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### Math 1010‘Mathematical Thought and Practice’

### Nell RayburnDavid Cochener

### Decisions

### Ordinal Ballots

### Arrow’s Theorem

An active learning approach to a liberal arts mathematics course

Department of Mathematics

Austin Peay State University

Clarksville, Tennessee

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!

Preference List

Ordinal Ballots

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

Ballots must be

- Complete (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.)

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.

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, Condorcetand 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?

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 winnerunless 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.

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