. The following problem was first considered by John von Neumann and is a fundamental result game theory. A and B play the following game: A writes down either number 1 or number 2, and B must guess which one. If the number that A has written down is i and B has guessed correctly, B receives i units from A. If B makes a wrong guess, B pays 4/5 of a unit to A. First we consider the expected gain of player B. Suppose B guesses 1 with probability p and 2 with probability 1 −p. Let X1 denote B’s gain (or loss) in a game where A has written down 1. Let X2 denote B’s gain (or loss) in a game where A has written down 2. (a) Find the pmf of X1 and X2 (b) Find B’s expected gain for these two cases, E[X1] and E[X2]. (c) What value of p maximizes the minimum possible value of B’s expected gain? Now consider the expected loss of player A
5. The following problem was first considered by John von Neumann and is a fundamental
result
A and B play the following game:
A writes down either number 1 or number 2, and B must guess which one.
If the number that A has written down is i and B has guessed correctly, B receives i units from A.
If B makes a wrong guess, B pays 4/5 of a unit to A.
First we consider the expected gain of player B.
Suppose B guesses 1 with probability p and 2 with probability 1 −p.
Let X1 denote B’s gain (or loss) in a game where A has written down 1.
Let X2 denote B’s gain (or loss) in a game where A has written down 2.
(a) Find the pmf of X1 and X2
(b) Find B’s expected gain for these two cases, E[X1] and E[X2].
(c) What value of p maximizes the minimum possible value of B’s expected gain?
Now consider the expected loss of player A
Suppose that A writes down 1 with probability q and 2 with probability 1 −q.
Let Y1 be A’s loss (or gain) if B chooses number 1.
Let Y2 be A’s loss (or gain) if B chooses number 2.
(a) Find the pmf of Y1 and Y2
(b) Find E[Y1] and E[Y2].
(c) What value of q minimizes the maximum possible value of A’s expected loss?
Given ,
A and B play the game
B guesses 1 with probability = p
B guesses 2 with probability = (1-p )
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
