Question

3 Abstention and costly voting Groucho Marx once said, "I'll never join any club that would have me for a member.” Well, Groucho is not interested in joining your investment club, but Julie is. Your club has 10 members, and the procedure for admitting a new member is simple: Each person receives a ballot that has two options: (1) admit Julie and (2) do not admit Julie. Each person can check one of those two options or abstain by not submitting a ballot. For Julie to be admitted, she must receive at least 6 votes in favor of admittance. Letting m be the number of ballots submitted with option 1 checked, assume that your payoff function is ſi if m=6,7,8,9, 10 10 ifm = 0,1,2,3,4,5 (a) Prove that checking option 1 (admit Julie) is not a dominant strategy. (b) Prove that abstaining is a weakly dominated strategy. (c) Now suppose you're tired at the end of the day, so that it is costly for you to attend the evening's meeting to vote. By not showing up, you abstain from the vote. This is reflected in the payoff function having the form 11 } O 1-1 ifm = 6,7,8,9 and you abstained ifm = 6, 7, 8, 9, 10 and you voted ifm = 0,1,2,3,4,5 and you abstained ifm = 0,1, 2, 3, 4, 5 and you voted Prove that abstaining is not a weakly dominated strategy.

Answer 1

PJNO:-1 Answe r;- checking option I (admit julle) a dominant strategy (a) is not congider the follo wing case graup mem ber the ☺ Lets take (say Alfce) F If only op tion one other mem ber check I (admit Julie) li) Then checking option Idoed not matter for Alice because be admittedin Julie Will not any cale, iv) Therefoore, checking option (admit gulie) does not always better leave Alice implies checking option () This iladmit fulie) is not dominant strategy.

9 NO:- 2 dominated (b) Abstaining is a strategy weakly congider The following Care, follo wi a member f the 1) Le ts take goroup (say AH Pce) () If Alice ladmit Sukie ) then. check3 option (i Either it (when at least 5 Others do not check option ) doesn't matter EV) or it ig staictly better than Checking option a (do not Admit gulie) or abstaining (when Exactly feve mem bers check option! other (V) there fo ge, checking option l a weakly dominant strategy.

PgiNo 3 (vi) checking option I weakly and it is dominates abstaining a strategy is Weakly another strtegy Known that domâinated if Weakly dominates it (vii) This weakly domånated strategy. implies abstaining is a (C) hew paydt citeria AS per the abst aining Jo dominated strategy wweakly is not a off coiteria, new pay per the following Case, abstained and, six other Confideg the i) you members check option il paygf=r) lii) Then abstaining is strictly Checking option /2) better than (pay off

PgiNo: 4 Li) There fore, checking option ! does not weakly dominate F abstaini ng dominate of Ne xt congide The following Cases selected option mem bers checked and You voted 2 and six othe 91 option 1 (pay off = /2) h) Abstaining wou ld have given a 1 in this of payoff Case you voted and selected option 2 and six check option i (pay off = -112) (iv) Abst aining would have given a other members did not %3D payoff g o in this case (v) There fore, abqtaining weakly dominates checki ng option 2 can be * There fore, it abstaining is not weakly dominatel any other strategy. concluded that by