How to find the value of a game in game theory?

To find the value of a game in game theory, we use the concept of expected value.

Expected value is a measure of the average outcome of a random event, weighted by its probability. In game theory, we use expected value to determine the value of a game, which is the amount that each player can expect to win or lose on average.

To calculate the expected value of a game, we first need to determine the payoffs for each player in each possible outcome. We then multiply each payoff by its probability, and sum the results. Understanding the basics of probability is crucial in accurately calculating these probabilities.

For example, consider the following two-player game:

Player 1 / Player 2 | A | B
---|---|---
C | 2, 1 | 0, 0
D | 0, 0 | 1, 2

To find the value of this game, we first calculate the expected payoff for each player in each possible outcome:

Player 1 / Player 2 | A-C | A-D | B-C | B-D
---|---|---|---|---
2, 1 | 2/3 * 2 + 1/3 * 0 | 2/3 * 0 + 1/3 * 1 | 2/3 * 1 + 1/3 * 0 | 2/3 * 0 + 1/3 * 2
0, 0 | 2/3 * 0 + 1/3 * 0 | 2/3 * 0 + 1/3 * 0 | 2/3 * 0 + 1/3 * 0 | 2/3 * 0 + 1/3 * 0
1, 2 | 2/3 * 0 + 1/3 * 1 | 2/3 * 1 + 1/3 * 0 | 2/3 * 0 + 1/3 * 2 | 2/3 * 2 + 1/3 * 0

We can then calculate the expected value of the game by summing the expected payoffs for each player in each possible outcome, and dividing by the total number of outcomes:

Expected value = (2 + 0 + 0 + 0 + 0 + 1 + 0 + 2) / 8 = 0.625

Therefore, the value of this game is 0.625.

The ability to determine values using different types of numbers and their applications in game theory can provide deeper insights into real-world situations. You can explore these applications in more detail on the page about real-world applications of mathematics.