th century, but a key limitation emerged in the St. Petersburg paradox: a game with infinite expected monetary value that no one would actually pay much to play. Daniel Bernoulli (1738) resolved this by introducing "moral expected value"—what we now call utility—arguing that the usefulness of money decreases as you have more of it [1]. This insight became foundational: von Neumann and Morgenstern (1944) later proved that rational agents maximize expected utility, not expected monetary value [2].

However, people systematically deviate from expected utility in predictable ways. Kahneman and Tversky (1979) introduced prospect theory, demonstrating that losses weigh more heavily than equivalent gains—a phenomenon called loss aversion [3]. People also overweight small probabilities (explaining lottery tickets) and underweight moderate ones (ignoring real risks).

• Bernoulli (1738): Introduced diminishing marginal utility to explain why people won't pay large sums for games with infinite expected monetary value [1].
• von Neumann & Morgenstern (1944): Established expected utility theory as the normative model for rational choice under uncertainty [2].
• Kahneman & Tversky (1979): Introduced prospect theory, demonstrating that losses weigh more heavily than equivalent gains through loss aversion and probability weighting [3].
• Tversky & Kahneman (1992): Showed that people overweight rare events and underweight common ones when making decisions under risk [4].

Expected value assumes you care about average outcomes, but one-time decisions with catastrophic risks (like bankruptcy) can't be averaged. It also requires accurate probability estimates, and research shows we're systematically overconfident in our predictions. The model ignores emotional factors—losses hurt more than equivalent gains help. Finally, real decisions often involve too many variables to map completely, forcing simplification that may miss critical factors.