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Expected Value vs Expected Utility

Why It Matters

Risk aversion is rational when utility is concave. Turning down positive expected value bets can be sensible. Kelly betting, insurance, and portfolio theory all depend on this distinction.

The Key Move

Don't just multiply probability by payoff. Ask: what's the utility of each outcome? People are typically risk-averse (diminishing marginal utility of wealth), which means equivalent expected values don't mean equivalent desirability.

Classic Application

The St. Petersburg paradox. A game with infinite expected value isn't worth infinite price because the utility of infinite wealth isn't infinite (and you might lose everything getting there). Expected utility resolves the paradox.

Surprising Application

Why rich people take more risks. Diminishing marginal utility means the next dollar matters less to a billionaire. At higher wealth, expected value approximates expected utility better. This makes expected-value-maximizing behavior more rational at higher wealth levels.

Failure Modes

  • Using expected value when expected utility applies - ignoring risk preferences
  • Assuming universal risk aversion - some domains show risk-seeking (gambling on longshots when behind)
  • Ignoring that utility functions differ across people and contexts
  • Treating utility as cardinal when it's ordinal - you can compare but not add utilities
  • Confusing risk aversion with loss aversion - they have different implications

Go Deeper

Von Neumann-Morgenstern utility theory. Kahneman and Tversky's prospect theory for modifications based on observed behavior.