Rationality of Voting and Voting Systems: Lecture II
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1 Rationality of Voting and Voting Systems: Lecture II Rationality of Voting Systems Hannu Nurmi Department of Political Science University of Turku Three Lectures at National Research University Higher School of Economics Hannu Nurmi (Turku) Rationality of Voting II November, / 28
2 Motivation Rules make a difference 4 voters 3 voters 2 voters A E D B D C C B B D C E E A A 5 options, 5 winners Hannu Nurmi (Turku) Rationality of Voting II November, / 28
3 Two winner intuitions Highest average ranking Borda Count 2 voters 2 voters 2 voters 1 voter D A B D C D A C B C D B A B C A This yields the ranking DABC. Now remove D. This gives: CBA, i.e. reversal of collective preference over A, B and C. Fishburn: it is possible that the Borda winner wins in only one of the proper subsets of the alternative set. Obviously, fiddling with the alternative set opens promising vistas for outcome control. Hannu Nurmi (Turku) Rationality of Voting II November, / 28
4 Two winner intuitions Pairwise victories Condorcet extensions Condorcet s paradox 4 voters 4 voters 4 voters A B C C A B B C A Surely, there is no winner here, or what? If so, then removing this kind of component from any larger profile or adding it to some profile should not change the winners, right? Hannu Nurmi (Turku) Rationality of Voting II November, / 28
5 Surprise? Two winner intuitions A profile with a strong Condorcet winner 7 voters 4 voters A B B C C A Adding the Condorcet paradox profile to this one results in a new Condorcet winner. N.B. the Borda winner remains the same in the 11- and 23-voter profiles. Hannu Nurmi (Turku) Rationality of Voting II November, / 28
6 Improving old systems Borda s paradox 4 voters 3 voters 2 voters A B C B C B C A A Borda s points: plurality voting results in a bad outcome a superior system exists (Borda Count) Hannu Nurmi (Turku) Rationality of Voting II November, / 28
7 Improving old systems Improving Borda Count: Nanson s rule How does it work? Compute Borda scores and eliminate all candidates with no more than average score. Repeat until the winner is found. Properties: Guarantees Condorcet consistency Is nonmonotonic Hannu Nurmi (Turku) Rationality of Voting II November, / 28
8 Improving old systems Nanson s rule is nonmonotonic C B A B A A A D B A C C D C D C B D B A C D D B The Borda ranking: A C B D with D s score 97 being the only one that does not exceed the average of 150. Recomputing the scores for A, B and C, results in both B and C failing to reach the average of 100. Thus, A wins. Suppose now that those 12 voters who had the ranking B A C D improve A s position, i.e. rank it first, ceteris paribus. Now, both B and D are deleted and the winner is C. Hannu Nurmi (Turku) Rationality of Voting II November, / 28
9 Improving old systems Improving plurality rule: plurality runoff Properties: Does not elect Condorcet losers Is nonmonotonic 6 voters 5 voters 4 voters 2 voters A C B B B A C A C B A C Hannu Nurmi (Turku) Rationality of Voting II November, / 28
10 Improving old systems Black s system: a synthesis of two ideas How does it work? Pick the Condorcet winner. If none exists, choose the Borda winner. Properties: Satisfies Cordorcet criteria Is monotonic Is inconsistent 4 voters 3 voters 3 voters 2 voters 2 voters A B A B C B C B C A C A C A B Hannu Nurmi (Turku) Rationality of Voting II November, / 28
11 Varieties of goodness Some systems and performance criteria Criterion Voting system a b c d e f g h i Amendment Copeland Dodgson Maximin Kemeny Plurality Borda Approval Black Pl. runoff Nanson Hare Hannu Nurmi (Turku) Rationality of Voting II November, / 28
12 Varieties of goodness Criteria a: the Condorcet winner criterion b: the Condorcet loser criterion c: the strong Condorcet criterion d: monotonicity e: Pareto f: consistency g: Chernoff property h: independence of irrelevant alternatives i: invulnerability to the no-show paradox Hannu Nurmi (Turku) Rationality of Voting II November, / 28
13 The no-show and related paradoxes The no-show paradox Theorem Moulin, Pérez: all Condorcet extensions are vulnerable to the no-show paradox. 26% 47% 2% 25% A B B C B C C A C A A B Hannu Nurmi (Turku) Rationality of Voting II November, / 28
14 The strong version The no-show and related paradoxes 2 seats 3 seats 2 seats 2 seats c b a a b a c b a c b c The amendment agenda: b vs. c and the winner vs. a results in b. Suppose that the two right-most voters abstain. Then a (the abstainers favorite) wins. Hannu Nurmi (Turku) Rationality of Voting II November, / 28
15 The no-show and related paradoxes Nanson s method and preference truncation 5 voters 5 voters 6 voters 1 voter 2 voters A B C C C B C A B B D D D A D C A B D A Here Nanson s method results in B. However, if the 2 voters with preference ranking CBDA reveal only their first-ranked option, C, the outcome is C, obviously a superior option from their point of view. Hannu Nurmi (Turku) Rationality of Voting II November, / 28
16 Comments The no-show and related paradoxes Abstention can obviously be regarded as an extreme form of preference truncation and, thus, these two paradoxes are closely related. To the same family of paradoxes belongs also the twins paradox or the twins not welcome phenomenon. This paradox occurs whenever adding k copies or clones of voter i leads to an outcome which is worse than the original one for i. Hannu Nurmi (Turku) Rationality of Voting II November, / 28
17 The no-show and related paradoxes Dodgson s rule and the twins paradox 42 voters 26 voters 1 voters 11 voters B A E E A E D A C C B B D B A D E D C C In this profile B is the (strong) Condorcet winner. Adding 20 copies of the one voter with ranking EDBAC leads to A being closest to Condorcet winner. This is worse than B from the point of view of the clones. Hence we have an instance of the twins paradox. Hannu Nurmi (Turku) Rationality of Voting II November, / 28
18 The no-show and related paradoxes Kemeny s rule and non-show paradox 5 voters 4 voters 3 voters 3 voters D B A A B C D D C A C B A D B C Here the Kemeny winner is D. Now, add 4 voters with DABC ranking. Then the resulting Kemeny ranking would have had A on top. Hence, we have an instance of the strong no-show paradox. Adding the DABC voters one by one to the 15-voter profile demonstrates the twins paradox. Hannu Nurmi (Turku) Rationality of Voting II November, / 28
19 The no-show and related paradoxes Maskin monotonicity Definition Maskin monotonicity. Let R N be a profile of n voters and and a procedure that, given this profile, results in alternative x being chosen. Let now another profile S N be constructed so that at least all those (and possibly some others) individuals who prefer x to y in R N do so in S N as well and this holds for all alternatives y x. Maskin monotonicity now requires that x be chosen in S N. Remark N.B. Maskin monotonicity is a very strong property. It implies monotonicity. Hannu Nurmi (Turku) Rationality of Voting II November, / 28
20 The no-show and related paradoxes Plurality fails on Maskin monotonicity 2 voters 1 voter 1 voter 1 voter A B C D B C B C C A A B D D D A Obviously, A is the plurality winner. Lift now B above C and D in the two left-most voters rankings and the new winner is B. Yet, A s position vis-à-vis the other alternatives has not been changed. (In fact one could improve A s position by lifting it above B in the right-most voter s ranking). Hannu Nurmi (Turku) Rationality of Voting II November, / 28
21 Monotonicity and no show paradox Does non-monotonicity imply no-show? monotonic systems non-monotonic systems vulnerable Copeland alternative vote invulnerable Borda count? Hannu Nurmi (Turku) Rationality of Voting II November, / 28
22 Monotonicity and no show paradox Campbell and Kelly s result Theorem Non-monotonicity does not imply the no-show paradox. Proof by way of a pretty implausible system. To wit, consider x X and J N, the active voters. Define the choice rule g so that g(j, P) = x if x is bottom-ranked by all i J. Otherwise, g(j, P) = y where y is top-ranked by smallest number of voters not ranking x at the bottom. Hannu Nurmi (Turku) Rationality of Voting II November, / 28
23 Monotonicity and no show paradox Campbell and Kelly, cont d This strange rule is non-monotonic since an improvement of the ranking of a winner if it is bottom-ranked makes it very often non-winning. Yet, this system is not vulnerable to no-show paradox since no group can improve the outcome from what it is by not voting. Remark N.B. This system is neither anonymous nor neutral. Hannu Nurmi (Turku) Rationality of Voting II November, / 28
24 Principles of system choice System choice in simple settings 1 A satisfies the criterion, while B doesn t, i.e. there are profiles where B violates the criterion, but such profiles do not exist for B. 2 in every profile where A violates the criterion, also B does, but not vice versa. 3 in practically all profiles where A violates the criterion, also B does, but not vice versa ( A dominates B almost everywhere ). 4 in a plausible probability model B violates the criterion with higher probability than A. 5 in those political cultures that we are interested in, B violates the criterion with higher frequency than A. Hannu Nurmi (Turku) Rationality of Voting II November, / 28
25 How often are the criteria violated? The role of culture impartial culture: each ranking is drawn from uniform probability distribution over all rankings impartial anonymous culture: all profiles (i.e. distributions of voters over preference rankings) equally likely unipolar cultures bipolar cultures Hannu Nurmi (Turku) Rationality of Voting II November, / 28
26 How often are the criteria violated? Lessons from probability and simulation studies cultures make a difference (Condorcet cycles, Condorcet efficiencies, discrepancies of choices) none of the cultures mimics reality IC is useful in studying the proximity of intuitions underlying various procedures Hannu Nurmi (Turku) Rationality of Voting II November, / 28
27 How often are the criteria violated? Literature Literature: Dummett, M. (1983)Voting Procedures. Oxford: Oxford University Press Fishburn P, and Brams S. (1983) Paradoxes of preferential voting, Mathematics Magazine 56, Moulin H (1988) Condorcet s principle implies the no show paradox, Journal of Economic Theory 45, Nurmi, H. (2004) Monotonicity and Its Cognates in the Theory of Choice, Public Choice 121, Hannu Nurmi (Turku) Rationality of Voting II November, / 28
28 How often are the criteria violated? Literature, cont d Nurmi, H. (2010) Voting Systems for Social Choice, pp in D. M. Kilgour and C. Eden (eds), Handbook of Group Decision and Negotiation. Berlin-Heidelberg-New York: Springer Verlag. Nurmi, H. (2011) Voting Procedures, pp (Vol. 5) in G. Kurian (ed.), The Encyclopedia of Political Science, Washington, D. C.: CQ Press Pérez, J.(2001) The strong no show paradoxes are common flaw in Condorcet voting correspondences, Social Choice and Welfare 18, Hannu Nurmi (Turku) Rationality of Voting II November, / 28
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