Game Theory: Consider a game in which each of two people simultaneously chooses an integer: 1 or 2. Find the expected value for player A and the expected value for player B. Is each game fair.
Question: If the numbers are the same, then player A wins 1 point from player B. If the numbers are different, then player B wins 1 point from player A.
Find probability of numbers matching (2/4) and not matching (2/4).
Construct table for A:
Outcome value,x P(outcome) x.P(x)
Win..... +1 ...... 0.5 ....... 0.5
Lose.....-1 ...... -0.5 .......-0.5
Sum(x.P(x))=0, so game is fair.
oops
The probability on the second line should read 0.5 (not negative).
what if player A wins 2 points from player B, and if the numbers are different, than player B wins 1 point from player A?
Games are random processes.
All we can say is that if he wins, he gets one point, and if he loses, he loses one point. We also know (or believe) that the probability of winning or losing is each 50%.
The actual outcomes depends naturally on chance. But in the long run, the number of points won or loses will be relatively close to zero, which is our calculated expectation.
Hope that answers your query.
To find the expected value for each player, we need to calculate the probability of each outcome multiplied by the value associated with that outcome.
Let's consider player A's expected value first. There are four possible outcomes in this game:
1. Both players choose 1: Player A wins 1 point.
2. Both players choose 2: Player A wins 1 point.
3. Player A chooses 1, while player B chooses 2: Player A loses 1 point.
4. Player A chooses 2, while player B chooses 1: Player A loses 1 point.
To calculate the probability of each outcome, we need to consider the choices of both players. As each player has 2 options (1 or 2), there are a total of 2 * 2 = 4 possible combinations of choices.
1. Both players choose 1: This occurs when player A chooses 1 with a probability of 1/2 and player B also chooses 1 with a probability of 1/2. The probability of this outcome is (1/2) * (1/2) = 1/4.
2. Both players choose 2: This occurs when player A chooses 2 with a probability of 1/2 and player B also chooses 2 with a probability of 1/2. The probability of this outcome is also 1/4.
3. Player A chooses 1, while player B chooses 2: This occurs when player A chooses 1 with a probability of 1/2 and player B chooses 2 with a probability of 1/2. The probability of this outcome is (1/2) * (1/2) = 1/4.
4. Player A chooses 2, while player B chooses 1: This occurs when player A chooses 2 with a probability of 1/2 and player B chooses 1 with a probability of 1/2. The probability of this outcome is also 1/4.
Now, let's calculate the expected value for player A by multiplying the value associated with each outcome by its probability:
Expected Value for Player A = (1 * 1/4) + (1 * 1/4) + (-1 * 1/4) + (-1 * 1/4)
= 1/4 + 1/4 - 1/4 - 1/4
= 0
Similarly, we can calculate the expected value for player B using the same approach:
1. Both players choose 1: Player B loses 1 point.
2. Both players choose 2: Player B loses 1 point.
3. Player A chooses 1, while player B chooses 2: Player B gains 1 point.
4. Player A chooses 2, while player B chooses 1: Player B gains 1 point.
Expected Value for Player B = (-1 * 1/4) + (-1 * 1/4) + (1 * 1/4) + (1 * 1/4)
= -1/4 - 1/4 + 1/4 + 1/4
= 0
The expected value for both player A and player B is 0. This means, on average, neither player gains nor loses any points. Therefore, the game is fair.