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<p>Maximize Affirmed Majorities (MAM) is another name for Ranked
Pairs (Winning Votes).<br>
<br>
Neither it or any other version Ranked Pairs or any other
Condorcet method meets Independence of Irrelevant Alternatives.<br>
<br>
Perhaps the Electowiki author confused it with "Local
Independence of Irrelevant Alternatives".<br>
</p>
<p>Chris Benham<br>
<br>
<a moz-do-not-send="true" class="moz-txt-link-freetext"
href="http://wiki.electorama.com/wiki/Independence_of_irrelevant_alternatives">http://wiki.electorama.com/wiki/Independence_of_irrelevant_alternatives</a><br>
<br>
</p>
<blockquote type="cite">
<h1 class="firstHeading" id="firstHeading" lang="en">Independence
of irrelevant alternatives</h1>
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<p>In <a moz-do-not-send="true" title="Voting system"
href="http://wiki.electorama.com/wiki/Voting_system">voting
systems</a>, <b>independence of irrelevant alternatives</b>
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. </p>
<p>Most <a moz-do-not-send="true" title="Condorcet method"
href="http://wiki.electorama.com/wiki/Condorcet_method">Condorcet
methods</a> fail this criterion, although <a
moz-do-not-send="true" title="Ranked Pairs"
href="http://wiki.electorama.com/wiki/Ranked_Pairs">Ranked
Pairs</a> satisfies it</p>
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</blockquote>
<br>
<br>
<a moz-do-not-send="true" class="moz-txt-link-freetext"
href="http://alumnus.caltech.edu/%7Eseppley/">http://alumnus.caltech.edu/~seppley/</a><br>
<p> </p>
<blockquote type="cite">
<title>The MAXIMIZE AFFIRMED MAJORITIES voting procedure</title>
<p><b>Some criteria not satisfied by MAM: </b></p>
<p><i>independence of irrelevant alternatives</i> (<i>IIA</i>,
the strong version for social <br>
ordering procedures): For all pairs of alternatives,
for instance <i>x</i> and <i>y</i>, <br>
their relative social ordering must not change if
voters raise or lower <br>
other alternatives in their votes. (This was proposed
by Kenneth Arrow <br>
and is similar in spirit to his <i>choice consistency</i>
criterion for social choice <br>
procedures, described below. It is too demanding for
any reasonable <br>
social ordering procedure to satisfy. See "<a
moz-do-not-send="true"
href="http://alumnus.caltech.edu/%7Eseppley/Arrow%27s%20Impossibility%20Theorem%20for%20Social%20Choice%20Methods.htm">Arrow's
Impossibility Theorem</a>.") </p>
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<br>
<blockquote type="cite">
<title>Arrow's Impossibility Theorem for Social Choice Methods</title>
<p align="center"><b>Implications of Arrow's "Impossibility
Theorem" for Voting Methods </b></p>
<blockquote>
<p><i>Kenneth Arrow proved no voting method can satisfy a
certain set of desirable <br>
criteria, implying no voting method is ideal. But this
does not mean we <br>
should abandon the search for the best (non-ideal) voting
method, and <br>
in particular, since the set of nominees is endogenous the
effect of the <br>
voting method on the set of nominees should be included in
the analysis. </i> </p>
</blockquote>
<p>There are often gains to be had by an organization or society
by making a collective choice <br>
from a set of alternatives available to them, rather than
having each individual act independently <br>
(uncertain how others will act). However, since there are
many ways to aggregate individuals' <br>
reports of their preferences in order to reach a collective
choice, the gain (or loss) may depend <br>
on the procedure by which the collective choice is made. To
study this we need to model the <br>
nature of individuals' preferences and consider various
criteria by which various aggregation <br>
methods can be compared. </p>
<p>We make some useful abbreviations. We use letters like <i>i</i>,
<i>j</i>, etc., to denote individuals who <br>
vote. Assume the group is choosing from a (possibly large)
set of possible alternatives, which <br>
we call <i>X</i>. We use letters like <i>x</i>, <i>y</i>,
<i>z</i>, etc., as abbreviations for alternatives in <i>X</i>.
Assume <br>
the alternatives are mutually exclusive, in that at most one
can be elected, and assume <i>X</i> is <br>
complete, in that it includes all possible outcomes. Thus one
and only one alternative in <i>X</i> <br>
will be elected. The individuals might not be asked to
consider every alternative in <i>X</i>, <br>
particularly if <i>X</i> is large, so we refer to the
alternatives under consideration as the "agenda" <br>
and call them <i>A</i>. We can also think of <i>A</i> as
the set of "nominated" alternatives, those which <br>
appear on the ballot. <i>A</i> is not determined by nature
but is affected by nomination decisions <br>
made by individuals--perhaps only a small number of
individuals are required to add an <br>
alternative to the agenda. Whether or not individuals have
incentives to nominate certain <br>
alternatives will depend on their beliefs about how the action
would affect the outcome <br>
in the short and long term. </p>
<p>We model each individual as behaving as if she has
"preferences" regarding alternatives. <br>
Every preference is a relative comparison of some pair of
alternatives. That is, for any <br>
pair of alternatives, say <i>x</i> and <i>y</i>, each
individual has a preference for <i>x</i> over <i>y</i> or
has a <br>
preference for <i>y</i> over <i>x</i> or is indifferent
between <i>x</i> and <i>y</i>. We assume each individual's <br>
preferences are "self-consistent": Each individual who prefers
<i>x</i> over <i>y</i> and <i>y</i> over <i>z</i> <br>
also prefers <i>x</i> over <i>z</i>, and each individual who
is indifferent between <i>x</i> and <i>y</i> and <br>
between <i>y</i> and <i>z</i> is also indifferent between <i>x</i>
and <i>z</i>. Such self-consistent preferences <br>
are called "orderings" of the alternatives, in the same sense
that numbers can be ordered <br>
from largest to smallest. Alas, no individual's preferences
can be directly observed; <br>
all we can observe are behaviors such as how they choose from
a set of options, <br>
or how they answer polls (not necessarily honestly), or how
they mark ballots. <br>
We don't attempt here to model the educational processes by
which individuals <br>
acquire preferences, nor how preferences may change with time;
we are concerned <br>
mostly with preferences as they are when society votes
(hopefully after due deliberation, <br>
but not necessarily). </p>
<p>Individuals' preferences may be intense, or mild, or
in-between. Depending on the <br>
criteria we impose on the voting method, information about
preference intensities <br>
might not be admissible when voting, or might be ignored when
tallying the outcome. </p>
<p>Without loss of generality, we assume that when society
votes, individuals mark ballots. <br>
The collection of all ballots is input to a tallying
procedure, called a "choice function," <br>
which we will call <i>C</i>. To avoid overly constraining
the analysis, we will not assume <i>C</i> <br>
always chooses a single winner; in the cases where <i>C</i>
chooses more than one we assume <br>
a subsequent procedure, such as flipping a coin, will be used
to pick one of those chosen <br>
by <i>C</i>. Thus our first criterion is simply the
following: </p>
<blockquote>
<p style="margin-top: -1px;"><i>Prime directive</i>: The
choice function must choose one or more of the <br>
nominated alternatives (if at least one alternative has been
nominated) <br>
and not choose any non-nominated alternatives. </p>
</blockquote>
<p style="margin-top: -1px;">The <i>prime directive</i> should
not be interpreted as banning "write-in" candidates, which <br>
we would treat as "just-in-time" nominees. Besides ensuring
that at least one of the <br>
nominees will be chosen, its purpose is to ensure that no
alternative left unranked <br>
by every voter will be chosen.</p>
<p>The next two criteria are straightforward and very mild
constraints: </p>
<blockquote>
<p style="margin-top: -1px;"><i>Unanimity</i>: No alternative
that is ranked by all voters below another <br>
alternative, say <i>x</i>, may be chosen if <i>x</i> is
one of the nominees.</p>
</blockquote>
<blockquote>
<p style="margin-top: -1px;"><i>Non-dictatorship</i>: No
voter may be so privileged that, regardless of the <br>
other voters' votes, the choice is always his top-ranked
nominee (or from <br>
among his top-ranked nominees, when he votes indifference at
the top).</p>
</blockquote>
<p>Our next criterion serves to limit the amount of information
that must be elicited from <br>
the voters, so they only need to express preferences regarding
nominated alternatives (<i>A</i>). <br>
This is justifiable since the set of possible alternatives <i>X</i>
might be very large, so a voting <br>
method that needs preference information regarding all of <i>X</i>
would exhaust all participants. <br>
Or, if the voting method needs info about some non-nominated
alternatives (in addition to <br>
info about the nominees) then it would not be obvious which
alternatives outside <i>A</i> should <br>
be included, and if any individuals are given the power to
decide which other alternatives <br>
will be voted on, they might be able to manipulate
the outcomes in their favor. Also, <br>
game theory predicts that this constraint is actually quite
mild, since if the voters know <br>
that alternatives outside <i>A</i> cannot be chosen then
their optimal voting strategies would <br>
elect the same alternatives as would be elected if those
outside alternatives could not <br>
appear in their votes. </p>
<blockquote>
<p style="margin-top: -1px;"><i>Independence from Irrelevant
Alternatives</i> (<i>IIA</i>): The choice function <br>
must neglect all information about non-nominated
alternatives. </p>
</blockquote>
<p>The next criterion further constrains the information that
may be considered by <i>C</i>. <br>
Specifically, we require <i>C</i> to ignore information about
the intensity of voters' preferences, <br>
so "mild" preferences will be treated the same as "intense"
preferences. In other words, <br>
two ballots that rank the alternatives in the same order must
be treated the same. The <br>
justification for this is that, if intensity information were
not ignored, it would create a <br>
strong incentive for each voter to exaggerate her intensities
by dividing the alternatives <br>
into two groups and voting the maximum possible intensity
between the two groups <br>
(and indifference within each group). To see this, suppose
pre-election polls indicate <br>
the two likely front-running candidates are <i>x</i> and <i>y</i>.
Then each voter who prefers <i>x</i> over <i>y</i> <br>
has an incentive to report the maximum possible intensity for
<i>x</i> over <i>y</i>, to avoid partially <br>
wasting the power of her vote. Similarly, those who prefer <i>y</i>
over <i>x</i> have an incentive to <br>
report the maximum possible intensity for <i>y</i> over <i>x</i>.
If the voters who prefer <i>x</i> over <i>y</i> <br>
believe those who prefer <i>y</i> over <i>x</i> will
vote the maximum intensity for <i>y</i> over <i>x</i>, they
would <br>
be foolish not to vote the maximum intensity for <i>x</i> over
<i>y</i>, etc. While doing so, it would <br>
be most effective for those voting <i>x</i> over <i>y</i> to
also cast the maximum possible vote for <br>
candidates preferred over <i>x</i> (in other words,
indifference between them and <i>x</i>, since <i>x</i> <br>
already is getting their maximal vote) and the minimal
possible vote for candidates less <br>
preferred than <i>y</i> (indifference between them and <i>y</i>),
etc. This may not be obvious at first, <br>
but we presume most voters would quickly learn the strategy
since it is so easy. The result <br>
would be that voters would express much less information in
their votes than if the choice <br>
function ignores all intensity information. Thus, we have our
next criterion: </p>
<blockquote>
<p style="margin-top: -1px;"><i>Ordinality</i>: The choice
function must neglect all "intensity" information. <br>
In other words, only "ordinal" information may affect the
choice. </p>
</blockquote>
<p>The next criterion requires that the choice function accept a
considerable amount and <br>
diversity of information from each voter about her
preferences, if she wishes to express it. <br>
Since Kenneth Arrow was analyzing whether and how voters'
preferences might be <br>
aggregated to reach a collective decision, and since there is
no <i>a priori</i> reason to expect <br>
voters' preferences to adhere to any pre-ordained pattern, it
makes sense to require the <br>
method of aggregating preferences to work no matter what the
voters' preferences may be. </p>
<blockquote>
<p style="margin-top: -1px;"><i>Universal Domain</i>: The
choice function must accept from each voter <br>
any ranking of the alternatives. </p>
</blockquote>
<p style="margin-top: -1px;">On the other hand, we are not
really limited to Arrow's framework, which was designed <br>
merely to try to aggregate voters' (sincere) preferences.
Although it is reasonable to require <br>
the voting method to work for any collection of preferences
the voters may have, it does <br>
not necessarily follow that no constraints should be placed on
the expressions voters may <br>
make when voting. For instance, the so-called Approval voting
method constrains each <br>
voter to partitioning the alternatives into two subsets, which
is equivalent to a non-strict <br>
ordering that has at most two "indifference classes." It is
not <i>a priori</i> obvious that the use <br>
of voting methods such as Approval, which constrain the
voters from completely expressing <br>
their preference orderings, are worse for society, so the <i>universal
domain</i> criterion should <br>
be considered controversial until other arguments not explored
by Arrow are examined <br>
(assuming those arguments support the conclusion that it is
better not to constrain the voters <br>
from expressing orderings). In other words, other criteria
for comparing voting methods, <br>
in addition to Arrow's criteria, need to be evaluated. (My own
conclusion is that there are <br>
solid reasons why it is better not to constrain the voters'
expressions, but that is beyond <br>
the scope of this document.) </p>
<p>Kenneth Arrow's theorem [1951, 1963] states that, if <i>X</i>
includes at least 3 alternatives <br>
then no choice function that satisfies all of the criteria
listed above also satisfies the <br>
following criterion: </p>
<blockquote>
<p style="margin-top: -1px;"><i>Choice consistency</i>: For
all pairs of alternatives, say <i>x</i> and <i>y</i>, if
the votes <br>
are such that <i>x</i> but not <i>y</i> would be chosen
from some set of nominees that <br>
includes both, then <i>y</i> must not be chosen from any
set of nominees that <br>
includes both. (The literature usually calls this <i>rationality</i>,
but I prefer <br>
the less loaded term <i>choice consistency</i>.) </p>
</blockquote>
<p style="margin-top: -1px;">(A <a moz-do-not-send="true"
href="http://alumnus.caltech.edu/%7Eseppley/Arrow%27s%20Impossibility%20Theorem%20for%20Social%20Choice%20Methods.htm#Proof%20of%20Arrow%27s%20theorem">
proof of Arrow's theorem</a> is provided in the appendix.)</p>
</blockquote>
<br>
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