Question

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.

Answer 1

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.

Answer 2

oops
The probability on the second line should read 0.5 (not negative).

Answer 3

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?

Answer 4

Games are random processes.
All we can say is that <i>if</i> 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.