A priori Smith fair extra practice

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1 From Homework 8. A priori Smith fair extra practice ontents 1 The exercises 2 The solutions 1 The exercises 1. Preference schedule is: (a) What is the Smith set? A A A A A (b) Which candidates have the power to be weak spoilers? (c) Let Ω P = plurality. Is a weak spoiler for Ω P? (d) Let Ω P = pairwise comparison. Is a weak spoiler for Ω P? (e) Let Ω R = Runoff. Is a weak spoiler for Ω R? (f) Let Ω E = instant runoff/elimination. Is a weak spoiler for Ω E? (g) Let Ω = orda count. Is a weak spoiler for Ω? (h) Which WSM s can we conclude are NOT a priori Smith fair from these computations? Which WSM s do we still need to check? (i) Now conduct a priori Smith fair WSM s on this example: (a) What is the winner set for plurality? (b) What is the winner set for a priori Smith fair plurality? (c) What is the winner set for a priori Smith fair orda ount? (d) What is the winner set for a priori Smith fair Runoff? (e) What is the winner set for a priori Smith fair Elimination? (f) What is the winner set for pairwise comparison? (g) What is the winner set for a priori pairwise comparison? 2. The next preference schedule:

2 (a) What is the Smith set? A A E A E E E A A E A (b) Which candidates have the power to be weak spoilers? (c) heck plurality for weak spoilers. (d) heck runoff for weak spoilers. (e) heck Instant runoff/elimination for weak spoilers. (f) heck orda count for weak spoilers. (g) Are there any weak spoilers for pairwise comparison? 3. The last preference schedule: E A A A A A A A (c) What is the Smith set? (d) What is the winner set for plurality? (e) What is the winner set for a priori Smith fair plurality? (f) What is the winner set for orda ount? (g) What is the winner set for a priori Smith fair orda ount? (h) What is the winner set for Elimination? (i) What is the winner set for a priori Smith fair Elimination? (j) What is the winner set for a priori Smith fair Runoff? (k) What is the winner set for pairwise comparison? (l) What is the winner set for a priori pairwise comparison? an we conclude any WSM s are NOT a priori Smith fair from these computations? If YES, you should find the weak spoilers in this election for that WSM.

3 2 The solutions 1. Preference schedule is: A A A A A (a) What is the Smith set? S = {A,, } (b) Which candidates have the power to be weak spoilers? (c) Let Ω P = plurality. Is a weak spoiler for Ω P? First, W P = {A}. When is eliminated, W P = {}. The winner set changed, so is indeed a weak spoiler for plurality. (d) Let Ω P = pairwise comparison. Is a weak spoiler for Ω P? No. One could check by computation, or just use that pairwise comparison is a priori Smith fair, which means it has no weak spoilers ever. (e) Let Ω R = Runoff. Is a weak spoiler for Ω R? First, the winner set is W R = {} because A and have the most first place votes, and when they enter the second runoff election, wins. Then is disqualified. Again, A and have the most first place votes, and enter the second election, which wins again. So W R = {} is the same as W R, meaning is NOT a weak spoiler for runoff. (f) Let Ω E = instant runoff/elimination. Is a weak spoiler for Ω E? First, wins by elimination. In the first elimination round, is removed with the fewest first place votes. In this second election, has the fewest first place votes and is eliminated. Now it s just A vs., and wins. So: W E = {}. Now is disqualified. We consider an election with only A,, and. With three candidates, runoff and instant runoff are the same, so we don t have to compute anything more to know that W E = W R = {} again. That means is NOT a weak spoiler for elimination. (g) Let Ω = orda count. Is a weak spoiler for Ω? ompute the orda scores for each candidate in this election to deduce that A wins by orda count, ie- W = {A}. (A) = 31 () = 24 () = 27 () = 18 (ps- we can check that this computation is right... N = 10 and n = 4, so the sum of the orda scores should be 10 ( ) = 100.) Now eliminate and compute scores again. It would probably be convenient to write out the new preference schedule:

4 orda scores are: (A) = 23 () = 16 () = A A A A A And A wins again by orda count! So is not a weak spoiler for orda count. (h) Which WSM s can we conclude are NOT a priori Smith fair from these computations? Which WSM s do we still need to check? is a weak spoiler for plurality, but not for anyone else. So plurality fails the a priori Smith fair criterion. We can t make any conclusions about the other WSM s from these computations!!! (i) Now compare the WSM s with their a priori Smith fair versions: For our convenience, the preference schedule with the Smith set is copied and consolidated here: A A A (a) What is was the winner set for plurality from before? W P = {}. (b) What is the winner set for a priori Smith fair plurality? If we do the plurality vote on only the Smith set, we see that wins with 5 first place votes. (c) What is the winner set for a priori Smith fair orda ount? {A} (d) What is the winner set for a priori Smith fair Runoff? {} (e) What is the winner set for a priori Smith fair Elimination? {} (f) What is the winner set for pairwise comparison? W P = {} (g) What is the winner set for a priori Smith fair pairwise comparison? {} (see graph below) A

5 2. The next preference schedule: A A E A E E E A A E A E (a) What is the Smith set? S = {A,, } (b) Which candidates have the power to be weak spoilers? E and (c) heck plurality for weak spoilers. is a weak spoiler (d) heck runoff for weak spoilers. is a weak spoiler (e) heck Instant runoff/elimination for weak spoilers. is a weak spoiler (f) heck orda count for weak spoilers. E is a weak spoiler. First A and tie with a orda score of 24. Without E, A wins alone with a orda score of 20 ( s new orda score is 18). If is removed then A and again have the same orda score of 20, so is not a weak spoiler for orda count. (g) Are there any weak spoilers for pairwise comparison? neverrrr (h) Which winner selection methods decidedly fail the a priori Smith fair criterion? from this example: plurality, runoff, elimination, and orda count! 3. The last preference schedule: A A A A A A A (c) What is the Smith set? S = {A,, } (d) What is the winner set for plurality? W P = {} (e) What is the winner set for a priori Smith fair plurality? {} (f) What is the winner set for orda ount? (A) = () = 21 () = () = 24

6 So the winner set is {, }. (g) What is the winner set for a priori Smith fair orda ount? (A) = 16 () = () = 19 So the winner set is again {, }. (h) What is the winner set for Elimination? W E = {} (i) What is the winner set for a priori Smith fair Elimination? {} (j) What is the winner set for Runoff? and tie for number of first place votes after, and then (if we just made up this tie breaker) they tie for number of second place votes, and number of third place votes, and number of last place votes, so this was not a well defined question at all and we should pretend it never happened. (k) What is the winner set for a priori Smith fair Runoff? {} (l) What is the winner set for pairwise comparison? {A,, } (m) What is the winner set for a priori pairwise comparison? {A,, } an we conclude any WSM s are NOT a priori Smith fair from these computations? just plurality so far, but we already knew that it wasn t so this isn t too exciting. learly is a weak spoiler for plurality because S wins by plurality.

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