# For Your Enjoyment

1. Aug 30, 2009

### wofsy

Here is a famous problem for you to enjoy.

You are betting on the World Series and want to make bets in such a way that if the Yankees win the series you win exactly a dollar and if the Red Sox win the series you lose exactly a dollar.

You can bet what ever you want on each game. If the Yankees win that game you win the amount that you bet. If the Red Sox win you lose the amount of your bet. So for instance if on game three you bet ten dollars and the Yankees win, you win ten dollars.

Problem: How much should you bet on the first game?

(The series ends ,of course, as soon as one team has won 4 games.)

Last edited: Aug 30, 2009
2. Aug 30, 2009

### mXSCNT

Betting against the Red Sox? I disapprove. Let's say instead that you win a dollar if the Red Sox win, and lose a dollar if the Yankees win.

Bet 5/16 dollars on the first game.

3. Aug 31, 2009

### MathematicalPhysicist

I think I should bet 0.25 dollars.
In the second game I should bet the same money as in the first game, if someone else wins the second game than in the first game bet on 0.5 dollars in the third game, if not then bet on 0.25 dollars in the third game. If someone else wins the third game than in the second then bet on 0.5 dollars in the fourth game, if not bet on 0.25.
Now if the team is over after 4 games then you have either won or lost 1 dollar, if not then it's tied 2-2 and in a net balance, and you are left with at most 3 games, you should be on the fith game 0.5, and so in the sixth game, if it's finished in sixth game then you have either won or lost 1 dollar, if not then on the last game bet 1 dollar on the last game.

4. Aug 31, 2009

### wofsy

OK So the Yankees win the first 3 games and you are up 75 cents. The Red Sox win the next two so now you are flat. What do you do then?

Last edited: Aug 31, 2009
5. Aug 31, 2009

### wofsy

do you have a proof?

6. Aug 31, 2009

### mXSCNT

Suppose the score is 3-3. Then your total winnings up to this point must be $0 and you must bet$1--there is no other way to succeed.

Suppose the score is 3-2, Red Sox ahead. Then you must bet an amount such that if the Red Sox lose, you lose an amount to set you at $0 (because the score is then 3-3), and such that if the Red Sox win, you win an amount to set you at$1. So if the score is 3-2, your total winnings must be +$0.50 and you must bet$0.50. Similarly if the score is 2-3, you must be at -$0.50 and bet$0.50.

Proceed backwards in this manner to reach the answer for when the score is 0-0.

7. Aug 31, 2009

### wofsy

Right. You can think of your methodology as pricing a contingent claim on the world series with pay off +- 1 depending on the outcome of the series. This backwards induction is standard in pricing options and other non-path dependent contingent claims.

8. Aug 31, 2009

### wofsy

but there is a probability proof as well.