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?
Question
- 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?
Solution
This question seems to be about voting systems and manipulation within those systems. Let's break it down:
(a) In the given profile, the plurality-with-elimination system would work as follows: In the first round, each candidate (a, b, c) gets 2 votes. According to the tie-breaking rule, candidate 'a' wins. However, if one of the voters who initially preferred 'c' changes their vote to 'b', the results would change. In the first round, 'a' and 'b' would each get 2 votes, and 'c' would get 1 vote and be eliminated. In the second round, 'b' would get 3 votes and 'a' would get 2 votes, making 'b' the winner. This is a manipulation by one of the voters who initially preferred 'c'.
(b) The mean rule is not strategy-proof because a voter can manipulate the outcome by misrepresenting their top choice. For example, if a voter's true top choice is at location 0.5, but they report it as 0.1, this could potentially lower the mean and cause a different alternative to be chosen.
(c) The suggested procedure for finding manipulation in the Borda rule involves calculating the Borda score for each alternative given truthful reports, then constructing a misreport that places the favored alternative at the top and ranks the others in descending order of their Borda scores. This is the optimal manipulation because it maximizes the Borda score of the favored alternative, increasing its chances of being elected. If there is any manipulation that can make the favored alternative win, this procedure will find it.
(d) The FindManipulation problem might be viewed as unrealistic because it assumes that voters have complete and perfect information about the preferences of other voters, and that they can calculate the exact outcomes of different manipulations. In reality, this information might not be available or the calculations might be too complex. Relaxing this assumption would likely make manipulation harder, as voters would have less information to base their manipulations on.
Similar Questions
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