7. Choice under Uncertainty

From expected utility through the measurement of risk aversion to the demand for insurance — choice in the face of uncertain prospects.

• Expected Utility — See how to score a lottery. Give each outcome a happiness score, weight it by probability, and add up. The key: 💰 money does not pile up happiness at a steady rate. The richer you are, the less one more coin delights you. So the average prize, paid for sure, feels better than the gamble itself. That difference is the Jensen gap. For example, flip a coin: heads pays 100 won, tails pays nothing. The average prize is 50 won. But offer a sure 40 won instead, and many friends pick the 40. The sure thing is easier on the heart.
• Measuring Risk Aversion — See how to put a number on fear of risk. One slider makes the fear bigger or smaller. The bigger the fear, the more the sure thing is worth. A gamble averaging 50 won gets sold for a sure 💰 money of 30 won. The 20 won shaved off is the risk premium. The 📏 ruler that measures this fear is the formula below. For example, one friend quits a coin-flip game right away for a sure 30 won. Another friend holds out for 45 won. The second friend is simply less afraid.
• Demand for Insurance — Compare two versions of life in one picture: sick and healthy. Insurance is a promise that fattens your share in the bad case. At a fair premium, covering the bad case completely is best. Both cases end up holding the same 💰 money. Add a loading to the premium and the math changes. Covering only part of the loss becomes the smarter deal. For example, 30 🧑 classmates each chip in 100 won every month. When someone gets hurt, the pot helps pay the hospital bill. If nobody gets hurt, the 100 won is simply gone. Still, the hurt friend is spared a scary bill.