Ranked Pairs "satisfies Independence of Irrelevant Alternatives", says Electowiki
Maximize Affirmed Majorities (MAM) is another name for Ranked Pairs
(Winning Votes).
Neither it or any other version Ranked Pairs or any other Condorcet
method meets Independence of Irrelevant Alternatives.
Perhaps the Electowiki author confused it with "Local Independence of
Irrelevant Alternatives".
Chris Benham
http://wiki.electorama.com/wiki/Independence_of_irrelevant_alternatives
Independence of irrelevant alternatives
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In voting systems http://wiki.electorama.com/wiki/Voting_system,
independence of irrelevant alternatives is the property some voting
systems have that, if one option (X) wins the election, and a new
alternative (Y) is added, only X or Y will win the election.
Most Condorcet methods
http://wiki.electorama.com/wiki/Condorcet_method fail this
criterion, although Ranked Pairs
http://wiki.electorama.com/wiki/Ranked_Pairs satisfies it
The MAXIMIZE AFFIRMED MAJORITIES voting procedure
*Some criteria not satisfied by MAM: *
/independence of irrelevant alternatives/ (/IIA/, the strong version
for social
ordering procedures): For all pairs of alternatives, for
instance /x/ and /y/,
their relative social ordering must not change if voters raise
or lower
other alternatives in their votes. (This was proposed by
Kenneth Arrow
and is similar in spirit to his /choice consistency/ criterion
for social choice
procedures, described below. It is too demanding for any
reasonable
social ordering procedure to satisfy. See "Arrow's
Impossibility Theorem
http://alumnus.caltech.edu/%7Eseppley/Arrow%27s%20Impossibility%20Theorem%20for%20Social%20Choice%20Methods.htm.")
Arrow's Impossibility Theorem for Social Choice Methods
*Implications of Arrow's "Impossibility Theorem" for Voting Methods *
/Kenneth Arrow proved no voting method can satisfy a certain set
of desirable
criteria, implying no voting method is ideal. But this does not
mean we
should abandon the search for the best (non-ideal) voting method, and
in particular, since the set of nominees is endogenous the effect
of the
voting method on the set of nominees should be included in the
analysis. /
There are often gains to be had by an organization or society by
making a collective choice
from a set of alternatives available to them, rather than having each
individual act independently
(uncertain how others will act). However, since there are many ways
to aggregate individuals'
reports of their preferences in order to reach a collective choice,
the gain (or loss) may depend
on the procedure by which the collective choice is made. To study
this we need to model the
nature of individuals' preferences and consider various criteria by
which various aggregation
methods can be compared.
We make some useful abbreviations. We use letters like /i/, /j/,
etc., to denote individuals who
vote. Assume the group is choosing from a (possibly large) set of
possible alternatives, which
we call /X/. We use letters like /x/, /y/, /z/, etc., as
abbreviations for alternatives in /X/. Assume
the alternatives are mutually exclusive, in that at most one can be
elected, and assume /X/ is
complete, in that it includes all possible outcomes. Thus one and
only one alternative in /X/
will be elected. The individuals might not be asked to consider every
alternative in /X/,
particularly if /X/ is large, so we refer to the alternatives under
consideration as the "agenda"
and call them /A/. We can also think of /A/ as the set of "nominated"
alternatives, those which
appear on the ballot. /A/ is not determined by nature but is affected
by nomination decisions
made by individuals--perhaps only a small number of individuals are
required to add an
alternative to the agenda. Whether or not individuals have incentives
to nominate certain
alternatives will depend on their beliefs about how the action would
affect the outcome
in the short and long term.
We model each individual as behaving as if she has
"preferences" regarding alternatives.
Every preference is a relative comparison of some pair of
alternatives. That is, for any
pair of alternatives, say /x/ and /y/, each individual has a
preference for /x/ over /y/ or has a
preference for /y/ over /x/ or is indifferent between /x/ and /y/. We
assume each individual's
preferences are "self-consistent": Each individual who prefers /x/
over /y/ and /y/ over /z/
also prefers /x/ over /z/, and each individual who is indifferent
between /x/ and /y/ and
between /y/ and /z/ is also indifferent between /x/ and /z/. Such
self-consistent preferences
are called "orderings" of the alternatives, in the same sense that
numbers can be ordered
from largest to smallest. Alas, no individual's preferences can be
directly observed;
all we can observe are behaviors such as how they choose from a set of
options,
or how they answer polls (not necessarily honestly), or how they mark
ballots.
We don't attempt here to model the educational processes by
which individuals
acquire preferences, nor how preferences may change with time; we are
concerned
mostly with preferences as they are when society votes (hopefully
after due deliberation,
but not necessarily).
Individuals' preferences may be intense, or mild, or in-between.
Depending on the
criteria we impose on the voting method, information about preference
intensities
might not be admissible when voting, or might be ignored when tallying
the outcome.
Without loss of generality, we assume that when society votes,
individuals mark ballots.
The collection of all ballots is input to a tallying procedure, called
a "choice function,"
which we will call /C/. To avoid overly constraining the analysis, we
will not assume /C/
always chooses a single winner; in the cases where /C/ chooses more
than one we assume
a subsequent procedure, such as flipping a coin, will be used to pick
one of those chosen
by /C/. Thus our first criterion is simply the following:
/Prime directive/: The choice function must choose one or more of
the
nominated alternatives (if at least one alternative has been
nominated)
and not choose any non-nominated alternatives.
The /prime directive/ should not be interpreted as banning "write-in"
candidates, which
we would treat as "just-in-time" nominees. Besides ensuring that at
least one of the
nominees will be chosen, its purpose is to ensure that no alternative
left unranked
by every voter will be chosen.
The next two criteria are straightforward and very mild constraints:
/Unanimity/: No alternative that is ranked by all voters
below another
alternative, say /x/, may be chosen if /x/ is one of the nominees.
/Non-dictatorship/: No voter may be so privileged that,
regardless of the
other voters' votes, the choice is always his top-ranked nominee
(or from
among his top-ranked nominees, when he votes indifference at the top).
Our next criterion serves to limit the amount of information that must
be elicited from
the voters, so they only need to express preferences regarding
nominated alternatives (/A/).
This is justifiable since the set of possible alternatives /X/ might
be very large, so a voting
method that needs preference information regarding all of /X/ would
exhaust all participants.
Or, if the voting method needs info about some non-nominated
alternatives (in addition to
info about the nominees) then it would not be obvious which
alternatives outside /A/ should
be included, and if any individuals are given the power to decide
which other alternatives
will be voted on, they might be able to manipulate the outcomes in
their favor. Also,
game theory predicts that this constraint is actually quite mild,
since if the voters know
that alternatives outside /A/ cannot be chosen then their optimal
voting strategies would
elect the same alternatives as would be elected if those outside
alternatives could not
appear in their votes.
/Independence from Irrelevant Alternatives/ (/IIA/): The choice
function
must neglect all information about non-nominated alternatives.
The next criterion further constrains the information that may be
considered by /C/.
Specifically, we require /C/ to ignore information about the intensity
of voters' preferences,
so "mild" preferences will be treated the same as "intense"
preferences. In other words,
two ballots that rank the alternatives in the same order must be
treated the same. The
justification for this is that, if intensity information were not
ignored, it would create a
strong incentive for each voter to exaggerate her intensities
by dividing the alternatives
into two groups and voting the maximum possible intensity between the
two groups
(and indifference within each group). To see this, suppose
pre-election polls indicate
the two likely front-running candidates are /x/ and /y/. Then each
voter who prefers /x/ over /y/
has an incentive to report the maximum possible intensity for /x/ over
/y/, to avoid partially
wasting the power of her vote. Similarly, those who prefer /y/ over
/x/ have an incentive to
report the maximum possible intensity for /y/ over /x/. If the voters
who prefer /x/ over /y/
believe those who prefer /y/ over /x/ will vote the maximum intensity
for /y/ over /x/, they would
be foolish not to vote the maximum intensity for /x/ over /y/, etc.
While doing so, it would
be most effective for those voting /x/ over /y/ to also cast
the maximum possible vote for
candidates preferred over /x/ (in other words, indifference between
them and /x/, since /x/
already is getting their maximal vote) and the minimal possible vote
for candidates less
preferred than /y/ (indifference between them and /y/), etc. This may
not be obvious at first,
but we presume most voters would quickly learn the strategy since it
is so easy. The result
would be that voters would express much less information in their
votes than if the choice
function ignores all intensity information. Thus, we have our next
criterion:
/Ordinality/: The choice function must neglect all "intensity"
information.
In other words, only "ordinal" information may affect the choice.
The next criterion requires that the choice function accept a
considerable amount and
diversity of information from each voter about her preferences, if she
wishes to express it.
Since Kenneth Arrow was analyzing whether and how voters' preferences
might be
aggregated to reach a collective decision, and since there is no /a
priori/ reason to expect
voters' preferences to adhere to any pre-ordained pattern, it makes
sense to require the
method of aggregating preferences to work no matter what the voters'
preferences may be.
/Universal Domain/: The choice function must accept from each voter
any ranking of the alternatives.
On the other hand, we are not really limited to Arrow's framework,
which was designed
merely to try to aggregate voters' (sincere) preferences. Although it
is reasonable to require
the voting method to work for any collection of preferences the voters
may have, it does
not necessarily follow that no constraints should be placed on the
expressions voters may
make when voting. For instance, the so-called Approval voting method
constrains each
voter to partitioning the alternatives into two subsets, which is
equivalent to a non-strict
ordering that has at most two "indifference classes." It is not /a
priori/ obvious that the use
of voting methods such as Approval, which constrain the voters from
completely expressing
their preference orderings, are worse for society, so the /universal
domain/ criterion should
be considered controversial until other arguments not explored by
Arrow are examined
(assuming those arguments support the conclusion that it is better not
to constrain the voters
from expressing orderings). In other words, other criteria for
comparing voting methods,
in addition to Arrow's criteria, need to be evaluated. (My own
conclusion is that there are
solid reasons why it is better not to constrain the voters'
expressions, but that is beyond
the scope of this document.)
Kenneth Arrow's theorem [1951, 1963] states that, if /X/ includes at
least 3 alternatives
then no choice function that satisfies all of the criteria listed
above also satisfies the
following criterion:
/Choice consistency/: For all pairs of alternatives, say /x/ and
/y/, if the votes
are such that /x/ but not /y/ would be chosen from some set of
nominees that
includes both, then /y/ must not be chosen from any set of
nominees that
includes both. (The literature usually calls this /rationality/,
but I prefer
the less loaded term /choice consistency/.)
(A proof of Arrow's theorem
http://alumnus.caltech.edu/%7Eseppley/Arrow%27s%20Impossibility%20Theorem%20for%20Social%20Choice%20Methods.htm#Proof%20of%20Arrow%27s%20theorem
is provided in the appendix.)