Question:
Team $A$ and team $B$, in a series of $7$ games, whoever wins $4$ games first wins. You want to bet $100$ that your team wins the series, in which case you receive $200$, or $0$ if they lose. However the broker only allows bets on individual games. You can bet $X$ on any individual game that day before it occurs to receive $2X$ if it wins and $0$ if it loses. How do you achieve the desired pay-out? In particular, what do you bet on the first match?
I tried to do this in excel but I do not understand what I am doing wrong here. Here is a screen shot of my 7-step binomial tree.
Start after the conclusion of game 6.
The series is 3-3. You must have worked to have a bank of $100 at this point and you are betting it all on Game 7.
Work backwards.
Going int Game 5, if you team is up, you want to be in a situation where if you win you will have \$200 and if you loose you have \$100. So you must have a bank of \$150 and will be betting \$50
use similar logic if you are down in the series.
Work back. At each step your bank needs to be the average of the two possible next outcomes, and the wager equal to have the difference between the next two possible outcomes.
I get that you should wager \$31.25 on games one and two.
Update
Below is my table
$\array{\text{game 1} &&&\text{game2}&&&\text{game3}&&&\text{game4}\\ \text {wager}&\text{record}&\text{bank}&\text {wager}&\text{record}&\text{bank}&\text {wager}&\text{record}&\text{bank}&\text {wager}\\ &&&&&&&3-0&187.5&12.5\\ &&&&2-0&162.5&25\\ &1-0&131.25&31.25&&&&2-1&137.5&37.5\\ 31.25&&&&1-1&100&37.5\\ &0-1&31.25&31.25&&&&1-2&62.5&37.5\\ &&&&2-0&62.5&25\\ &&&&&&&3-0&12.5&12.5}$
$\array{&&\text{game 5} &&&\text{game6}&&&\text{game7}\\ \text{record}&\text{bank}&\text {wager}&\text{record}&\text{bank}&\text {wager}&\text{record}&\text{bank}&\text {wager}&\text{record}&\text{bank}\\ 4-0&200\\ &&&4-1&200\\ 3-1&175&25&&&&4-2&200\\ &&&3-2&150&50&&&&4-3&200\\ 2-2&100&50&&&&3-3&100&100\\ &&&2-3&50&50&&&&3-4&0\\ 1-3&25&25&&&&2-4&0\\ &&&1-4&0\\ 0-4&0}$
To build it, I started at the end and figured what my bank had to be at that point, and then what wagers I must have made on the previous game that could have gotten me to that level of bank.
Your bank at any point is the expected value of your bank after the next game.