Vol. 16: Puzzle This

A baseball wager in North Korea.

By Michael H. Pryor

Illustrations by Roy Doty



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Solution

Create a matrix to represent the pile of money you want to have at any point in time.

You can start by filling in all the places where the Yanks win 4 games. You know you want $2 million at that point. You can also fill in all the places where the Sox win 4 games, as you want to have $0 at that point. Starting at the bottom right, fill in the matrix by adding the cell to the right and the cell below and dividing by 2.

For example, if the series is tied up at 3 to 3, you know you want to have either $0 or $2 million after the next game (since the series will be over). So you want to have half that amount before the final game, or $1 million, so that if you lose you end up with nothing, but if you win, you have the $2 million.

Work your way up the matrix, from the bottom right, until it is filled in.

If i is the number of games the Yanks have won, and j is the number the Sox have won, then

P[i, j] = (P[i + 1, j] + P[i, j + 1]) / 2

The next step is to create a matching matrix to help you figure out what to bet at each point.

Starting at the top left this time, you can fill this matrix by looking at your pile of money matrix.

B[i, j] represents your bet when the Yanks have won i games and the Sox j games.

B[i, j] = P[i + 1, j] - P[i, j]

In English, that just translates to the difference between what you want to have in your pile if the Yanks win the next game, versus what you should have in your pile now. Since it's an even money bet, you'll win exactly that amount.

For example, if the Yanks win the first game, you know you want to have $1.31 million. So you bet $1.31 million - $1 million (what you have when the series is at 0, 0) or $0.31 million, i.e. $310,000.

In the image attached, the numbers are rounded, and not exactly accurate.


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