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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.

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