World Series Betting with Dynamic Programming

Replicate a $100 even-odds bet on the Yankees to win a best-of-seven series using only even-odds bets on individual games.

Answer

$31.25

Bet this amount on the Yankees in Game 1.

The target terminal payoff is +$100 if the Yankees win the series and −$100 if they lose.

Series wager size

Change this amount to scale every hedge proportionally.

1. Dynamic-programming value matrix

Let V(i,j) be the required profit position when the Yankees have i wins and the Giants have j wins.

V(i,j) = ½[V(i+1,j) + V(i,j+1)]

Boundary values: V(4,j)=+$100 and V(i,4)=−$100.

Y \ G	0	1	2	3	4

2. Why Game 1 is $31.25

If Yankees win Game 1+$31.25

If Giants win Game 1−$31.25

At even odds, betting B on the Yankees moves the position from 0 to +B after a win and −B after a loss.

B = (31.25 − (−31.25))/2 = $31.25

3. Probability interpretation

After a Game-1 Yankees win, the score is 1–0. Assuming each game is a fair 50–50 event, the Yankees' probability of eventually winning the series is:

P(series win | 1–0) = 21/32 = 65.625%

Therefore the value of the original series bet at that state is:

100 × [2(21/32) − 1] = $31.25

4. Walk through a possible series

Click game outcomes to see how much you would bet before each subsequent game.

Current score

0 – 0

Before Game 1, bet $31.25 on the Yankees.

Betting path

Start at 0–0 → bet $31.25 on Yankees.

Interview shortcut

Work backward from +100 and −100. At each state, the required value is the average of the two next-state values, and the next-game bet is half their difference. From 0–0, the next states are +31.25 and −31.25, so the first bet is $31.25.