Question

In problems 23-25 discuss the validity of each statement. Explain why it is true or give a counter example.

If half of the payoffs are positive and half of the payoffs are negative then the game is fair.

Answer 1

In problems 23-25 discuss the validity of each statement. Explain why it is true or give a counter example. Solution: 23. If If a matrix game is fair then all payoffs are zero. A fair matrix game between R and C: Fair matrix indicates the R and C have fair chance in the game. One-way of getting this is when all payoffs of a matrix game are zero but for others none zero payoffs also game can be fair. Assume a game matrix as below: 0 -1 0 0 2 -1 -2 3 Here the saddle value is 0, so the value of matrix game is 0. Therefore, the game is fair but all payoffs are not zero. So the provided statement is false. 24. If half of the payoffs are positive and half of the payoffs are negative then the game is fair. 0 6 -2 -4 5 2 1 3 -8 -1 -2 -7 Here the saddle value is 1, so the value of matrix game is not zero, but half payoffs are positive and half are negative. Therefore the statement is false 25. If a matrix game is fair then some of the payoffs are positive and some of the payoffs are negative. Consider a fair matrix game, when a matrix has all 0 payoffs, the matrix game is said to be fair, so the statement is false.