Voting and Elections

V. Approval Voting, IBI, and Range Voting

Approval Voting has been adopted by a number of scientific and technical organizations, such as the American Mathematical Society, and the America Statistical Association to elect officers and make other important decisions. Also, approval voting is used to elect the Secretary-General of the United Nations and to elect new members into the National Academy of Sciences. It was proposed independently by several policital analyst in the 1970's and has many advantagous qualities. It works in the following way:

Note that, the system accepts as input only those preference orders in which the symbol > appears exactly once. So the only preference orders allowed are those that have one or more candidates tied for first place, followed by all of the remaining candidates, who are tied for last place

Approval voting is clearly not a dictatorship and also preserves Citizen Sovereignty. Howevery, to see that it is monotone, just consider a voter changing there preference order (for candidate A) to go from dissapproval to approval, this can only increase the candidates ranking in the societal preference order. Hence, approval voting is satisfies monotonicity.

Problem Set 9 :Three friends, Peter, James, and John, are trying to decide who among them is the greatest. To do so, they ask 9 of there friends to cast approval ballots. The friends agree and the following preference schedule is the result of their votes:
Number of Voters

Where "x" denotes approval. Under approval voting, who would be declared the greatest? What societal preference order would be produced? Do you think that the outcome of the election under approval voting accuretly reflects the will of the voters? Why or why not?

Consider the two voters who approved of both Peter and James but not John. Which of the following individual preference orders could be consistent with these two voters' approval ballots?

  1. Peter>James>John
  2. James>Peter>John
  3. Peter=James>John
  4. James=Peter=John
  5. Peter>James=John
By definition, approval voting violates one of the important fairness criteria of Arrow (see the previous section). Which one is it and why is it violated?

So approval voting satisfies Conditions 2,4,5 of Arrow, and violates Condition 1. What about Condition 3, IIA? Suppose that an election is held using approval voting, but that, due to voter irregularities, a revote is necessary. Suppose also that, in this revote, some voters change their ballots, but never in a way that affects there individual preferences between just candidates A and B? Explain why, in the revote, the difference in the number of approval votes received by A and B will be exactly the same as it was in the original elections. What does this allow you conclude about approval voting and IIA?
End of Problem Set 9

We saw in the last section that there is no voting system that satisfies Arrow's five conditions. Donald Saari has interpreted Arrow's Theorem as saying: we require voting systems to produce transitive societal preference orders from individual transitive preference orders, but transitivity forces certain connections between pairwise comparisons in individual voter preferences (i.e. if a voter prefers A over B, and B over C, then transitivity requires the voter to also prefer A over C). Since IIA requires voting systems to determine the societal preference between any pair of candidates based solely on the individual voters' preferences between those two candidates, this requirements forces voting systems to throw away the connecting information supplied by transitivity, making it impossible for systems that satisfy IIA to distinguish between voters with rational transitive preferences and voters with irrational, cyclic preferences (remember cyclic preference just circle around A>B>C>A). Saari's suggestion was to weaken IIA by allowing the voting system to take into account the intensity of peoples preference also.

Supposing that a voter prefers candidate A over B in an election, the intensity of this preference is the number of candidates listed between A and B on the voter's individual preference order. A voting system is said to satisfy the intensity of binary independence (IBI) criterion if it takes into account voter's preferences and the intensity of those preferences.

Along a similar line of thought, one can be led to the idea of range voting, which is where each voter gives each candidate a score (typically 0-99, 0 being completely disapprove and 99 being complete approve). Voters can choose not to give a score to any candidate, and the societal preference order is determined by the average scores of the candidates. For any candidate that a particular voter did not provide a score, the average is computed without any input from that voter. This system was used in all public elections in ancient Sparta (with the average score being how loudly the crowd yelled), and in a limited form by members of the Electoral College for the first four Presidents (until being made unconstitutional by the XII amendment). Also, amazingly Honey Bees have been using range voting for around 20-50 million years in hundreds of trillions of elections they use a complicated system to range vote of where to build the new nest each season. See Honey Bees use Range Voting! for more information.

Notice that approval voting is just range voting with a scale of 0-1. In some competitons (such as figure skating) instead of using averages, truncated means are used

Problem Set 10 :Suppose Greg, Sharon, Dean, and Carolyn are the last four competitors of the new reality TV show, Starvation Island. The show uses the Borda count to determine the player eliminated during each round of the contest. After the ballots were cast by the four competitors, Greg breaks down and begins to eat the ballots. The other members manage to restrain him, but only in time to recover the following information:
Using only this information, what can you conclude about the resulting societal preference order between Sharon and Greg?

Does the Borda count satisfy IBI? Why or why not?

Is there a voting system that satisfies universality, unanimity, and IBI, that is not a dictatorship? Would such a voting system be a contradiction to Arrow's Theorem?

Does range voting satisfy IIA, monotonicity, citizen sovereignty? Also, since it's not a dictatorship, why is it not a contradiction to Arrow's Theorem?

End of Problem Set 10

References and Further Reading:
[1]Jonathan K. Hodge and Richard E. Klima.The Mathematics of Voting and Elections: A Hands-On Approach. American Mathematical Society, Providence, R.I., 2005.
[2]'s entry on Honey Bees:
[3]Wikipedia's entry on Range Voting: