Duncan Black, in 1958, proposed a new voting system for multi-candidate elections (called Black's System):
- Each voter submits their entire preference order.
- If a Condorcet winner exist, then election that candidate.
- If not, then use the Borda count to determine the societal preference order.
Problem Set 7 : Which of the following criteria (anonymity, neutrality, monotonicity, the majority criterion and the CWC) does Black's system satisfy? Justify your answers.
Also, suppose Dale, Paul,and Wayne are the three finalist in the "World's Sexiest Man" contest held aboard a luxury cruise ship. The 15 judges will be using Black's method and the preference schedule they vote for is:
Number of Voters
Rank 7 6 2 1 P D W 2 D W P 3 W P D
Under Black's system, what societal preference order would be produced?
Suppose that after votes are cast, but before the winner is announced, Wayne is kicked off the ship for disorderly conduct, thus he is ineligible to be named the "World's Sexiest Man". Given your answer to the previous question, should Wayne's exclusion from the contest change its outcome?
Suppose that Wayne's name is removed from each of the 15 ballots shown above, with the remaining contestant moved up wherever is appropriate. What would be the outcome, under Black's system, with this new two candidate preference schedule? Is this odd to you?
End of Problem Set 7
We've come across this type of problem before (remember Jesse "The Body" Ventura in Minnesota).
A voting system in which the societal preference between two candidates depends only on the individual voters' preferences between those two candidates is said to satisfy the independence of irrelevant alternatives criterion or (IIA for short).
The "World's Sexiest Man" contest shows us that Black's system (and the Borda counting system) violates IIA. The example involving "The Body" shows us that plurality also violates IIA. Does sequential pairwise voting violate IIA? Let's see...
Suppose in a rematch for the "World's Sexiest Man" the judges rank Dale, Paul, and Wayne with the preference schedule:
Number of Voters
Rank 5 5 5 1 P D W 2 D W P 3 W P D
Then under SVP with the agenda D,P,W, the winner would be Wayne. However, had the judges in the last column switched (knowingly, since they do disapprove of Wayne's behavior) there preferences between Dale and Paul, who would win the contest under SVP with agenda D,P,W? Dale would win. This shows us that SVP is not IIA. Verify this using the definition of IIA.
Is there another "fair" voting system that satisfies IIA?
Let's take a look at the voting system called Instant Runoff (This voting system was hailed by the famous philosopher John Stuart Mill) where preference schedules produce the societal preference order by following algorithm:
- The candidate with the least number of first-place votes (or candidates in the case of a tie) is eliminated from each voter's preference order, and the remaining candidates are moved up on each preference order, yielding a new preference schedule.
- Repeat the first step until only one candidate remains. This is the winner.
- Form the societal preference order for the election by listing the candidates in the reverse of the order in which they were eliminated.
Problem Set 8 :Suppose that the three finalist to the "World's Wittiest Woman" contest are name Katie, Pam and Rachel and again that there are fifteen judges who produce preference schedule:
Number of Voters
Rank 6 3 3 3 1 P K R K 2 R P P R 3 K R K P
Using instant runoff, what societal preference order would be produced?
Suppose that in a rematch amonge the three finalist, the six judges in the left column swap Pam and Rachel in their rankings. What would the new societal preference order be?
Does instant runoff violate IIA?
Which other criterion does instant runoff violate among (anonymity, neutrality, monotonicity, MC, CWC)? Provide proof for your answers.
End of Problem Set 8
Are you starting to notice something? We have yet to find a single voting system that satisfies some reasonable criterion. We only need a few more definitions before we can get to the point here.
We say that a preference order (or societal preference order) is transitive, if whenever A>B and B>C, then we have the A>C. Also, a voting system is a function (i.e. a procedure) that receives as input a collection of transitive preference ballots and outputs a transitive societal preference order. Note that ties are perfectly acceptable.
Arrow thought that every reasonable voting system should satisfy these five properties.
- Condition 1-Universality. Votion systems should not place any restrictions other than transitivity on how voters can rank the candidates in an election. Specifically, systems should not dictate that some preference orders are acceptable while others are not; every possible colletion of transitive preference ballots must yield a transitive societal preference order.
- Condition 2-Positive Association of Individual Values.Voting systems should be monotone.
- Condition 3-Independence of Irrelevant Alternatives. Voting systems should satisfy IIA.
- Condition 4-Citizens Sovereignty. Voting systems should not be imposed in any way. That is, there should never be a pair of candidates in an election, say A and B, such that A is preferred over or tied with B in the resulting societal preference order regardless of how any of the voters vote.
- Condition 5-Nondictatorship. Voting systems should not be dictatorial. That is, there should never be a voter v such that, for any pair of candidates A and B, if v prefers A over B, then society will also prefer A over B.
Arrow was trying to find a voting system that satisfies all five conditions, after a few days of experimenting and finding no such system. He began to think that perhaps no such voting system exist. He was able to prove this in only a fews days, and the following "impossibility theorem" would come to be regarded as the single most important result in the history of voting theory (and also help earn him the 1972 Nobel Prize in economice science).
Arrow's Theorem. For an election with more that two candidates, it is impossible for a voting system to satisfy all five of Arrow's conditions.
If you are interested in learning how to prove this (with studying the last four sections you are now perfectly well prepared) see [Hodge & Klima, 2005] chapter five. Or you could read Arrow's original proof in [Arrow, 1951].
References and Further Reading:
[1]Kenneth J. Arrow. Social Choice and Individual Values. John Wiley and Sons, New York, 1951.
[2]Jonathan K. Hodge and Richard E. Klima.The Mathematics of Voting and Elections: A Hands-On Approach. American Mathematical Society, Providence, R.I., 2005.
[3]Wikipedia's entry on