Market Making the Population Guessing Game
You are making a market on the population of a country -- say, Indonesia. The interviewer knows the true population and will trade against you: if they think the true value is above your ask, they buy; if it's below your bid, they sell.
Your job is to quote a bid and ask, observe whether the interviewer buys or sells, update your estimate, and repeat. Each round, the interviewer's trade tells you something about where the true value lies relative to your quotes.
**Rules:**
1. Your spread (ask minus bid) can be at most
0$ million at any point during the game.
2. The interviewer always trades optimally -- they will buy if the true population is above your ask, sell if it is below your bid, and do nothing if the true value is inside your spread.
3. You lose money on every trade: if they buy at your ask and the truth is higher, you are short at too low a price. If they sell at your bid and the truth is lower, you are long at too high a price.
**Questions:**
1. What strategy should you use to set and update your bid-ask spread to minimize your total losses?
2. How many rounds does it take to pin down the population to within your spread?
3. What is your expected total loss under the optimal strategy?