Expected Value in Decisions
Trevor L Taylor, a Miami private investor whose investing grew out of systematic and quantitative models, works in a field where the most basic tool for weighing uncertain choices is expected value. The idea is plain arithmetic. For each possible outcome of a decision, multiply its value by its probability, then add the results; the option with the highest total is, on this measure, the better bet. Few concepts are more useful, and few are more often misapplied, because the sum hides as much as it reveals.
The arithmetic and where it came from
Encyclopaedia Britannica defines1 expected utility as the expected value of an action to an agent, calculated by multiplying the value of each possible outcome by the probability that it occurs and summing those numbers, and notes that the concept is used to explain decisions made under risk. A simple case shows the logic. A wager that pays $100 with a probability of one in four and nothing otherwise has an expected value of $25. Paying $20 to play is favorable on average; paying $30 is not.
The same arithmetic applies to choices with no casino in sight. Suppose a project has a 60 percent chance of returning $50,000, a 30 percent chance of returning nothing and a 10 percent chance of losing $40,000. Its expected value is $30,000 minus $4,000, or $26,000, and that figure can be set beside an alternative with a smaller but surer payoff. The discipline lies less in the multiplication than in the inputs: writing down every outcome, including the unpleasant ones, and attaching a probability to each one forces assumptions into view that a confident forecast would otherwise hide.
The arithmetic is old. Christiaan Huygens set it out in De Ratiociniis in Ludo Aleae, which a reprint hosted by Dartmouth’s mathematics department2 describes as the first book on probability theory ever published, issued in Latin in 1657. Its English translation carries a title that states the program plainly: the value of all chances in games of fortune, from cards and dice to wagers and lotteries.
Modern decision theory refined the idea by replacing money with utility. The Stanford Encyclopedia of Philosophy explains3 that the expected utility of an act is a weighted average of the utilities of its possible outcomes, where the utility of an outcome measures how strongly it is preferred to the alternatives. The entry stresses a distinction that matters in practice: evidence that people often fail to choose this way does not answer whether they should.
The method moved from philosophy into management largely through decision analysis. The New York Times obituary of Howard Raiffa4 credited the Harvard professor with pioneering decision science, a discipline spanning negotiation, conflict resolution and risk analysis, and recalled that his formulas were applied to the search for a missing nuclear bomb and to siting an airport for Mexico City. The same reasoning can frame public questions with high stakes. A Guardian article5 borrowed the structure of Pascal’s wager to argue about climate policy: when the possible losses from inaction are grave, the comparison of outcomes and their likelihoods can favor acting even under real uncertainty.
Where the simple sum misleads
The first warning came within a century of Huygens. The Stanford Encyclopedia’s entry on the St. Petersburg paradox6 describes a game introduced by Nicolaus Bernoulli in 1713. A fair coin is flipped until it lands heads, and the payout doubles with every flip. Because each additional flip halves the probability but doubles the prize, every term in the expected value sum equals one, and the total is infinite. Yet no one would pay more than a modest sum to play. Daniel Bernoulli’s response, published in 1738, was that the value of money to a person shrinks as wealth grows, so the game’s utility is finite even if its expected payout is not. That move from money to utility is the foundation of the expected utility theory described above.
A second warning concerns repetition. Expected value describes an average across many independent trials, but a person or a firm lives through one sequence of results, and a single ruinous outcome ends the sequence. In a paper for the Bell System Technical Journal7, J. L. Kelly Jr. showed that a gambler with an edge at fair odds can make capital grow exponentially, and he defined the quantity to maximize as the long run exponential rate of growth, computed with a logarithm of capital rather than its raw amount. The practical consequence is that a positive expected value does not license any bet size. Staking too much on a favorable bet can still lead to ruin, which is why position sizing comes before picking.
A third warning is psychological. When the Royal Swedish Academy of Sciences honored Maurice Allais8, its announcement noted that he is best known outside economics for his studies of risk and the Allais paradox, showing that the theory of maximizing expected utility, accepted for more than forty years, does not apply to many realistic decisions under risk and uncertainty. Kahneman and Tversky’s later work on loss aversion documented further departures, such as giving losses more weight than equal gains.
Even people with every incentive to calculate struggle in practice. The Economist9 reported on a betting exercise built around a biased coin and concluded that even a simple probability exercise shows people do not follow an optimal investment strategy. The lesson for decision makers is not to abandon expected value but to use it with three guardrails. Estimate probabilities honestly, ideally against base rates rather than hopes. Weigh outcomes by what they would mean, not only by their dollar size, so that a catastrophic loss counts for more than its arithmetic share. And size each commitment so that no single bad draw removes the chance to keep playing. Seen that way, the calculation is where a decision begins, the opportunity cost of each alternative is part of the comparison, and survival marks the edge of the playing field.
