Modified Beauty Contest

An oracle accepts two possible answers (1 and 2). There are three reporters, A, B, and C, who have to answer. The winning answer is the answer that gains the majority of votes. We assume that we play this game only once.

All reporters that voted for the winning answer receive a reward. We call them eligible reporters. The total amount of the reward is fixed. The total amount is divided among the eligible reporters evenly. So, if two out of three reporters voted for answer 1, each receives ½ of the reward. If all three reporters vote for the winning answer, each reporter receives 1/3 of the reward.

What would be the options for the participants if:

• a.I All reporters have the same number of votes, and no communication is possible.
• a.II All reporters have the same number of votes, and communication is possible.
• b.I Reporter A has 50 % + e votes. B and C have the same amount of votes, and no communication is possible
• b.II Reporter A has 50 % + e votes. B and C have the same amount of votes, and communication is possible?

a.I: A can only randomly choose one answer. The expected pay-off is 1/3.

a.II A could announce to play 1. This would be a Nash equilibrium. But he could try to increase his reward by telling B to play 1 and C to play 2. The problem is that also B and C can communicate.

b.I Without communication A cannot do anything to influence the pay-off. The expected pay-off is 0.583. The expected pay-off for A and B is 0.2083 each.

b.II A could publicly announce to play 1. But this would not be credible since this would reduce his pay-off. Therefore, it would be beneficial for B and C not to follow this announcement and play 2. But A could anticipate that and still play 1.

A could privately talk to B and C and tell them different choices. Now, B and C could communicate and learn what A told them. But they cannot trust each other as they could lie.