Prove or disprove the following statements by either giving a formal proof or a counter-example.(a) Given any number of agents and their complete preference profile over a set of outcomesO, if |O| = 2 then Plurality voting will always select the Condorcet winner.(b) Borda voting will always select the Condorcet winner if the possible number of outcomesis exactly 3, ie. |O| = 3, and n > 999 where n is the number of agents.

Consider the following preference orderings of 1000 agents for candidates O = {a, b, c}147 agents: a b c401 agents: b a c32 agents: c b a420 agents: a c b(a) Who wins under Plurailty? Explain.(b) Who wins in the Borda voting? Explain.(c) Who is the Condorcet winner? Explain.(d) If you are the agenda setter and strongly prefer b. What ordering would you chose forpairwise elimination such that b wins.

Which of the following statements described preference theory?

Consider majority voting over pairwise alternatives subject to agenda manipulation. Use thefollowing voting rules and preference profiles. There are three propositions to choose amongcandidates, A, B, and C. There are three voters, 1, 2, 3. The notation indicates strictpreference.Rules: There is a designer who sets the agenda, the order of voting. She announces twopropositions to choose between; the winner of that vote faces a runoff against the remainingalternative.Profile I:Voter 1: A B CVoter 2: B C AVoter 3: C A BProfile II:Voter 1: A B CVoter 2: B C, B A, (C vs. A preference is unspecified)Voter 3: C B AClaim: Under Profile I the designer can arrange that any one of the three propositions bethe winner by the chair’s choice of the order of voting.Under Profile II, the choice is independent of the order of the agenda.(a) Demonstrate the claim.(b) Show that Profile II fulfills the concept of single peaked preferences and that Profile Idoes not.

Which of the following is TRUE? A. Every agent in every 2-person game has a strongly dominant action B. An agent has a strongly dominant action if the agent has a best action for each of opponent's action C. An agent can have at most one strongly dominant action

5. Strategic Voting(a) Consider plurality-with-elimination, and profile2 agents : a b c2 agents : b c a2 agents : c a bAssume tie breaking in favor of a then b then c. Find a manipulation for one of thevoters.(b) Given that the median rule for single-peaked preferences is strategy-proof. For single-peaked preferences, suppose that a location l(j) ∈ [0, 1] is associated with each alter-native j ∈ A, with the alternatives arranged in order of increasing location. Define themean rule to choose the alternative that is closest to the mean of the locations of thetop choice of each voter. Explain why this rule is not strategy-proof.(c) A suggested procedure for F indM anipulation in the Borda rule is: Consider agent i, andfix reports of others. Compute the Borda score of each alternative given truthful reporti∈ P. To construct misreport 0i: place the favored alternative a in top position, andrank the other alternatives in descending order of their Borda scores at truth.Prove that this procedure is the optimal manipulation in Borda, in the sense that it willcause a to be elected whenever any manipulation can succeed.(d) What could be viewed as unrealistic about the way the F indM anipulation problem isdefined? Would relaxing this assumption be expected to make manipulation easier orharder?