HPC › Decisions › Glossary
Expected value is the probability-weighted average of all possible outcomes of a decision, computed by multiplying each outcome's magnitude by its probability of occurring and summing the results. The formula reduces any uncertain choice to a single representative number, establishing a normative benchmark that rational decision theory uses to identify which option a fully informed decision-maker should prefer.
Expected value treats outcomes as raw magnitudes; expected utility replaces those magnitudes with subjective utilities, accommodating risk aversion and individual preferences.
How it works
The formula is EV = Σ(pᵢ × vᵢ): sum, across all possible outcomes, the product of each outcome's probability and its value. An investment with a 50% chance of returning £2,000 and a 50% chance of returning nothing has an expected value of £1,000. Von Neumann and Morgenstern demonstrated in 1944 that maximising expected value follows necessarily from four axioms of rational preference: completeness, transitivity, continuity, and independence 1. These axioms constitute the normative standard against which actual human behaviour is measured.
Human beings do not use raw probabilities at face value. Kahneman and Tversky established a systematic probability weighting function: small probabilities are overweighted, making lottery tickets feel more attractive than their EV warrants, while moderate-to-high probabilities are underweighted 2. The certainty effect compounds this: a guaranteed outcome is valued disproportionately above a probabilistically equivalent gamble. In canonical experiments, 80% of subjects preferred a certain gain of 3,000 over an 80% chance of 4,000, even though the gamble's EV of 3,200 exceeds the certain sum 2. The pattern persists under replication and across populations.
Salience theory extends this account: when one attribute of a choice option is contextually extreme or unusual, it captures disproportionate attention, distorting the effective weight given to that attribute independent of its objective probability or magnitude 4. EV departures cluster around predictable cognitive fault lines rather than appearing as random noise, which means they can be partially anticipated and corrected.
Example
A portfolio manager evaluating two investment options assigns probabilities to each return scenario. Option A offers a 60% chance of a 20% return and a 40% chance of a 5% loss, giving an EV of 10%. Option B offers a certain 8% return. The EV framework selects Option A. A manager anchored to certainty selects Option B and consistently forgoes two percentage points of expected return per decision cycle.
The maths identifies which choice maximises long-run returns; the bias identifies which choice feels safer, and the two rarely align.
Why it matters
Systematic departures from EV maximisation carry measurable economic costs. Loss aversion, probability weighting, and salience effects together account for significant mispricings in financial markets, suboptimal insurance uptake, and predictable errors in business investment decisions 2 4. People who score higher on the Cognitive Reflection Test are substantially more likely to select the EV-maximising option in lottery and time-preference tasks, indicating that deliberate analytical reasoning is the mechanism connecting EV literacy to improved decision quality 3.
Insurance markets offer a concrete illustration. Premiums are set above the actuarial EV of a loss, so policyholders consistently accept negative-EV contracts 1 2. This is not always a cognitive error: when the disutility of a large uninsured loss exceeds its probability-weighted cost, purchasing coverage can be utility-maximising even if EV-negative. Recognising where EV applies directly and where it requires adjustment for diminishing marginal utility is the core practical skill.
How do you calculate expected value?+
Multiply each possible outcome's value by its probability of occurring, then sum all the products. For a coin flip paying £10 on heads and £0 on tails, EV = (0.5 × 10) + (0.5 × 0) = £5. This result is the average payoff you would receive if you repeated the decision under identical conditions many times.
Why do people systematically ignore expected value in real decisions?+
The brain applies a probability weighting function rather than using raw probabilities: small chances are overweighted (lottery tickets) and moderate-to-high chances are underweighted. The certainty effect compounds this, causing a guaranteed outcome to be valued disproportionately above a probabilistically equivalent gamble, even when the gamble carries a higher EV.
Is expected value the same as expected utility?+
No. Expected value multiplies outcomes by their objective numerical magnitudes. Expected utility replaces those magnitudes with subjective utilities that reflect individual risk preferences, such as risk aversion. Von Neumann and Morgenstern developed expected utility theory to accommodate the fact that rational agents' preferences over uncertain outcomes need not be proportional to raw outcome values.
How do high-performers apply expected value thinking in practice?+
Professionals in poker, options trading, and clinical trial design explicitly track EV per decision rather than per outcome, separating process quality from result quality. This overrides the default tendency to judge decisions by their outcomes, a bias the cognitive reflection literature identifies as the primary obstacle to systematic EV use.
