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You can take a coin flip for 100 dollars or nothing, or a sure 40 dollars. The expected value of the flip is 50, more than 40. Most people still take the 40. Are they wrong?

Preferences and expected utility

To compare risky options we need utilities, not just money. A rational player maximises expected utility, and a curved utility can make a sure thing beat a better-on-average gamble.

From earlier lessons

One quick one before we start.

A payoff table lists utilities that mainly capture...

Predict, then play

Bend the utility curve and watch the coin flip lose its shine. The gamble pays $0 or $100, so it averages $50. Drag the risk attitude and see the sure amount this person would accept instead.

Why did that happen?

Take a guess, then reveal the reason

The two outcomes are equally likely, but they are not equally weighted in utility. Going from $50 down to $0 costs more happiness than going from $50 up to $100 adds, because each extra dollar is worth a little less than the last. So the gamble’s expected utility sits below the utility of its $50 average, and the sure thing that matches it, the certainty equivalent, is less than $50.

Value the risk, not the cash

Assign each amount a utility $u(x)$ . The value of a gamble is its expected utility, the probability-weighted average of $u$ . If $u$ is concave (each extra dollar is worth a little less than the last), the player is risk-averse and will pay to avoid a fair bet. That is exactly why insurance exists.

Definition, expected utility

The von Neumann and Morgenstern theorem says a player whose preferences over lotteries obey a few consistency axioms acts as if maximising $\mathbb{E}[u(x)]$ for some utility $u$ , unique up to a positive affine transformation.

Find the certainty equivalent

Take $u(x)=\sqrt{x}$ and the gamble (50% of $0, 50% of $100). Its expected utility is $0.5\sqrt{0}+0.5\sqrt{100}=5$ . The sure amount worth the same is $x$ with $\sqrt{x}=5$ , so $x=25$ , that is $25. This player would swap the gamble for a guaranteed $25, well below its $50 average. That gap is the risk premium.

Key idea

Rational does not mean chasing the highest average payout. It means maximising expected utility, which can be cautious by design.

Check yourself

Check yourselfconcept

A person with a concave (risk-averse) utility can keep $50 for sure or take a 50-50 lottery paying $0 or $100. What do they do?

Check yourselfconcept

You multiply every payoff in a game by 10 and add 5. Do the players' best choices change?

Check yourselfbrings back: what is a game

Given all this, what do the numbers in a payoff table really represent?

Sources

von Neumann, J., and Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton University Press.
Osborne, M. J. (2004). An Introduction to Game Theory. Oxford University Press.