Question

4. (Voter Participation): 2 candidates A and B compete in an election. Of the n citizens, k support candidate A and the remaining (n − k) support B. Each citizen chooses whether to abstain, or to vote at a cost. A citizen who abstains receives payoff 2 if the candidate she supports wins, 1 if this candidate ties for 1st place, and 0 if the candidate loses. A citizen who votes receives the same payoffs, minus a voting cost 0 < c < 1. Assume that k ≤ n . 2 Find p such that the following is a NE: •
each candidate who supports A votes with probability p

• k of the B-supporters each vote with probability 1 •

the remaining n − 2k B-supporters all abstain.

Note: if every A-supporter votes w.p. p, then the probability that all k of them vote is pk, and the probability that exactly (k − 1) of them vote is kpk−1(1 − p).

Answer 1

Anstoer te voren the given stata wehave, They can be says 4 types of Possible ways. ů If every one votes the Each votes gets a pay off of 1-c. If a voltes does not vote then the he gets 0. Thes there is no profitable deviation. 1 c? If Every one did not vote & Candidates are tied then people " If Every who does not vote gets. ² voting person has a profitable deviation to voting to get - which is larger than 1 since (<1. If every one does not vote and one Candidate wins by 1 vote then a votes who does not vote for the losing candidates would Profitably deviate to voting, Since Osl-c . (iv) If Every one If Every one does not vote and 1 Candidate wins by more than one vote, then a voter who votes for the winning - Candidate would Profitably device to not voting Since g-c<2. Therefore the only Nash Eauilibrium is the one where Every one votes. o ~ Dear Student please Yatestet Thank You.