Betting on game of the World Series

gambling

You and a friend are betting on individual games of the World Series. For each game, if your team wins, you win a certain (positive) amount of money, and if your team loses, you lose that amount. You and your friend create a betting scheme such that you will win $\$1000$ if your team wins and lose $\$1000$ if your team loses, regardless of the game score (it could be 4-3, 2-4, etc.). How much do you bet on the first game?

I was thinking about the Kelly Criterion but I don't know if this is the case to use it. Anyway, I am not sure if I have correctly understood the question, but If I can choose not to put all my money in one single game I would bet just a fraction of it every game, a fraction given by Kelly. I am assuming also $50\%$ chance for every team but I don't know if this is the case.

Best Answer

Step 0: Work backwards from the end

Step 1: If the score is $3-3$ you want the net position so far to be $0$, and you bet $1000$

Step 2: If the score is $3-2$ you want the net position so far to be $+500$, and you bet $500$; if the score is $2-3$ you want the net position so far to be $-500$, and you bet $500$

Step 3: If the score is $3-1$ you want the net position so far to be $+750$, and you bet $250$; if the score is $1-3$ you want the net position so far to be $-750$, and you bet $250$; if the score is $2-2$ you want the net position so far to be $0$, and you bet $500$ etc.

You want the position at a particular score to be the average of the two possible positions after the next game. You want to bet half the difference of the two possible positions after the next game. You should end up with this:

Bets:
=====
Score     0       1       2      3  
                        
0       312.5   312.5   250    125  
1       312.5   375     375    250  
2       250     375     500    500  
3       125     250     500   1000  

Positions:
==========                  
Score      0       1       2       3       4
                        
0          0     312.5   625     875    1000
1       -312.5     0     375     750    1000
2       -625    -375       0     500    1000
3       -875    -750    -500       0    1000
4      -1000   -1000   -1000   -1000
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