Example — Lottery against a sure alternative

Comparing a lottery with a certain gain, one discovers that the expected value is not enough to explain real choices: one’s attitude towards risk comes into play.

Example — Choice between a lottery and a sure alternative

I am offered two bets:

• Lottery $L$: I toss a fair coin; if heads I win $200$ EUR, if tails I lose $80$ EUR.
• Sure $S$: I receive $50$ EUR in hand, with no uncertainty.

Comparison via expected value. $\bar V_L = \tfrac{1}{2}\cdot 200 + \tfrac{1}{2}\cdot(-80) = 100 - 40 = 60 \text{ EUR},\qquad \bar V_S = 50 \text{ EUR}.$ The lottery is “better” on average: $\bar V_L=60 > 50 = \bar V_S$. Yet many people still choose $S$: this preference for “the certain” even at the cost of a lower expected value is called risk aversion. It is an empirical phenomenon (and modelled with the utility function of Bernoulli/Von Neumann–Morgenstern, but here we limit ourselves to comparing the expected value).

Links

Topics: Probability
Concepts: Risk aversion · Expected value · Random variable
Methods: Expected value
Skills: Probability calculation · Modelling