Daniel Bernoulli

Daniel Bernoulli stands at the beginning of the book’s history of behavioral economics as the theorist who made risk a psychological problem rather than a purely mathematical one. His central insight was that people evaluate gambles through the subjective value of outcomes, not through money alone. By proposing a concave utility function, he explained why people commonly prefer a sure amount to a gamble with an equal or even slightly higher expected value. The later development of prospect theory both inherited this achievement and identified the crucial variable his model lacked: the reference point from which gains and losses are judged.

Utility and the Psychology of Wealth

Bernoulli argued that the psychological value of money depends on a person’s existing wealth. A gift of 10 ducats to someone who already possessed 100 ducats could have the same utility as a gift of 20 ducats to someone whose wealth was 200 ducats. The relevant comparison was therefore proportional rather than absolute: a 30 percent raise could produce a broadly similar psychological response for rich and poor people, whereas an additional $100 would not have the same meaning at different levels of wealth.

This reasoning led Bernoulli toward a logarithmic account of utility. As wealth increased, each additional unit produced a smaller psychological increment. The same psychological distance could therefore separate $100,000 from $1 million as separated $10 million from $100 million. In modern terms, Bernoulli identified diminishing marginal value: an additional million mattered greatly to someone with one million, but much less to someone who already had nine million.

Risk Aversion and the Value of Gambles

Before Bernoulli, mathematicians generally assessed a gamble by its expected value, the probability-weighted average of its monetary outcomes. A prospect offering an 80 percent chance of $100 and a 20 percent chance of $10 has an expected value of $82. Yet most people prefer $80 for sure. Bernoulli treated this apparent departure from arithmetic as evidence that decisions are based on utilities rather than dollar totals.

His model explained risk aversion by replacing the expected value of money with the expected value of utility. People usually dislike the possibility of receiving the lowest outcome, and a risk-averse decision maker may accept a sure amount below a gamble’s expected monetary value in order to avoid uncertainty. The sure payment is, in effect, worth a premium. Because a concave utility function makes the psychological difference between lower amounts larger than the difference between comparable higher amounts, a sure gain of $800 can be preferred to an 80 percent chance of winning $1,000 even when the two prospects have the same monetary expectation.

Bernoulli’s contribution was therefore both mathematical and psychological. He did not deny that probabilities matter; he changed what was being averaged. The value of a gamble became the probability-weighted average of subjective utilities, allowing a formal model to represent an ordinary human preference that expected monetary value alone could not explain.

The Missing Reference Point

The book presents Bernoulli’s theory as powerful but incomplete. In Bernoulli’s framework, the current state of wealth is sufficient to determine utility. Prospect theory, developed by Daniel Kahneman and Amos Tversky, requires an additional moving part: the earlier state relative to which an outcome is evaluated.

That distinction explains why objectively equivalent choices can produce different preferences. If a person is given $1,000 or $2,000 before choosing an outcome, the gift may barely enter conscious attention because it has already been absorbed into the reference point. Being richer by $1,500 can then be experienced as a $500 gain in one formulation and a loss in another, depending on whether the implied reference point is $1,000 or $2,000 higher. The person’s attitude toward the outcome changes even though the final wealth is the same.

This is not simply a refinement of the old utility calculation. The psychological value of winning $100 and losing $100 is not derived only from their effects on total wealth. People like winning and dislike losing, and the dislike of losing is usually stronger. Prospect theory consequently evaluates changes relative to a reference state rather than treating ultimate wealth as the sole carrier of value.

From Bernoulli to Prospect Theory

Daniel Kahneman and Amos Tversky spent five years examining intuitive choices between gambles, often testing simple decision problems on themselves before seeking broader verification. Their eventual essay, “Prospect Theory: An Analysis of Decision under Risk,” was modeled closely on utility theory but was explicitly descriptive: it sought to document systematic violations of the axioms of rationality rather than prescribe the most rational choice.

Prospect theory retained Bernoulli’s psychological orientation while adding three features associated with automatic judgment. Outcomes are evaluated relative to a reference point; sensitivity to changes diminishes as the size of the gain or loss grows; and losses loom larger than corresponding gains. Together, these principles produce an S-shaped value function with a sharp asymmetry at the reference point. The result preserves Bernoulli’s explanation of diminishing sensitivity while showing why the same amount can feel different according to whether it is framed as a gain or a loss.

The contrast also clarifies the book’s distinction between Econs and Humans. The idealized Econ evaluates choices coherently through comprehensive incentives and stable preferences. Humans, by contrast, use System 1, attend to the information available at the moment, and often judge outcomes through narrow mental accounts. Richard Thaler’s later account of mental accounting extends this behavioral tradition: people separate spending money, savings, earmarked funds, and emotional balances in ways that can be useful for self-control but can also produce sunk-cost errors and other costly decisions.

Historical and Intellectual Significance

Bernoulli’s importance lies in having opened a conversation between economic theory and psychological observation. In the early 1970s, Daniel Kahneman received a dark-red mimeographed essay by Bruno Frey stating that economic theory assumed an agent who was rational, selfish, and stable in taste. The contrast with psychological evidence made the two disciplines appear to be studying different species, a divide Richard Thaler later named the difference between Econs and Humans.

Bernoulli’s theory supplied the earlier bridge across that divide. It accepted the formal language of economics while insisting that the evaluation of outcomes must reflect how wealth is experienced. Prospect theory later showed that even this psychologically informed model was too simple because it omitted reference points, framing, loss aversion, and the changing sensitivity of gains and losses.

His legacy is thus double-edged. Bernoulli made risk aversion intelligible by giving utility a psychological basis, but the later theory made clear that people do not merely value states of wealth. They react to transitions, expectations, entitlements, and perceived losses. The intellectual movement from Bernoulli to prospect theory marks a shift from asking what wealth a person possesses to asking how the person experiences a change—and from treating that experience as a deviation from rationality to modeling it as a systematic feature of human judgment.