Problem Overview

The New York Yankees and the San Francisco Giants are playing in a best-of-seven World Series. You want to bet $100 on the Yankees winning the series, but you are only allowed to place bets on individual games, each at even odds. The question is: how much should you bet on the first game?

To solve this, we utilize dynamic programming and model the game state using matrices to determine the optimal amount to bet at each stage of the series.

State Representation

We define a matrix \( P \) that holds the net payoff in each state \( (i, j) \), where:

  • \( i \): number of games won by the Yankees,
  • \( j \): number of games lost by the Yankees,
  • \( 0 \leq i, j \leq 4 \).

Boundary conditions are set as follows:

  • \( P(4, j) = 100 \) for \( 0 \leq j \leq 3 \),
  • \( P(i, 4) = -100 \) for \( 0 \leq i \leq 3 \),
  • \( P(4,4) \) is not used as it is not a reachable state in a best-of-seven format.

We also define a matrix \( B \) that contains the bet sizes for each intermediate state. It satisfies:

\[ P(i+1, j) = P(i, j) + B(i, j) \quad \text{(3.205)} \] \[ P(i, j+1) = P(i, j) - B(i, j) \quad \text{(3.206)} \]

Adding and subtracting these gives us:

\[ P(i, j) = \frac{1}{2}(P(i+1, j) + P(i, j+1)) \quad \text{(3.207)} \] \[ B(i, j) = \frac{1}{2}(P(i+1, j) - P(i, j+1)) \quad \text{(3.208)} \]

These recursive equations allow us to compute \( P \) and \( B \) by working backward from the terminal states.

Matrices

The computed matrices are:

\[ P = \begin{pmatrix} 0 & -31.25 & -62.5 & -87.5 & -100
31.25 & 0 & -37.5 & -75 & -100
62.5 & 37.5 & 0 & -50 & -100
87.5 & 75 & 50 & 0 & -100
100 & 100 & 100 & 100 & 0 \end{pmatrix} \]

\[ B = \begin{pmatrix} 31.25 & 31.25 & 25 & 12.5
31.25 & 37.5 & 37.5 & 25
25 & 37.5 & 50 & 50
12.5 & 25 & 50 & 100 \end{pmatrix} \]

First Game Betting Strategy

We start at state \( (0, 0) \) since no games have been played. From matrix \( B \), we find:

\[ B(1,1) = 31.25 \]

This means we should bet $31.25 on the Yankees in the first game.

The strategy is robust: regardless of the outcome of individual games, the betting adjustments according to matrix \( B \) ensure that we end up with exactly $100 if the Yankees win the series.

Conclusion

Using dynamic programming, we calculate the optimal strategy for betting on a best-of-seven series. The key takeaway is that we should bet $31.25 on the first game, and adjust subsequent bets according to matrix \( B \) to maintain an optimal expected outcome.

Reference