### Fairness and Equity Glossary (9/25/06)

Prepared by:

Joseph Malkevitch

Mathematics and Computing Department

York College (CUNY)

Jamaica, New York 11451-0001

Email: malkevitch@york.cuny.edu (for additions, suggestions, and corrections)

**Adams Method**

A method of solving the apportionment problem based on the rounding rule which always
rounds any fraction up.

**Apportionment problem**

Given a positive integer h which represents the number of identical indivisible items
available (often seats in a legislature), determine the number of items (non-negative
integer) to give to the claimants.

** Approval voting**

An election decision method where the winner is the candidate who gets the largest
number of votes based on a ballot where each voter votes for as many of the candidates
running as he/she is willing to have serve in the office (i.e. each voter votes for
the candidates who have the voters approval).

** Ballot**

A way for voters to express their preferences for alternatives from which a choice has to be made. Sometimes a ballot is used to get information when a single alternative is to be chosen and sometimes when many alternatives are to be selected. Many new ballot types have recently been explored. The most common ballot types ask voters to pick a single alternative, to rank all the alternatives with no ties allowed, or to rank a subset of the alternatives with ties allowed or not allowed. A more novel type of ballot would require the alternatives be placed in to groups: those that are acceptable and these are all ranked with ties allowed and another group of those that are not acceptable.

**Bankruptcy problem**

Given an "estate" to determine the amount to be given to each claimant. It is assumed
that the total claims exceed the value of the estate.

**Borda Count**

The Borda Count is an election decision method where the winner is decided on the
basis of points which are assigned to the candidates based on how far up on the ballots
of the voters the candidates appear. The number of points assigned to candidate X
as a consequence of a voter i's ballot is the number of candidates who lie below X on
i's ballot. The Borda Count belongs to the family of scoring rule methods for deciding
elections.

**Cardinal ballot**

A ballot in which a voter is allowed to assign a fixed number of points to candidates
in any way the voter likes. (Using the cardinal ballot, an associated ordinal ballot
can be constructed.)

**Coalition**

A group of players in a game who work together with each other.

**Condorcet method**

The election decision method where the winner is the candidate who can beat all the
other candidates in a two way race. Not all elections can be decided by this method.

**Condorcet loser**

A candidate in an election situation who would lose to any other candidate in a two-way
race. Some elections methods guarantee that a Condorcet loser can not be elected.
For other election methods it is possible for a condorcet loser to be elected.

**Condorcet winner**

A candidate in an election situation who can beat any other candidate in a two-way
race. Some elections have no Condorcet winners. Some election decision methods do
not guarantee the election of a Condorcet winner even when there is one.

**Coomb's method**

An election decision method where if no candidate has a majority, the candidate with
the largest number of last place votes is eliminated and the procedure is repeated.
Coomb's method may not elect a Condorcet candidate when there is one.

**Cooperative game**

A game in which the players are allowed to communicate and interact prior to the each
player's decision as to how to play the game.

**Copeland's method**

An election method in which one first considers the number of two-way races each of
the candidates can win. Now one computes the number of wins minus the number of loses.
The candidates are ranked on the basis of the resulting numbers. For small numbers
of candidates the method often leads to many ties. When there is a Condorcet winner,
Copeland's method will select this candidate.

**Dean's Method**

The apportionment method in which the seats are assigned on the basis of a rounding
rule that involves the harmonic mean.

**Gini index**

The Gini Index, named for allows one to use one number to compare the equity of
income or wealth between different countries at a given time or the trend in equity
within one country at different times.

**Hamilton's Method**

The apportionment method in which each claimant is initially assigned the greatest
integer less than or equal to its exact quota. If there are additional "seats" to
be assigned these are assigned in order of the size of the remaining fractional parts.

**Huntington-Hill Method**

A method of solving the apportionment problem based on the rounding rule which round
fractions based on the geometric mean.

**Jefferson's Method (in Europe, D'Hondt's Method)**

A method of solving the apportionment problem based on the rounding rule which rounds
any fraction down.

**Lorenz Curve**

The Lorenz Curve, named for Lorenz plots for the group of people who are the lowest
i percent of wage earners, the percent of the total income earned in the society
by them. If a society were perfectly equitable, then the lowest 20 percent of wage
earners should earn 20 percent of the total income. A Lorenz Curve can also be drawn for
wealth rather than income. Lorenz curves from different countries at a given time
can be used to compare the equity of the distribution of wages in those countries.
For one country, drawing two Lorenz curves enables one to determine if things are becoming
more or less equitable.

**Maimonides Method**

The method of solving the bankruptcy problem in which each claimant is assigned as
equal an amount possible but never more than what is claimed.

**Majority voting**

The election decision method where the winner is the candidate who received the largest
number of first place votes. In many elections this approach will not yield a winner.

**Majority Principle**

An election method obeys the majority principle if it selects as the winner a candidate
who received the majority of first place votes when there is such a candidate.

**Minimal winning coalition**

A winning coalition but one in which if any player decides not to participate in the
coalition then the remaining players can no longer win.

**Nanson's method**

A method of conducting elections which is based on a sequential run-off using the
Borda Count. When there is a Condorcet winner, Nanson's method will select such a
candidate.

**Ordinal ballot**

A ballot in which the voter is asked to rank the candidates in order from highest
ranked to lowest. The voter may be required to do this with no provision for ranking
candidates at the same level or the voter may be allowed to rank candidates at the
same level. A practical problem arises when a voter for whatever reason does not list all
the choices (candidates) on the ballot.

**Pareto optimality (Pareto improvement)**

Solution of an equity problem where each person is at least as well of they would
be in any other solution to the problem. Sometimes the phrase "Pareto improvement"
is used for an outcome which is superior for at least one person, and no worse for
any of the other players than the current outcome.

**Perfect information game**

A game in which each player knows exactly what actions are available to him/her as
well as of the his/her opponents in the game. Furthermore, the payoffs associated
with all possible actions the players may take are known to all all of the players.

**Plurality voting**

The method of deciding an election based on the winning candidate having the largest
number of first place votes.

**Proportional method**

The method of solving the bankruptcy problem in which each claimant is assigned an
amount proportional to the person's claim.

**Run-off method**

The method of deciding an election where if no candidate gets a majority all but the
two candidates with the highest number of first place votes are eliminated and a
new election is conducted between these two candidates.

**Scoring rule methods**

A scoring rule method is election decision method in which each candidate is assigned
points (i.e. a score) depending on where a voter's ballot the candidate appears.
The scores chosen have the property that more points are usually given for being
higher up on a voter's preference schedule, however, some point assignments are allowed to
be equal. It is usual not to permit all the scores to be the same and that the scores
decrease from top rank to lowest rank.

**Sequential run-off method**

The method of deciding an election where if no candidate gets a majority the candidate
with the lowest number of first place votes is eliminated and the procedure is repeated
until some candidate gets a majority.

**Sincere voting**

The situation when the voters vote in a decision making situation in accordance with
their true preferences for the candidates.

**Single transferable vote**

A method of conducting elections which when only one seat is to be filled is similar
to sequential run-off, but when several seats are to be filled can used as a proportional
representation method. The method establishes a quota for election and transfers
votes for candidates who exceeded this quota to other candidates. If seats are yet
to be filled but no candidate has achieved a quota, the candidate with the lowest
number of votes is eliminated and these ballots are transferred to other lower ranked
candidates of those voters. (The system was developed by Thomas Hare.)

**Sophisticated voting**

The situation in a voting situation when voters do not vote in accordance with their
true preferences but vote instead for other candidates in the hope of achieving a
more preferred outcome than if they voted in a sincere manner.

**Utilities**

Numbers (called utilities) which a person can assign to the outcomes of a game (or
election) where when outcome X is preferred to outcome Y by the person, the size
of the number assigned to X is larger than the size of the number assigned to Y.
(It is important to note that the preferences are represented via the utilities and are assumed
to come first.)

**Utilitarianism**

An approach to social justice where the "utility" for society is obtained by adding
up the utilities for all its members. The goal for society is to maximize society's
utility.

**Webster's method**

A method of solving the apportionment problem based on the usual round rule that rounds
up fractions only when they are .5 or more.

**Weighted voting (games)**

A voting situation where each of the individuals who votes casts a block of votes
and the number of such votes varies from one voter to another. Examples include the
Electoral College and stockholders meetings. Some voting situations not explicitly
described as involving weighted voting (e.g. the Security Council of the United Nations)
can be none-the-less thought of as being a weighted voting game.

**Winning coalition**

A group of players in a game who can "win" the game by acting together.

You can find another glossary dealing with voting and election issues on the
__Range Voting Website__.

Acknowledgements

I would like to thank Professor Hope Young of the York College (CUNY) Library for
her assistance in obtaining many books and articles which helped enrich my understanding
of the concepts discussed above.

This work was prepared with partial support from the National Science Foundation (Grant
Number: DUE 9555401) to the Long Island Consortium for Interconnected Learning (administered
by SUNY at Stony Brook).

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