Friedman–Savage utility function¶

A wealth-dependent expected-utility curve with alternating concave and convex regions intended to explain simultaneous insurance purchase and lottery play.

Version: v2 · 2026-09-19 · History
Domain-specific #: 4622
Domain group: Social Sciences
Origin domain: Economics & Finance
Subdomain: Decision Theory → Economics & Finance

Core Idea¶

The curve is concave at lower wealth, convex across an intermediate range and concave again at higher wealth, so local risk attitudes change as outcomes cross wealth regions. Expected utility weights outcome utilities by probability; changes in curvature alter whether a mean-preserving spread is disliked or preferred and can make insurance and gambling rational for the same person. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

How would you explain it like I'm…

The Wiggly Happiness Line

Some grown-ups pay to protect their house from fire, and the same grown-ups also buy lottery tickets. That seems odd! Two economists drew a picture of how much happiness each amount of money gives, and the line bends one way, then the other way, then back again. Where the line bends one way, people play it safe; where it bends the other way, taking a chance looks good.

Insure-and-Gamble Curve

Economists sometimes imagine a curve showing how much 'happiness value' a person gets from different amounts of money. The Friedman–Savage curve has three parts. At low and high amounts, each extra dollar adds a bit less happiness than the one before, which makes people dislike risk—so buying insurance makes sense. In the middle, extra dollars add more and more happiness, which makes a gamble that could jump you up look attractive. That is how one person can sensibly both buy insurance and play the lottery.

Concave-Convex-Concave Wealth Utility

The Friedman–Savage utility function is a utility-of-wealth curve that is concave at low wealth, convex in a middle range, and concave again at high wealth. In expected-utility theory, you weight each outcome's utility by its probability. Where the curve is concave, spreading outcomes out while keeping the average the same lowers expected utility, so the person avoids risk. Where it is convex, such a spread raises expected utility, so the person welcomes risk. Because the curvature changes, the same rational person can buy insurance against losses and also accept gambles that might push them into a higher wealth region. To apply it, you must specify the person's starting wealth, where the bends are, and the exact lotteries or insurance contracts involved.

The Friedman–Savage utility function is a utility-of-wealth curve with three curvature regions: concave at lower wealth, convex over an intermediate range, and concave again at higher wealth. In expected-utility theory a lottery is valued by the probability-weighted sum of outcome utilities. The sign of curvature determines the response to a mean-preserving spread: concavity implies the spread is disliked (risk aversion), while convexity implies it is preferred (risk seeking). Local risk attitude therefore changes as outcomes cross between wealth regions. This lets the same agent rationally buy insurance against downside losses while accepting gambles whose prizes reach into or across the convex segment. Applying the model properly requires stating the wealth variable, the utility normalization, where the curvature changes sign, the specific lotteries or insurance contracts, the initial wealth, and the expected-utility calculation. A prediction should only be made from those explicit ingredients.

Scope of Application¶

Friedman–Savage utility function belongs to decision theory and is useful where the analyst can specify the typed decision theory carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the wealth variable and utility normalization, locations and signs of curvature, lotteries or insurance contracts, initial wealth, expected-utility calculation and behavioral prediction are explicit. The scope is broad within that domain but bounded by the need for the wealth variable and utility normalization, locations and signs of curvature, lotteries or insurance contracts, initial wealth, expected-utility calculation and behavioral prediction are explicit. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the wealth variable and utility normalization, locations and signs of curvature, lotteries or insurance contracts, initial wealth, expected-utility calculation and behavioral prediction are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Friedman–Savage utility function can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Friedman–Savage utility function. Friedman–Savage utility function compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed decision theory carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express the wealth variable and utility normalization, locations and signs of curvature, lotteries or insurance contracts, initial wealth, expected-utility calculation and behavioral prediction are explicit independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of decision theory because they reuse the typed decision theory carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets, Expected utility weights outcome utilities by probability; changes in curvature alter whether a mean-preserving spread is disliked or preferred and can make insurance and gambling rational for the same person., and type the carrier, state every parameter and convention in the definition, test that the wealth variable and utility normalization, locations and signs of curvature, lotteries or insurance contracts, initial wealth, expected-utility calculation and behavioral prediction are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Friedman–Savage utility function Domain-specific

Parents (1) — outgoing typed relations

Friedman–Savage utility function is a kind of Expected Utility Prime

It remains its own entry because its identity is fixed by the wealth variable and utility normalization, locations and signs of curvature, lotteries or insurance contracts, initial wealth, expected-utility calculation and behavioral prediction.

Hierarchy paths (3) — routes to 3 parentless roots

Friedman–Savage utility function → is a kind of Expected Utility → is a kind of Expected Value → is a kind of Aggregation → is a kind of Transformation → is a kind of, typical Function (Mapping)

Show alternative paths (2)

Neighborhood in Abstraction Space¶

Friedman–Savage utility function sits in a crowded region of the domain-specific corpus (23^rd percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Welfare Economics & Utility Theory (29 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25