Market Making and Betting Games

Games about odds, fair prices and bet sizing, with the expected value of each choice worked out.

1. 01Paid the squareEasyAbout 2 min

   You roll a fair six-sided die and receive the square of the number in dollars. What is the fair price of this game? Would you pay 15 dollars to play it once?

   Hint

   Average the six squares. The answer is different from the square of the average roll.

   Answer

   The fair price is 91/6, about 15.17 dollars. At 15 dollars the game has a small edge of about 0.17 dollars for you, so you would play.

   Solution

   The six squares are 1, 4, 9, 16, 25 and 36, which add up to 91.

   Each face has probability 1/6, so the expected payout is 91/6, about 15.17 dollars.

   Squaring the average roll would give 3.5^2 = 12.25, which is too low, because the square rewards high rolls more.

   At 15 dollars the edge is 1/6 of a dollar per game. With a single play the spread of outcomes is wide, so the edge only shows over many games.

   Mark as done
2. 02A market on three diceMediumAbout 6 min

   Three fair dice will be rolled. Make a two-sided market on their sum, 2 dollars wide. Then one die is shown to everyone, and it is a 6. Where is your new market? Finally, a counterparty who you think saw a second die buys from you at your ask. How do you react?

   Hint

   Center the market on the expected value. After the reveal, only the two hidden dice are uncertain.

   Answer

   Quote 9.50 at 11.50 around the fair value of 10.5. After the 6, the fair value is 6 + 7 = 13, so quote 12 at 14. After a buy from someone who may know more, move the market up and keep it wide until you learn more.

   Solution

   Each die has mean 3.5, so the sum has mean 10.5 and a standard deviation of sqrt(3 × 35/12), about 2.96. A market 2 dollars wide is 9.50 bid, 11.50 offered.

   Once a 6 is known, the sum is 6 plus two hidden dice with mean 7. The fair value moves to 13, and the market to 12 at 14.

   A buyer who has seen a die is more likely to have seen a high one. Treat the trade as information: raise both prices, and widen until the risk of trading against better information is paid for.

   Interviewers watch whether you update at once and whether you can say by how much.

   Mark as done
3. 03Quoting against a better informed traderHardAbout 15 min

   A card shows a whole number from 1 to 10, each equally likely, and it will be revealed later. You quote a bid of 5.5 - h and an ask of 5.5 + h for one unit. Each request comes from an informed trader with probability 1/2, who knows the card and trades only when it pays strictly, or from a noise trader, who buys or sells with probability 1/2 each. What is your expected profit per request for h = 2.5? What is the smallest h at which you do not lose on average?

   Hint

   The noise trader pays you h on average. The informed trader takes the amount by which the card lies beyond your quote. Set the two parts equal.

   Answer

   For h = 2.5 the expected profit is 0.95 per request. The break-even half spread is h = 4/3, so the market has to be at least 8/3, about 2.67, wide.

   Solution

   Noise trader: a sale at the ask earns 5.5 + h - V and a buy at the bid earns V - 5.5 + h. Both average to h, because the card has mean 5.5.

   Informed trader with h = 2.5: the ask is 8 and the bid is 3. The trader buys on 9 or 10 and sells on 1 or 2. Your losses are 1 or 2 in each case, which averages to (1 + 2 + 1 + 2)/10 = 0.6.

   Expected profit: 0.5 × 2.5 - 0.5 × 0.6 = 0.95 per request.

   In general the profit is 0.5h - 0.5 L(h), where L(h) is the informed loss. For an ask between 6 and 7, the cards 7 to 10 are above it, so L(h) = 2 × (34 - 4(5.5 + h))/10 = (24 - 8h)/10.

   Setting h = (24 - 8h)/10 gives h = 4/3. The ask is then 6.83, which lies in the assumed range, so the solution holds.

   Mark as done

Compare your own answer with the worked solution.