[EM] Ranked Pairs "satisfies Independence of Irrelevant Alternatives", says Electowiki

[C.Benham]

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
>
> From Electowiki
> Jump to: navigation 
> <http://wiki.electorama.com/wiki/Independence_of_irrelevant_alternatives#mw-head>, 
> search 
> <http://wiki.electorama.com/wiki/Independence_of_irrelevant_alternatives#p-search> 
>
>
> 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
>


http://alumnus.caltech.edu/~seppley/

> 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.)
>

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