Who knows what?
Nine used cars are for sale, each equally likely. Their quality Q runs from 0, a lemon, to 2, a plum, in steps of ¼. Each owner knows their own car’s quality. Buyers know only that every quality is equally likely.
Owner’s value = $5,000 × Q Buyer’s value = $7,500 × Q
Every buyer values every car (with Q above 0) more than its owner does. If buyers could see quality, every one of these cars would sell.
Try this: click any car on the graph. How much more is it worth to a buyer than to its owner?
The auctioneer lowers the price
An auctioneer calls out a price, P. An owner offers a car only if it is worth less to them than the price: 5,000 Q ≤ P. Better cars stay home. Buyers cannot tell the offered cars apart, so they will pay for the average car offered.
Try this: press “Next round” several times. Which cars leave each time, and does the market ever settle?
Why no price works
Each round, the auctioneer tries what buyers offered last time. Follow the rounds down the table. Select a row to see that round on the graph.
At price P, only cars with Q ≤ P/5,000 are offered. Quality is spread evenly from 0, so the average car offered is about half the cutoff:
average Q ≈ P / 10,000 ⇒ buyers pay at most 7,500 × P / 10,000 ≈ 0.75 P
The bad drives out the good
Buyers always offer about three quarters of the asking price, so no price satisfies both sides, even though buyers value every car (with Q above 0) more than its owner does. The market unravels.
Try this: compare the last two columns in each row. Is there any round where buyers offer as much as the asking price?
Health insurance: the mirror image
People with the same demographics expect different health spending, spread evenly from $0 to $M. The insurer must break even, so the premium must cover the pool’s average spending. Now the buyer knows more: each person knows their own expected spending, while the insurer knows only the spread.
Anyone who expects to spend less than the premium won’t pay it. They leave, the pool gets costlier, and the premium must rise.
Simplifications: people know their spending only roughly, and risk aversion makes some low risks willing to pay a bit more than their expected loss (Chapter 13). The tendency is still real, and it shapes how insurance markets are regulated.
Try this: run four rounds. What share of people is still insured, and who are they?