Question

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 1/3 , independently of other players. Explicitly, the player sees a signal that depends on her status, according to the following conditional probabilities:

	Congrats	No signal
If winner	1/3	2/3
if not winner	0	1

1. If Player 1 receives no signal, what’s the probability that she ascribes to being a winner?
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?

Answer 1