Voting. Hannu Nurmi. Game Theory and Models of Voting. Public Choice Research Centre and Department of Political Science University of Turku

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1 Hannu Nurmi Public Choice Research Centre and Department of Political Science University of Turku Game Theory and Models of

2 points the history of voting procedures is highly discontinuous, early contributions typically aim at fixing a specific flaw in the prevailing system the main winner intuitions can be traced back to the 18 th the work on voting and electoral systems has proceeded in a parallel fashion, i.e.largely independently

3 Plato, around , B. C. practically no attention to voting and elections in Republic in Laws the issue of constitutional design are squarely confronted the prevailing system in Athens: Assembly (all male citizens), Council (500 elected by lot), Courts of Law (201 citizens or 501 citizens elected by lottery from a pool of 6000) Plato s proposal: 37 guardians elected through successive ballots (first 300, then out of these 100 and finally 37; all citizens entitled to vote) plurality of votes is required in final elections candidates (proposed by the guardians) may be objected to and then a pairwise majority determines the final candidate Szpiro: Plato s system biased against poorer and less

4 Pliny the Younger, 61 or 62 - ca. 113 A. D. faction 1 faction 2 faction 3 freedom death banishment banishment banishment freedom death freedom death Assume the nda: guilty vs. not guilty (i.e. freedom vs. banishment or death) if the former wins, then: banishment vs. death If the groups are of roughly equal size, banishment wins. Pliny thought that faction 1 is somewhat larger than the others and therefore proposed a plurality vote simultaneously on all alternatives. It turned out that banishment won. Undoubtedly Pliny was not the only strategic actor in the Senate.

5 Lull Ramon Lull ( ) Blanquerna, 1283, Ch. 24 electing the electors: one-person-one-vote principle the ultimate election (by electors): pairwise comparison where the winning alternative survives in each comparison and the loser is eliminated controversial remark: The candidate to be elected should be the one with the most votes in the most compartments. De Arte Eleccionis, 1299 pairwise comparison method matrix notation proposed a Copeland-like system

6 : Cusanus Nicolaus Cusanus, De concordantia catholica, 1434 the conciliaristic principle: the councils are entitled to elect the popes and emperors proposed the Borda Count was apparently unaware of the details of Blanquerna, even though refers to it.

7 Borda Jean-Charles de Borda,

8 Borda s contributions Jean-Charles de Borda, On Elections by Ballot (1770; 1781; 1784) the plurality winner may be the Condorcet loser introduces a point counting system (method of marks) can be implemented using binary comparisons

9 the following condition guarantees that the plurality winner is also the Borda winner y > E m 1 m where m is the number of candidates, y the number of those voters who rank the Borda winner first, and E is the number of voters. I.e. only unanimity guarantees that the plurality winner is the same as the Borda winner. Borda s argument runs as follows: suppose that A is the plurality winner with y votes and B is the runner-up with z votes. Now, the worst situation for A is when all E y voters rank A last, while B ranked second by y voters and first by E y voters. Hence the following condition guarantees that A will also be the Borda winner. (1)

10 my + E y > mz + [(m 1) (E z)] (2) When z = E y we get Equation 1.

11 Borda s example B C A C B C A A B Here A would win in plurality voting even though it has a plurality of voters against it (i.e. it would be defeated in pairwise contests by all the other candidates wit a majority of votes. Therefore, the voters have to be able to rank all candidates in the order of merit.there are two ways in which this can be accomplished: 1 the method based on preferences 2 the method based on pairwise comparisons

12 Condorcet Marie Jean Antoine Nicolas de Caritat, Marquis de Condorcet,

13 Condorcet s example and successive reversal 4 voters 3 voters 2 voters A B C B C A C A B The corresponding outranking matrix is: A B C A B 3-7 C here we have a majority cycle A B C A. Condorcet advises to look for the smallest number that exceeds the majority threshold > 4.5. Here it is 5. By inverting the majority relation that corresponds to this number - C A we get: A B, B C and A C. Condorcet s successive

14 Maximal agreement In the example above the support of the 6 different rankings: 1 ABC : 17 2 ACB : 12 3 BAC : 14 4 BCA : 15 5 CAB : 13 6 CBA : 10 I.e. the same outcomes as in successive reversal.

15 Successive reversal and maximal agreement may differ 7 voters 6 voters 4 voters 1 voter A B C B B C D C C D A A D A B D The pairwise majorities in order of magnitude are: C D, B C (and B D), A B (and C A), D A (next slide). Hence, the cycle A B C D A should according to the system of successive reversals be resolved by reversing D A to end up with: A B C D.

16 A B C D A B C D Maximal agreement (Kemeny), however, ends up with a different outcome. The order A B C D has the support: = 72. The order B C D A, in turn, has the support: = 74, i.e. larger.

17 Yet another Condorcet method Condorcet s practical method: The voters reveal their rankings. If a candidate has been ranked first by more than half of the electorate, he/she is elected. Otherwise, the candidate with the largest sum of first and second ranks is elected. The practical method differs from both of the preceding Condorcet methods. In the preceding example is differs from the reversal method. The following example shows that also differs from the maximal agreement one.

18 5 voters 5 voters 2 voters 2 voters 2 voters A C A B C B B C A A C A B C B Practical methods ranks B first, while the agreement one yields: A C B.

19 Simon Lhuilier ( ) mathematician who worked in Poland and in 1786 won a prize of the Science Academy of Berlin for an essay on the theory of infinity. in worked in University of Geneva and was a member of the legislative council (until Napoleon closed the council in 1798). analyzed Condorcet s practical method then in use in Geneva. L. showed that the method does not always elect the Condorcet winner and can lead to arbitrary outcomes when preference cycles occur. also showed that the method is non-monotonic and manipulable. supported the method proposed by a council member that would assign 1/2 votes to the candidates ranked second.

20 Dodgson Charles Lutwidge Dodgson (a.k.a. Lewis Carroll),

21 Dodgson wrote about proportional systems and majority elections invented the matrix notation in describing preference profiles and pairwise votes showed in A Discussion on the Various Methods of Procedure in Conducting Elections (1873) that different methods may lead to widely different outcomes under identical profiles recommends in a pamphlet called Discussion the Borda Count augmented with the no choice option.

22 Dodgson, cont d switched later to support Condorcet extensions. in the pamphlet A Method of Taking Votes on More Than Two Issues (1876) proposes a method that (i) elects the candidate voted for by more than a half of the electorate, (ii) if no such candidate exists the voters are to reveal their entire preference rankings and (iii) the Condorcet winner is elected if one exists (iv) If it doesn t, the top cycle is elected and other alternatives are rejected. (vi) From the top cycle the alternative is elected that can be made the Condorcet winner with fewer binary preference switches than any other top cycle alternative.

23 Nanson Edward John Nanson,

24 E. J. Nanson, Methods of Election, 1882 in contrast to Dodgson, Nanson was well aware of Borda and Condorcet The object of... an election is to select, if possible, some candidate who shall, in the opinion of a majority of the electors, be most fit for the post. Accordingly, the fundamental condition which must be attended to in choosing a method of election is that the method adopted must not be capable of bringing about a result which is contrary to the wishes of the majority. (The Condorcet winner criterion) Nanson shows that some widely used systems do not satisfy this criterion

25 Nanson, cont d recommends a method that satisfies the Condorcet criterion and, yet, is in the spirit of the Borda Count divides voting methods into: (i) those that need only one count of ballots to determine the winner, (ii) methods that require several rounds of balloting, and (iii) those that require only one round of balloting but several rounds of computing. class (i): plurality system, vote for two - method, Borda Count

26 class (ii): plurality runoff class (iii): Ware s method or alternative vote, Venetian method : st 1: vote for two, st 2: eliminate all but two largest vote-getters and conduct a plurality election between these two. Condorcet s practical method. Nanson s own proposal.

27 Some I D. Black. The Theory of Committees and Elections. Cambridge: Cambridge University Press 1958 I. McLean and A. Urken, eds. Classics of Social Choice. Ann Arbor: The University of Michigan Press G. Szpiro. Numbers Rule. Princeton: Princeton University Press I. Todhunter The History of Probability from the Time of Pascal to That of Laplace. Cambridge and London: Macmillan 1865.