math 1010 ‘mathematical thought and practice’

Math 1010‘Mathematical Thought and

Practice’An active learning approach to

a liberal arts mathematics course

Nell RayburnDavid Cochener

Department of Mathematics

Austin Peay State University

Clarksville, Tennessee

Decisions

How do we make choices?

Types of Decisions

• Individual—our opinion is our decision.

• Group—Individual opinions are expressed by voting(at least in a democratic society)and some procedure is used to combine these individual preferences for a group decision.

Two Basic Questions

• What type of election decision procedure should we use to combine individual decisions (preferences) into a group decision?

• How can we be sure that what is decided is really what the group wants?

Desirable Properties of Majority Rule

(Two Alternative Case)• All voters are treated equally. (Swapping

marked ballots gives no change)

• Both alternatives are treated equally. (If all votes are reversed, so is the winner.)

• If a new election were held and a single voter changed from a vote for the previous loser to the previous winner, then the outcome would be the same as before.

May’s Theorem

• If the number of voters is odd, and if we are interested only in voting procedures that never result in a tie, then majority rule is the only voting system for two alternatives that satisfies the conditions listed on the previous slide.

Kenneth O. May (1915-1977)

• Mathematician, Political Activist

• PhD, Univ. of California, 1946

• Mathematics plays a crucial role in social science!

Ordinal Ballots

Preference List

Ordinal Ballots

• List your choices in order with the favorite on top and ‘least favorite’ on bottom

Ballots must beComplete (you must rank all candidates)Linear (no ties)Transitive (If you prefer A to B and B to C,

then you must prefer A to C.)

Preference Schedule

18 12 10 9 4 2

Allen Evert Davis Baker Childs Childs

Baker Childs Evert Davis Evert Davis

Childs Baker Childs Childs Baker Baker

Davis Davis Baker Evert Davis Evert

Evert Allen Allen Allen Allen Allen

Five Election Decision Methods

• Plurality

• Standard Runoff

• Sequential Runoff

• Borda Count

• Condorcet Winner Criterion

Plurality

• Whoever has the most votes wins!

• Problems? Remember Jesse Ventura??

1998 Minnesota Governor’s Race

• Jesse ‘The Body’ Ventura: 38%

• Hubert Humphrey III: 33%

• Norm Coleman: 29%

• The latter two were highly experienced, but somewhat dull compared to Ventura.

Standard Runoff

• If there is no majority, the two candidates receiving the most votes compete head to head.

Sequential Runoff

• If no one has a majority, eliminate the candidate(s) with the fewest first place votes, and count again. Continue in this way until someone has a majority.

• Also known as Hare elimination.

Borda Count

• If there are n candidates, assign n – 1 points to a first place choice, n – 2 points to a second place choice,…, 0 points to a last place choice. Sum the points for each candidate – the one with the most points wins.

• Problems? Borda Count can violate majority rule!

A surprising result!

• Borda count may violate majority rule!

A has a majority, but B wins Borda count by 21 – 19 over C, with A getting 18 points and D getting 8 points.

6 2 3

A B C

B C D

C D B

D A A

Condorcet Winner Criterion

• Conduct head – to – head contests between each pair of candidates. If any candidate can beat each of the others, he is a Condorcet candidate. Under this method a Condorcet candidate is declared the winner.

• May not always be decisive! (produce a winner)

Fairness Criteria• 1. Majority Criterion. If there is a choice that has a majority of the first-

place votes, then that choice should be the winner of the election.• 2. Condorcet Criterion. If there is a choice that is preferred by the voters

over each of the other choices (in a head-to-head matchup), then that choice should be the winner of the election.

• 3. Monotonicity Criterion. If choice X is a winner of an election and, in a re-election, all the changes in the ballots are favorable to X, then X should still be a winner.

• 4. Independence-of-Irrelevant-Alternatives Criterion. If choice X is a winner of an election, and one(or more) of the other choices is disqualified and the ballots recounted, then X should still be a winner. (Also called Binary Independence.)

• IIA can also be stated as: It is impossible for an alternative B to move from non-winner to winner unless at least one voter reverses the order in which he/she had ranked B and the winning alterative.

• 5. Pareto Criterion. If every voter prefers alternative X over alternative Y, then the voting method should rank X above Y.

Fairness Criteria• **Every method we have studied can violate one or more of

these!• Plurality: violates Condorcet, IIA• Standard Runoff: violates Monotonicity, Condorcet• Hare Elimination(Sequential Runoff): violates Monotonicity,

Condorcet and IIA• Coombs’ Method: violates Monotonicity and Condorcet• Sequential Pairwise Runoff: violates Pareto• Condorcet’s Method: May not even produce a winner!• Borda Count: violates Majority (and hence Condorcet) and IIA• Is there any election decision procedure we could devise which

satisfy these fairness criteria if we have 3 or more candidates and use ordinal ballots to rank the candidates?

Arrow’s Theorem

The search for the perfect election decision procedure

The story so far…

• We have studied several election decision procedures designed to produce one or more winners from a slate of 3 or more candidates.

• Each procedure has had some desirable features and some undesirable ones (‘quirks’).

• We’ve even seen that these methods can give different winners using exactly the same set of ordinal ballots!

Enter Kenneth Arrow

• Arrow, an economist, wanted to find a completely ‘fair’ election decision procedure.

• He began by making a list of a few basic properties that he believed any good election decision procedure should have:

Arrow’s Properties

• Universality—The decision procedure must be any to process any set of ordinal ballots to produce a winner, and must be able to compare any two alternatives.

• Non-dictatorship (no one voter can determine the outcome)

• Independence-of-Irrelevant-Alternatives Criterion (Binary Independence)

• Pareto Criterion

IIA or Binary Independence

• It is impossible for an alternative B to move from non-winner to winner unless at least one voter reverses the order in which he/she had ranked B and the winning alternative.

• In other words, whether A or B wins should depend only on how the voters compare A to B, and not on how other alternatives are ranked relative to A or B.

Pareto Criterion

• If every voter prefers alternative X to alternative Y, then the decision procedure should rank X above Y.

Asking the Impossible

• In 1951 Arrow published a book Social Values and Individual Choice in which he proved that there does not exist an election procedure which ranks for society 3 or more candidates based on individual preferences and which satisfies the fairness criteria we have listed.