Maximizing the Expected Number of Games in a Two-Player Series

In game theory, it is often essential to calculate the expected value of a particular outcome. This concept is also applicable to a two-player series in which the first player to win two games wins the series. In this blog post, we will explore how to find the expected number of games in a two-player series when i = 2 and show that this number is maximized when p = 1/2.

To begin, let's define some terms. Let X be the random variable that represents the number of games played in a two-player series when i = 2. Let p be the probability of the first player winning a game, and q = 1 - p be the probability of the second player winning a game.

We can calculate the expected value of X using the formula:

E(X) = Σ x * P(X = x)

where Σ represents the sum, x represents the number of games played, and P(X = x) is the probability of X taking on the value x.

Let's break down the possible outcomes for a two-player series with i = 2:

The first player wins both games: This outcome has probability p^2 and takes two games to complete.

The second player wins both games: This outcome has probability q^2 and takes two games to complete.

The series goes to a third game: This outcome has probability 2pq and takes three games to complete.

Using these outcomes and probabilities, we can calculate the expected value of X as:

E(X) = 2p^2 + 2q^2 + 3(2pq) = 2p^2 + 2q^2 + 6pq

Simplifying this equation gives:

E(X) = 2(p + q)^2 - 2p^2 - 2q^2 = 4p(1-p)

To maximize E(X), we can take the derivative of the equation with respect to p and set it equal to 0:

d/dp (4p(1-p)) = 4(1-2p) = 0

Solving for p gives:

p = 1/2

Therefore, the expected number of games played in a two-player series when i = 2 is maximized when the probability of the first player winning a game is 1/2.

In summary, calculating the expected number of games played in a two-player series can help us understand the potential outcomes and probabilities in a game. To maximize this value, we need to set the probability of the first player winning a game to 1/2. This concept is crucial in game theory and can be applied in various situations where probabilities and outcomes are involved.