ECON 402 HW 5- Consider the following probability

Subject: Economics / General Economics
Question
Homework 5
Question 1
Let X = {A, B, C, D}. Consider the following probability distributions on X
Q
R
S A B C D 1
4
1
5 1
4
1
5
1
6 1
4
2
5
1
3 1
4
1
5
1
2 0 For P = q, r and s, calculate
1. P (B| {B, C})
2. P ({A, D} | {C, D})
3. Suppose that an agent starts with the belief Q and then sees a signal s = 1 with probability 12
and a signal s = 2 with probability 21 . After seeing the signal s = 1, the agent updates the belief
to R. What’s the belief of the agent after seeing the signal s = 2? Question 2
Consider the following dice game: two players throw two dice each. Player 1 beats Player 2 if the
smallest result of Player 1 is strictly higher than the smallest result of Player 2 and the highest result
of Player 1 is strictly higher than the highest result of Player 2. If Player 1 throws and gets a four and
a five, what’s the probability that he will win against Player 2? Question 31
Five players must vote on a trade policy, which is approved if at least three of them vote in favor.
It is known that this trade policy is favorable to exactly three of those players (the “winners”) and
unfavorable to exactly two of them (the “losers”). At the start, nature randomly picks three players
to be winners and two to be losers in a fair way (the probability of being a winner is the same for all
players). Then, if a player is a winner she learns that fact with probability 13 , independently of other
players. Explicitly, the player sees a signal that depends on her status, according to the following
conditional probabilities:
If winner
If not winner Congrats! No signal 1
3 2
3 0 1 1. If Player 1 receives no signal, what’s the probability that she ascribes to being a winner?
1 This Siga. question is based on “The perverse politics of polarization”, by Nageeb Ali, Maximiliahm Mihm and Lucas 2. Notice that a Player’s vote is only really relevant when exactly two of the other players are
voting in favor. Conditional on exactly two of the other four players having learned that they
are winners and on Player 1 having not learned any signal, what’s the probability that Player 1
ascribes to being a winner? Question 4
Suppose that nature draws either game A or B with probability 1/2 U
D A
L
2,2
0,0 R
0,0
4,4 U
D B
L
0,2
4,0 R
2,0
0,4 Player 1 observes Nature’s draw before playing, whereas Player 2 does not.
1. Draw this game in Extensive form
2. How many strategies does each Player have in this game? Compute the expected payoff for each
strategy profile and represent that as a normal form game.
3. Find all the Bayes Nash equilibria of this game.

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