http://wiki.electorama.com/w/api.php?action=feedcontributions&user=12.73.132.31&feedformat=atomElectowiki - User contributions [en]2020-02-29T02:56:56ZUser contributionsMediaWiki 1.27.3http://wiki.electorama.com/w/index.php?title=CDTT&diff=284CDTT2005-03-23T05:26:55Z<p>12.73.132.31: </p>
<hr />
<div>The '''Condorcet doubly-augmented top tier''' or '''CDTT''' is defined by Douglas Woodall as the union of all minimal nonempty sets of candidates such that no candidate in each set has a majority-strength pairwise loss to any candidate outside of the set.<br />
<br />
Equivalently it can be defined as the set containing each candidate ''A'' who has a majority-strength beatpath to every other candidate ''B'' who has a majority-strength beatpath to ''A''. That is, a candidate ''A'' is in the CDTT unless some candidate ''B'' has a majority-strength beatpath to ''A'' while ''A'' has no such beatpath to ''B''.<br />
<br />
(The term ''doubly-augmented'' refers to Woodall's notion of the ''doubly-augmented gross score'' of one candidate against another. This score, for ''X'' against ''Y'', is defined as the number of voters ranking ''X'' above ''Y'', plus the full number of voters abstaining from this pairwise contest. Then the CDTT can be defined as the union of all minimal nonempty sets such that no candidate in each set has a doubly-augmented gross score of less than half the number of votes, against any candidate outside the set.)<br />
<br />
== Uses ==<br />
<br />
Limiting an election method's selection to the CDTT members can permit it to satisfy the [[Strong Defensive Strategy criterion]] (or [[Minimal Defense]]) and [[Mutual majority criterion|Majority]], while coming close to satisfying the [[Later-no-harm criterion]]. Specifically, the CDTT completely satisfies [[Later-no-harm criterion|Later-no-harm]] in the three-candidate case, and failures can only occur in the general case when there are majority-strength cycles.<br />
<br />
In order to maximize Later-no-harm compliance, the CDTT should be paired with a method that itself fully satisfies Later-no-harm. In order to ensure that [[Monotonicity criterion|Mono-raise]] is not failed, the paired method should be used to generate a ranking of the candidates which is not influenced by which candidates make it into the CDTT. Then the CDTT member who appears first in this ranking is elected.<br />
<br />
Some methods which can be paired in this way with the CDTT:<br />
*'''[[Random Ballot]]''': This can be very indecisive, but it is conceptually simple, and it satisfies [[Monotonicity criterion|Mono-raise]] and Clone Independence.<br />
*'''[[Plurality voting|First-Preference Plurality]]''': This is decisive, simple, and [[Monotonicity criterion|monotone]], but fails Clone Independence.<br />
*'''[[Instant-runoff voting|Instant Runoff Voting]]''': This is more complicated. It satisfies Clone Independence but not [[Monotonicity criterion|monotonicity]].<br />
*'''[[Descending Solid Coalitions]]''': This is also somewhat complicated, but it's the only non-random option which satisfies Clone Independence and [[Monotonicity criterion|Mono-raise]].<br />
*'''[[Minmax|MinMax (Pairwise Opposition)]]''': This has the advantage that it is calculated based on the pairwise matrix, just as the CDTT itself is. However, it is somewhat indecisive and fails Clone Independence. It satisfies [[Monotonicity criterion|Mono-raise]].<br />
<br />
Regardless of the method paired with the CDTT, it should be noted that the combined method necessarily fails the [[Plurality criterion]] and [[Condorcet criterion]].</div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Later-no-harm_criterion&diff=302Later-no-harm criterion2005-03-23T05:18:45Z<p>12.73.132.31: </p>
<hr />
<div><h4 class=left>Statement of Criterion</h4><br />
<br />
<p><em>Adding a preference to a ballot must not decrease the probability of election of any candidate ranked above the new preference.</em></p><br />
<br />
The reasoning behind this criterion is that the voter should feel free to vote his complete ranking of the candidates, without fear that he is "giving away" information about his lower choices that the method may use against him.<br />
<br />
<h4 class=left>Complying Methods</h4><br />
<br />
<p>Later-no-harm is satisfied by [[Plurality voting|First-Preference Plurality]], [[IRV|Instant Runoff Voting]], [[Minmax|Minmax(pairwise opposition)]], Douglas Woodall's [[Descending Solid Coalitions]] method, and [[Random Ballot]]. It is failed by virtually everything else.</p></div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Plurality_criterion&diff=300Plurality criterion2005-03-23T05:14:44Z<p>12.73.132.31: </p>
<hr />
<div><h4 class=left>Statement of Criterion</h4><br />
<br />
<p><em>If the number of ballots ranking ''A'' as the first preference is greater than the number of ballots on which another candidate ''B'' is given any preference, then ''A''<nowiki>'</nowiki>s probability of election must be greater than ''B''<nowiki>'</nowiki>s.</em></p><br />
<br />
The reasoning behind this criterion is that, if A has more first preferences than B has any kind of preferences, it's intuitively implausible that there could be a good reason to elect B instead of A.<br />
<br />
<h4 class=left>Complying Methods</h4><br />
<br />
<p>[[Plurality voting|First-Preference Plurality]], [[Approval voting]], [[IRV]], and many [[Condorcet method|Condorcet methods]] (using winning votes as defeat strength) satisfy the Plurality criterion. [[Condorcet method|Condorcet methods]] using margins as the measure of defeat strength fail it, as do [[Raynaud]] regardless of the measure of defeat strength, and also [[Minmax|Minmax(pairwise opposition)]].</p></div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Raynaud&diff=275Raynaud2005-03-23T05:06:10Z<p>12.73.132.31: </p>
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<div>'''Raynaud''' or '''Pairwise-Elimination''' is a [[Condorcet criterion|Condorcet method]] in which the loser of the strongest pairwise defeat is repeatedly eliminated until only one candidate remains. Usually, defeat strength is measured as the absolute number of votes cast for the winning side.<br />
<br />
In contrast to most other Condorcet methods using winning votes as the measure of defeat strength, Raynaud fails the [[Plurality criterion]]. It also fails the [[Monotonicity criterion]].</div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Participation_criterion&diff=5150Participation criterion2005-03-23T05:01:36Z<p>12.73.132.31: </p>
<hr />
<div><h4 class=left>Statement of Criterion</h4><br />
<br />
<p><em>Adding one or more ballots that vote X over Y should never change<br />
the winner from X to Y.</em></p><br />
<br />
<h4 class=left>Complying Methods</h4><br />
<br />
<p>[[Plurality voting]], [[Approval voting]], [[Cardinal Ratings]], [[Borda count]], and Woodall's [[Descending Acquiescing Coalitions|DAC]] and [[Descending Solid Coalitions|DSC]] methods all pass the Participation Criterion. [[Condorcet method | Condorcet methods]], [[Majority Choice Approval]], and [[IRV]] fail.</p><br />
<br />
''Some parts of this article are derived with permission from text at http://electionmethods.org''<br />
<br />
== See Also ==<br />
<br />
*[[Voting system]]<br />
*[[Monotonicity criterion]]<br />
*[[Condorcet Criterion]]<br />
*[[Generalized Condorcet criterion]]<br />
*[[Strategy-Free criterion]]<br />
*[[Generalized Strategy-Free criterion]]<br />
*[[Strong Defensive Strategy criterion]]<br />
*[[Weak Defensive Strategy criterion]]<br />
*[[Favorite Betrayal criterion]]<br />
*[[Summability criterion]]<br />
<br />
== External Links ==<br />
<br />
* [http://electionmethods.org/ Election Methods Education and Research Group]<br />
* [http://www.mcdougall.org.uk/VM/ISSUE6/P4.HTM Woodall's DAC method]<br />
<br />
[[Category:Voting system criteria]]<br />
<br />
{{fromwikipedia}}</div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Descending_Solid_Coalitions&diff=283Descending Solid Coalitions2005-03-23T04:57:34Z<p>12.73.132.31: /* Procedure */</p>
<hr />
<div>'''Descending Solid Coalitions''' (or '''DSC''') is a [[voting system]] devised by Douglas Woodall for use with ranked ballots.<br />
<br />
== Procedure ==<br />
<br />
Every possible set of candidates is given a score equal to the number of voters who are ''solidly committed'' to the candidates in that set. A voter is solidly committed to a set of candidates if he ranks every candidate in this set strictly above every candidate not in the set.<br />
<br />
Then the sets are considered in turn, from those with the greatest score to those with the least. When a set is considered, every candidate not in the set becomes ineligible to win, unless this would cause all candidates to be ineligible, in which case that set is ignored.<br />
<br />
When only one candidate is still eligible to win, that candidate is elected.<br />
<br />
A variation of this method is [[Descending Acquiescing Coalitions]].<br />
<br />
== Properties ==<br />
<br />
DSC satisfies the [[Plurality criterion]], the [[Mutual majority criterion|Majority criterion]], [[Monotonicity criterion|Mono-raise]], [[Mono-add-top criterion|Mono-add-top]], the [[Participation criterion]], [[Later-no-harm criterion|Later-no-harm]], and Clone Independence.<br />
<br />
DSC fails the [[Condorcet criterion]] and [[Smith set|Smith criterion]].<br />
<br />
DSC can be considered a [[Plurality voting|First-Preference Plurality]] variant that satisfies Clone Independence.<br />
<br />
===Example===<br />
{{Tenn_voting_example}}<br />
<br />
The sets have the following strengths:<br />
100 {M,N,C,K},<br />
58 {N,C,K},<br />
42 {M,N,C},<br />
42 {M,N},<br />
42 {M},<br />
32 {C,K},<br />
26 {N,C},<br />
26 {N},<br />
17 {K},<br />
15 {C}.<br />
<br />
{N,C,K} is the strongest set that excludes a candidate. Memphis becomes ineligible.<br />
<br />
No matter in which order we consider the sets with 42% of the voters solidly committed to them, we will arrive at the same result, which is that Nashville will be the only candidate remaining. So Nashville is the winner.<br />
<br />
Notice that more than half of the votes held Memphis to be the worst alternative, yet the Memphis supporters' votes were still useful in securing their second choice, Nashville. If the Memphis voters had not listed any preferences after Memphis, the winner would have been Chattanooga.</div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Monotonicity_criterion&diff=320Monotonicity criterion2005-03-23T04:56:38Z<p>12.73.132.31: </p>
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<div>A [[voting system]] is '''monotonic''' if it satisfies the ''monotonicity criterion'':<br />
<br />
:''If an alternative X loses, and the ballots are changed only by placing X in lower positions, without changing the relative position of other candidates, then X must still lose.''<br />
<br />
This criterion is also called '''Mono-raise'''.<br />
<br />
A looser way of phrasing this is that in a non-monotonic system, voting for a candidate can cause that candidate to lose. Systems which fail the monotonicity criterion suffer a form of [[tactical voting]] where voters might try to elect their candidate by voting against that candidate.<br />
<br />
[[Plurality voting]], [[Majority Choice Approval]], [[Borda count]], [[Cloneproof Schwartz Sequential Dropping]], [[Maximize Affirmed Majorities]], and [[Descending Solid Coalitions]] are monotonic, while [[Coombs' method]] and [[Instant-runoff voting]] are not. [[Approval voting]] is monotonic, using a slightly different definition, because it is not a preferential system: you can never help a candidate by not voting for them.<br />
<br />
''Some parts of this article are derived from text at http://condorcet.org/emr/criteria.shtml''<br />
{{fromwikipedia}}</div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Strong_Defensive_Strategy_criterion&diff=277Strong Defensive Strategy criterion2005-03-23T04:53:31Z<p>12.73.132.31: </p>
<hr />
<div><h4 class=left>Definitions</h4><br />
<br />
<p>A voter votes X equal to Y if the voter doesn't vote X over Y, and<br />
doesn't vote Y over X, but votes X over someone, and votes Y over<br />
someone.</p><br />
<br />
<p>A sincere vote is one with no falsified preferences or preferences<br />
left unspecified when the election method allows them to be specified<br />
(in addition to the preferences already specified).</p><br />
<br />
<p>One candidate is preferred over another candidate if, in a one-on-one<br />
competition, more voters prefer the first candidate than prefer the<br />
other candidate.</p><br />
<br />
<h4 class=left>Statement of Criterion</h4><br />
<br />
<p><em>If a majority prefers one particular candidate to another, then<br />
they should have a way of voting that will ensure that the other cannot<br />
win, without any member of that majority reversing a preference for one<br />
candidate over another or falsely voting two candidates equal.</em></p><br />
<br />
<h4 class=left>Complying Methods</h4><br />
<br />
<p>[[Cloneproof Schwartz Sequential Dropping]] (with winning votes as the measure of defeat strength), [[Maximize Affirmed Majorities]], and [[CDTT]] methods comply with the Strong Defensive Strategy Criterion, while [[Approval voting]], [[Cardinal Ratings]], [[Borda count]], [[Plurality voting]], [[Instant-Runoff Voting]], and [[Descending Solid Coalitions]] do not comply.</p><br />
<br />
<h4 class=left>Commentary</h4><br />
<br />
<p>Compliance with SDSC means that a majority never needs any more than<br />
[[truncation strategy]] to defeat a particular candidate, even when<br />
countering [[offensive order reversal]] by that candidate's voters.<br />
Offensive order reversal is the only strategy that can create the need<br />
for defensive strategy in [[Cloneproof Schwartz Sequential Dropping]].</p><br />
<br />
''Some parts of this article are derived with permission from text at http://electionmethods.org''<br />
<br />
== External Links ==<br />
<br />
* [http://electionmethods.org/ Election Methods Education and Research Group]<br />
<br />
[[Category:Voting system criteria]]<br />
<br />
{{fromwikipedia}}</div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Talk:Strong_Defensive_Strategy_criterion&diff=5248Talk:Strong Defensive Strategy criterion2005-03-23T04:50:32Z<p>12.73.132.31: </p>
<hr />
<div>I edited this, since MCA doesn't satisfy SDSC. Limited slots force you to rank candidates equally. -Kevin Venzke</div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Summability_criterion&diff=317Summability criterion2005-03-23T04:47:28Z<p>12.73.132.31: /* Summable Methods */</p>
<hr />
<div>Each vote should map onto a summable array, where the summation operation is associative and commutative, and the winner should be determined from the array sum for all votes cast. An election method is ''k-summable'' (or "passes the k-Summability Criterion") if there exists a constant c such that in any election with n candidates, the required size of the "array" is at most c*n^k. An election method is "non-summable" if there is no k for which it is k-summable.<br />
<br />
== Summable Methods ==<br />
<br />
{|align=center|border=1<br />
|+ Methods and their summability levels.<br />
! k=1 !! k=2 !! k=3 !! non-summable<br />
|-<br />
|<br />
*[[Borda count]]<br />
*[[Plurality voting]]<br />
*[[Cardinal Ratings]]<br />
*[[Approval voting]]<br />
||<br />
*most [[Condorcet method]]s, <br />
*[[Bucklin voting|Bucklin]]<br />
*[[Descending Solid Coalitions]]<br />
||<br />
*[[Iterative Ranked Approval Voting]]<br />
||<br />
*[[Instant-Runoff Voting]] <br />
|}<br />
<br />
== Examples ==<br />
<br />
In [[plurality voting]], each vote is equivalent to a one-dimensional array with a 1 in the element for the selected candidate, and a 0 for each of the other candidates. The sum of the arrays for all the votes cast is simply a list of vote counts for each candidate. [[Approval voting]] is the same as plurality voting except that more than one candidate can get a 1 in the array for each vote. Each of the selected or "approved" candidates gets a 1, and the others get a 0.<br />
<br />
In [[Cloneproof Schwartz Sequential Dropping]], each vote is equivalent to a two-dimensional array referred to as a pairwise matrix. If candidate A is ranked above candidate B, then the element in the A row and B column gets a 1, while the element in the B row and A column gets a 0. The pairwise matrices for all the votes are summed, and the winner is determined from the resulting pairwise matrix sum.<br />
<br />
IRV does not comply with the summability criterion. In the IRV system, a count can be maintained of identical votes, but votes do not correspond to a summable array. The total possible number of unique votes grows factorially with the number of candidates. <br />
<br />
== Importance of summability ==<br />
<br />
The summability criterion addresses implementation logistics. Election methods with lower summability numbers are substantially easier to implement with integrity than those that do not.<br />
<br />
Suppose, for example, that the number of candidates is ten. Under first-order summable methods like [[plurality voting|plurality]] or [[Approval voting]], the votes at any level (precinct, ward, county, etc.) can be compressed into a list of ten numbers. For [[Cloneproof Schwartz Sequential Dropping]], a 10x10 matrix is needed. In an [[IRV]] system, however, the number of possible unique votes is over ten factorial--a very large number. The larger the number of candidates, the more error-prone and less practical it becomes to maintain counts of each possible unique vote. Under IRV, therefore, every individual vote (rank list) must be available at a central location to determine the winner. In a major public election, that could be millions or even tens of millions of votes. The votes cannot be compressed by summing as in other election methods because votes may need to be transferred according to which candidates are eliminated in each round.<br />
<br />
IRV therefore requires far more data transfer and storage than the other methods. Modern networking and computer technology can handle it, but that is beside the point. The biggest challenge in using computers for public elections will always be security and integrity. If many thousands of times more data needs to be transferred and stored, verification becomes more difficult and the potential for fraudulent tampering becomes substantially greater.<br />
<br />
To illustrate this point, consider the verification of a vote tally for a national office. In a plurality election, each precinct verifies its vote count. This can be an open process where The counts for each precinct in a county can then be added to determine the county totals, and anyone with a calculator or computer can verify that the totals are correct. The same process is then repeated at the state level and the national level. If the votes are verified at the lowest (precinct) level, the numbers are available to anyone for independent verification, and election officials could never get away with "fudging" the numbers. <br />
<br />
=== Recounts ===<br />
<br />
In first-order summable election systems, adding new ballots to the count (say, ballots that were found after the initial count, or late absentee ballots, or ballots that were initially ruled invalid) is as simple as "summing" the original result with the newly-found ballots. Under non-summable systems, though, finding new ballots means all ballots must be recounted. This is not a big problem for computer recounts, but manual recounts can be extremely time-consuming and expensive.<br />
<br />
[[Category:Voting system criteria]]<br />
<br />
''Some parts of this article are derived with permission from text at http://electionmethods.org''<br />
{{fromwikipedia}}</div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Mutual_majority_criterion&diff=278Mutual majority criterion2005-03-23T04:43:31Z<p>12.73.132.31: </p>
<hr />
<div>The '''Mutual majority criterion''' is a criterion for evaluating [[voting system]]s. It applies to [[ranked ballot]] elections. It can be stated as follows:<br />
<br />
:''If there is a majority of voters for which it is true that they all rank a set of candidates above all others, then one of these candidates must win.''<br />
<br />
This is often simply called the '''Majority criterion'''.<br />
<br />
Systems which pass:<br />
<br />
[[Borda-Elimination]], [[Bucklin]], [[Coombs]], [[IRV]], [[Kemeny-Young]], [[Nanson (original)]], [[Raynaud|Pairwise-Elimination]], [[Ranked Pairs]], [[Schulze]], [[Smith//Minmax]], [[Descending Solid Coalitions]]<br />
<br />
Systems which fail: <br />
<br />
[[Black]], [[Borda]], [[Dodgson]], [[Minmax]], [[Sum of Defeats]]<br />
<br />
[[Category:Voting system criteria]]</div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Descending_Acquiescing_Coalitions&diff=2656Descending Acquiescing Coalitions2005-03-23T04:35:14Z<p>12.73.132.31: </p>
<hr />
<div>'''Descending Acquiescing Coalitions''' or '''DAC''' is a [[voting system]] devised by Douglas Woodall for ranked ballots. It is equivalent to [[Descending Solid Coalitions]], except that sets are scored not by the number of voters solidly committed to them, but by the number of voters ''acquiescing'' to them. A voter "acquiesces" to a set of candidates if he does not strictly prefer any candidate outside of the set to any candidate within the set.<br />
<br />
Unlike DSC, DAC does not satisfy the [[Later-no-harm criterion]].<br />
<br />
When no voter uses equal rankings or truncation, then DSC and DAC give the same results.</div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Condorcet_Criterion&diff=496Condorcet Criterion2005-03-23T04:26:54Z<p>12.73.132.31: /* Complying methods */</p>
<hr />
<div>The '''Condorcet candidate''' or '''Condorcet winner''' of an [[election]] is the candidate who, when compared in turn with each of the other candidates, is preferred over the other candidate. Mainly because of Condorcet's [[voting paradox]], a Condorcet winner will not always exist in a given set of votes.<br />
<br />
The '''Condorcet criterion''' for a [[voting system]] is that it chooses the Condorcet winner when one exists. Any method conforming to the Condorcet criterion is known as a [[Condorcet method]].<br />
<br />
==Complying methods==<br />
<br />
Black, Smith/IRV, [[Copeland's method|Copeland]], [[Minmax]], Smith/Minmax, [[ranked pairs]] and variations ([[maximize affirmed majorities]], [[maximum majority voting]]), [[Cloneproof Schwartz Sequential Dropping|Schulze]] and variations ([[Cloneproof Schwartz Sequential Dropping|Schwartz sequential dropping]], [[Cloneproof Schwartz Sequential Dropping|cloneproof Schwartz sequential dropping]]) comply with the Condorcet criterion.<br />
<br />
[[Approval voting]], [[Range voting]], [[Borda count]], [[plurality voting]], and [[instant-runoff voting]] do not.<br />
<br />
==Commentary==<br />
<br />
Non-ranking methods such as [[plurality voting|plurality]] and [[approval voting|approval]] cannot comply with the Condorcet criterion because they do not allow each voter to fully specify their preferences. But instant-runoff voting allows each voter to rank the candidates, yet it still does not comply. A simple example will prove that IRV fails to comply with the Condorcet criterion.<br />
<br />
Consider, for example, the following vote count of preferences with three candidates {A,B,C}:<br />
<br />
<blockquote><table><br />
<tr><td align=right>499:</td><td align=left>A,B,C</td></tr><br />
<tr><td align=right>498:</td><td align=left>C,B,A</td></tr><br />
<tr><td align=right>3:</td><td align=left>B,C,A</td></tr><br />
</table></blockquote><br />
<br />
In this case, B is preferred to A by 501 votes to 499, and B is<br />
preferred to C by 502 to 498, hence B is preferred to both A and C. So<br />
according to the Condorcet criteria, B should win. By contrast, according to the rules of IRV, B is ranked first by the fewest voters and is eliminated, and C wins with the transferred voted from B; in plurality voting A wins with the most first choices.<br />
<br />
[[Category:Voting system criteria]]<br />
<br />
{{fromwikipedia}}</div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Voting_system_criterion&diff=314Voting system criterion2005-03-23T04:25:45Z<p>12.73.132.31: </p>
<hr />
<div>A formally defined pass/fail criterion by which a [[voting system]] may be assessed.<br />
<br />
Examples for such criteria are:<br />
<br />
:[[Monotonicity criterion]]<br />
<br />
:[[Pareto efficiency|Pareto criterion]]<br />
<br />
:[[Condorcet Criterion|Condorcet criterion]]<br />
<br />
:[[Smith set|Smith criterion]] ([[Aka_(initialism)|aka]] [[Generalized Condorcet criterion]])<br />
<br />
:[[Independence of irrelevant alternatives|independence from irrelevant alternatives]]<br />
<br />
:[[Independence of irrelevant alternatives|local independence from irrelevant alternatives]]<br />
<br />
:[[Schwartz set|Schwartz criterion]]<br />
<br />
:[[Plurality criterion]]<br />
<br />
:[[Later-no-harm criterion]]<br />
<br />
:[[Strategy-Free criterion]]<br />
<br />
:[[Generalized Strategy-Free criterion]]<br />
<br />
:[[Strong Defensive Strategy criterion]]<br />
<br />
:[[Weak Defensive Strategy criterion]]<br />
<br />
:[[Summability criterion]]<br />
<br />
:[[Strategic nomination|Independence of clones]]<br />
<br />
:[[Participation criterion]]<br />
<br />
:[[Consistency|Consistency criterion]]<br />
<br />
:[[Tactical voting|invulnerability to compromising]]<br />
<br />
:[[Tactical voting|invulnerability to burying]]<br />
<br />
:[[Favorite Betrayal criterion]]<br />
<br />
[[Category:Voting_system_criteria|Voting system criteria]]<br />
{{stub}}<br />
<br />
{{fromwikipedia}}</div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Random_Ballot&diff=274Random Ballot2005-03-23T04:23:38Z<p>12.73.132.31: /* Example */</p>
<hr />
<div>'''Random Ballot''', also known as '''Random Dictatorship''', is a [[voting system]] in which the first preference candidate of a ballot drawn at random is elected.<br />
<br />
When the drawn ballot is not decisive, then additional ballots are drawn and used only to resolve the indecision of previously drawn ballots.<br />
<br />
== Properties ==<br />
<br />
Random Ballot satisfies the [[Plurality criterion]], [[Monotonicity criterion]], [[Participation criterion]], [[Later-no-harm criterion]], Clone Independence, [[Favorite Betrayal criterion]], and [[Pareto efficiency|Pareto criterion]].<br />
<br />
However, Random Ballot fails the [[Mutual majority criterion|Majority criterion]], [[Condorcet criterion]], [[Smith set|Smith criterion]], and [[Strong Defensive Strategy criterion]].<br />
<br />
===Example===<br />
{{Tenn_voting_example}}<br />
<br />
Memphis wins with 42% probability, Nashville with 26%, Chattanooga 15%, and Knoxville 17%. If the Knoxville and Chattanooga voters has instead ranked Knoxville and Chattanooga equally, then Knoxville would win with 0% probability, since it would be impossible to draw a ballot which prefers Knoxville to Chattanooga.</div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Random_Ballot&diff=257Random Ballot2005-03-23T04:22:48Z<p>12.73.132.31: </p>
<hr />
<div>'''Random Ballot''', also known as '''Random Dictatorship''', is a [[voting system]] in which the first preference candidate of a ballot drawn at random is elected.<br />
<br />
When the drawn ballot is not decisive, then additional ballots are drawn and used only to resolve the indecision of previously drawn ballots.<br />
<br />
== Properties ==<br />
<br />
Random Ballot satisfies the [[Plurality criterion]], [[Monotonicity criterion]], [[Participation criterion]], [[Later-no-harm criterion]], Clone Independence, [[Favorite Betrayal criterion]], and [[Pareto efficiency|Pareto criterion]].<br />
<br />
However, Random Ballot fails the [[Mutual majority criterion|Majority criterion]], [[Condorcet criterion]], [[Smith set|Smith criterion]], and [[Strong Defensive Strategy criterion]].<br />
<br />
===Example===<br />
{{Tenn_voting_example}}<br />
<br />
Memphis wins with 42% probability, Nashville with 26%, Chattanooga 15%, and Knoxville 17%. If the Knoxville and Chattanooga voters has instead ranked Knoxville and Chattanooga equally, and one of their ballots had been drawn, then Knoxville would win with 0% probability, since it would be impossible to draw a ballot which prefers Knoxville to Chattanooga.</div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Majority&diff=279Majority2005-03-23T03:51:34Z<p>12.73.132.31: /* Criteria 1,2, and 3 - Intermediate Majority Rule Methods */</p>
<hr />
<div>A '''majority''' means, literally, "more than half". Compare this with [[plurality]], which means "the most of the group". When applied to specific situations, majority can take on different meanings, depending on how you apply it:<br />
<br />
*'''relative majority''' usually means "plurality" <br />
*'''simple majority''' means "more than half of cast votes"<br />
*'''absolute majority''' means "more than half of eligible voters"<br />
*a '''supermajority''' is a fraction of the voters between half and all (e.g. 2/3)<br />
*'''consensus''' usually means complete agreement or "all voters"<br />
<br />
== Majority rule/Majority winner - Four Critera==<br />
<br />
Many methods claim to elect the "majority winner" or work by "majority rule" (See, for example, the [[Center for Voting and Democracy|CVD]]'s talking points re: IRV: [http://www.fairvote.org/irv/talking.htm]). However, [[Condorcet's paradox]] raises an issue: with some groups of voters, no matter which candidate wins, ''some'' majority of the voters will prefer a different candidate. Below is a list of criterion, in ascending order of strictness, which could be used to rank the relative strengths of a "majority."<br />
<br />
<div class="blist"> <br />
*Criterion 1: If a majority of the electorate coordinates their efforts, they can assure that a given candidate is elected, or that another given candidate is not elected. <br />
*Criterion 2: [[Mutual majority criterion]]<br />
*Criterion 3: [[Condorcet criterion]]<br />
*Criterion 4: Minimal dominant set ([[Smith set|Smith]], GeTChA) efficiency </div><br />
<br />
In pseudo-majority methods (like plurality and range voting), a given majority of the electorate '''can''' coordinate their intentions and decide the winner, but this merely postpones the question of how they do this. The stronger majority methods not only enable firmly coordinated majorities to assert themselves, but they allow un-coordinated majorities to '''reveal''' themselves, without any need for prior coordination. Voting methods that facilitate this process of revelation are considered superior to those that do not.<br />
<br />
The remaining three categories allow mutual majorities to reveal themselves (in the absence of a self-defeating strategy by supporters of this majority). Strong majority rule methods not only reveal mutual majorities, but they reveal [[minimal dominant set]]s and Condorcet winners (in the absence of a severe [[burying strategy]]). This is considered especially valuable because it means revealing possible compromises on divisive issues, thus avoiding a lot of political polarization and strife.<br />
<br />
=== Criterion 1 only - Pseudo-Majority Rule Methods===<br />
<br />
Methods which pass criterion 1 only include [[First-past-the-post electoral system|Plurality]], [[approval voting|Approval]], [[Cardinal Ratings]], and the [[Borda count]]. Although it is always '''possible''' in these systems for a coordinated majority to elect their preferred candidate, coordination may be difficult. For example, take an electorate with preferences as follows:<br />
<div class="votesABC"><br />
:31 A > B > C <br />
:29 B > A > C<br />
:20 C > B > A<br />
:20 C > A > B</div><br />
<br />
In a plurality election, a clear majority (60-40) prefer both A and B to C. But unless A and B voters know whether to vote for A or whether to vote for B, C may win a plurality of votes. In addition, voters for A and B voters may play a game of "chicken", refusing to vote for the other, because they believe their candidate should win.<br />
<br />
===Criteria 1 and 2 - Weak Majority Rule Methods===<br />
<br />
[[Instant-runoff voting]] (aka IRV, Single-winner STV) passes the mutual majority criterion. In the example above, IRV enables A and B to coordinate. If all voters voted their sincere preferences, B would be eliminated first, but their votes would transfer to A, resulting in a majority for A.<br />
<br />
However, IRV doesn't pass the [[Condorcet criterion]]. In an election with preferences as follows:<br />
<div class="votesABC"><br />
:31 A > B > C<br />
:29 B > C > A<br />
:40 C > B > A</div><br />
<br />
Looking at this election pairwise, there are three majorities: a majority (69 to 31) prefer B to A, a majority (69-31) prefer C to A, and a majority (60-40) prefer B to C. If you were to award the title "majority winner" to any candidate, B has the fairest claim to that title, as (different) majorities of voters prefer B to each other candidate. However, in IRV, B is eliminated first and does not win.<br />
<br />
=== Criteria 1,2, and 3 - Intermediate Majority Rule Methods===<br />
<br />
Methods that pass the Condorcet criterion would always elect B, the Condorcet winner, in that election. <br />
<br />
[[Minmax|Minimax]] (aka SD, PC, etc.), [[Nanson]], [[Black]], etc.<br />
<br />
=== Criteria 1,2,3, and 4 - Strong Majority Rule Methods===<br />
<br />
[[ranked pairs]], [[beatpath]], [[river]], <br />
<br />
''Derived from an e-mail by James Green-Armytage''<br />
<br />
[[Category:Voting theory]]<br />
{{fromwikipedia}}</div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Ranked_Approval_Voting&diff=438Ranked Approval Voting2005-03-23T03:48:49Z<p>12.73.132.31: /* Procedure */</p>
<hr />
<div>'''Ranked Approval Voting''' is an election method combining a ranked ballot with an approval measure. Possibly Kevin Venzke was the first to suggest it on the election methods mailing list, in 2003. It was given the name "Ranked Approval Voting" by Russ Paielli.<br />
<br />
== Ballot Format ==<br />
<br />
The voter ranks the candidates. Approval could be indicated by a cutoff placed by the voter, or it can be implicit that the approved candidates are all those that the voter chooses to rank.<br />
<br />
== Procedure ==<br />
<br />
Ranked Approval Voting is a '''[[Condorcet method]]''', which means it always elects a "Condorcet winner" if one exists. A Condorcet winner is a candidate whom more voters rank above ''Y'' than vice versa, given any other candidate ''Y''.<br />
<br />
If a Condorcet winner doesn't initially exist, then the candidate with the least approval is eliminated such that his pairwise contests are no longer considered. These eliminations continue until a Condorcet winner is created; that is, until some non-eliminated candidate has pairwise wins over every other non-eliminated candidate. Then this candidate is elected.<br />
<br />
== Advantages ==<br />
<br />
Ranked Approval Voting inherently satisfies the [[Smith set|Smith criterion]], without requiring an explicit step to reduce to the Smith set.</div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Condorcet_method&diff=286Condorcet method2005-03-23T03:42:02Z<p>12.73.132.31: /* Different ambiguity resolution methods */</p>
<hr />
<div>Any election method conforming to the [[Condorcet criterion]] is known as a '''Condorcet method'''. The name comes from the 18th century mathematician and philosopher [[Marquis de Condorcet]], although the method was previously described by [[Ramon Llull]] in the 13th century. <br />
<br />
'''Condorcet''' is sometimes used to indicate the family of Condorcet methods as a whole.<br />
<br />
=== Casting ballots ===<br />
<br />
Each voter fills out a [[preferential voting|ranked ballot]]. The voter can include less than all candidates under consideration. Usually when a candidate ''is not listed'' on the voter's ballot they are considered less preferred than listed candidates, and ranked accordingly. However, some variations allow a "no opinion" default option where no for- or against- preference is counted for that candidate. Write-ins are possible, but are somewhat more difficult to implement for automatic counting than in other election methods. This is a counting issue, but results in the frequent omission of the write-in option in ballot software.<br />
<br />
=== Counting ballots ===<br />
<br />
Ballots are counted by considering all possible sets of two-candidate elections from all available candidates. That is, each candidate is considered against each and every other candidate. A candidate is considered to "win" against another on a single ballot if they are ranked higher than their opponent. All the votes for candidate Alice over candidate Bob are counted, as are all of the votes for Bob over Alice. Whoever has the most votes in each one-on-one election wins.<br />
<br />
If a candidate is preferred over all other candidates, that candidate is the [[Condorcet Criterion|Condorcet candidate]]. However, a Condorcet candidate may not exist, due to a fundamental [[Voting paradox|paradox]]: It is possible for the electorate to prefer A over B, B over C, and C over A simultaneously. This is called a circular tie, and it must be resolved by some other mechanism.<br />
<br />
==== Counting with matrices ====<br />
<br />
A frequent implementation of this method will illustrate the basic counting method. Consider an election between A, B, and C, and a ballot (B, C, A, D). That is, a ballot ranking B first, C second, A third, and D forth. This can be represented as a matrix, where the row is the runner under consideration, and the column is the opponent. The cell at (runner,opponent) has a one if runner is preferred, and a zero if not.<br />
<br />
{| border="1"<br />
! !! A !! B !! C !! D<br />
|-<br />
! A || &mdash; || 0 || 0 || 1<br />
|-<br />
! B || 1 || &mdash; || 1 || 1<br />
|-<br />
! C || 1 || 0 || &mdash; || 1<br />
|-<br />
! D || 0 || 0 || 0 || &mdash;<br />
|}<br />
<br />
Cells marked "&mdash;" are logically zero, but are blank for clarity&mdash;they are not considered, as a candidate can not be defeated by himself. This binary matrix is inversely symmetric: (runner,opponent) is &not;(opponent,runner). The utility of this structure is that it may be easily added to other ballots represented the same way, to give us the number of ballots which prefer each candidate. The sum of all ballot matrixes is called the sum matrix&mdash;it is not symmetric.<br />
<br />
When the sum matrix is found, the contest between each candidate is considered. The number of votes for runner over opponent (runner,opponent) is compared the number of votes for opponent over runner (opponent,runner). The one-on-one winner has the most votes. If one candidate wins against all other candidates, that candidate wins the election.<br />
<br />
The sum of all ballot matrices, the '''Condorcet pairwise matrix''', is the primary piece of data used to resolve circular ties (also called circular ambiguities).<br />
<br />
=== Key terms in ambiguity resolution ===<br />
<br />
Handling cases where there is not a single Condorcet winner is called ambiguity resolution in this article, though other phrases such as "cyclic ambiguity resolution" and "Condorcet completion" are used as well.<br />
<br />
The following are key terms when discussing ambiguity resolution methods:<br />
* '''[[Smith set]]''': the smallest set of candidates in a particular election who, when paired off in pairwise elections, can beat all other candidates outside the set.<br />
* '''[[Schwartz set]]''': the union of all possible sets of candidates such that for every set:<br />
*# every candidate inside the set is pairwise unbeatable by any other candidate outside the set, i.e., ties are allowed<br />
*# no proper (smaller) subset of the set fulfills the first property <br />
* '''Cloneproof''': a method that is immune to the presence of '''clones''' (candidates which are essentially identical to each other). In some voting methods, a party can increase its odds of selection if it provides a large number of "identical" options. A cloneproof voting method prevents this attack. See [[strategic nomination]].<br />
<br />
=== Different ambiguity resolution methods ===<br />
<br />
There are a countless number of "Condorcet methods" possible that resolve such ambiguities. The fact that Marquis de Condorcet himself already spearheaded the debate of which particular Condorcet method to promote has made the term "Condorcet's method" ambiguous.<br />
Indeed, it can be argued that the large number of different competing Condorcet methods has made the adoption of any single method extremely difficult.<br />
<br />
Examples of Condorcet methods include:<br />
* '''[[Black]]''' chooses the Condorcet winner when it exists and otherwise the [[Borda count|Borda winner]]. It is named after Duncan Black.<br />
* '''[[Smith/IRV]]''' is [[instant-runoff voting]] with the candidates restricted to the Smith set.<br />
* '''[[Copeland's method|Copeland]]''' selects the candidate that wins the most pairwise matchups. Note that if there is no Condorcet winner, Copeland will often still result in a tie.<br />
* '''[[Minmax|Minimax]]''' (also called '''Simpson''') chooses the candidate whose worst pairwise defeat is less bad than that of all other candidates.<sup>1</sup><br />
* '''Smith/Minimax''' restricts the Minimax algorithm to the Smith set.<sup>1</sup><br />
* '''[[Ranked Pairs]]''' (RP) or '''Tideman''' (named after [[Nicolaus Tideman]]) with variations such as '''[[Maximize Affirmed Majorities]]''' (MAM) and '''[[Maximum Majority Voting]]''' (MMV)<sup>1</sup><br />
* '''Schulze''' with several reformulations/variations, including '''Schwartz Sequential Dropping (SSD)''' and '''[[Cloneproof Schwartz Sequential Dropping]] (CSSD)'''<sup>1</sup><br />
* '''[[Approval-Condorcet Hybrids]]''', such as '''[[Definite Majority Choice]]''', use an [[Approval Cutoff]] to augment the Condorcet pair wise array. Many believe that such a method would make a good first-round public proposal.<br />
<br />
<sup>1</sup> There are different ways to measure the strength of each defeat in some methods. Some use the margin of defeat (the difference between votes for and votes against), while others use winning votes (the votes favoring the defeat in question).<br />
Electionmethods.org argues that there are several disadvantages of systems that use margins instead of winning votes.<br />
The website argues that using margins "destroys" some information about majorities, so that the method can no longer honor information about what majorities have determined and that consequently margin-based systems cannot support a number of desirable voting properties.<br />
<br />
Ranked Pairs and Schulze are procedurally in some sense opposite approaches:<br />
* Ranked Pairs (and variants) starts with the strongest information available and uses as much information as it can without creating ambiguity<br />
* Schulze (and variants) repeatedly removes the weakest ambiguous information until ambiguity is removed.<br />
<br />
The text below describes (variants of) these methods in more detail.<br />
<br />
=== Ranked Pairs, Maximize Affirmed Majorities (MAM), and Maximum Majority Voting (MMV) ===<br />
<br />
In the Ranked Pairs (RP) voting method, as well as the variations Maximize Affirmed Majorities (MAM)<br />
and Maximum Majority Voting (MMV), pairs of defeats are ranked (sorted)<br />
from largest majority to smallest majority.<br />
Then each pair is considered, starting with the defeat supported by the largest majority.<br />
Pairs are "affirmed" only if they do not create a cycle with the pairs already affirmed.<br />
Once completed, the affirmed pairs are followed to determine the winner.<br />
<br />
In essence, RP and its variants (such as MAM and MMV)<br />
treat each majority preference as evidence that the majority's more preferred <br />
alternative should finish over the majority's less preferred alternative, the weight of <br />
the evidence depending on the size of the majority.<br />
<br />
The difference betweeen RP and its variants is in the details of the ranking approach.<br />
Some definitions of RP use margins, while other definitions use winning votes (not margins).<br />
Both MAM and MMV are explicitly defined in terms of winning votes, not winning margins.<br />
In MAM and MMV, if two defeat pairs have the same number of votes for a victory, the defeat with<br />
the smaller defeat is ranked higher.<br />
If this still doesn't disambiguate between the two, MAM and MMV perform slightly differently.<br />
In MAM, information from a "tiebreaker" vote is used<br />
(which could be a distinguished vote such as the vote of a "speaker",<br />
but unless there is a distinguished vote, a randomly-chosen vote is used).<br />
In MMV all such conflicting matchups are ignored (though any non-conflicting matchups of that size are still included).<br />
<br />
=== Cloneproof Schwartz Sequential Dropping (CSSD) ===<br />
<br />
The "[[Cloneproof Schwartz Sequential Dropping]]" (CSSD) method resolves votes as follows:<br />
<br />
# First, determine the Schwartz set (the innermost unbeaten set). If no defeats exist among the Schwartz set, then its members are the winners (plural only in the case of a tie, which must be resolved by another method).<br />
# Otherwise, drop the weakest defeat information among the Schwartz set (i.e., where the number of votes favoring the defeat is the smallest). Determine the new Schwartz set, and repeat the procedure.<br />
<br />
In other words, this procedure repeatedly throws away the narrowest defeats, until finally the largest number of votes left over produce an unambiguous decision.<br />
<br />
The "Beatpath Winner" algorithm produces equivalent results.<br />
<br />
== Related terms ==<br />
<br />
Other terms related to the Condorcet method are:<br />
* '''Condorcet loser''': the candidate who is less preferred than every other candidate in a pair wise matchup.<br />
* '''weak Condorcet winner''': a candidate who beats or ties with every other candidate in a pair wise matchup. There can be more than one weak Condorcet winner.<br />
* '''weak Condorcet loser''': a candidate who is defeated by or ties with every other candidate in a pair wise matchup. Similarly, there can be more than one weak Condorcet loser.<br />
<br />
== An example ==<br />
<br />
Imagine an election for the capital of Tennessee, a state in the United States that is over 500 miles east-to-west, and only 110 miles north-to-south. Let's say the candidates for the capital are Memphis (on the far west end), Nashville (in the center), Chattanooga (129 miles southeast of Nashville), and Knoxville (on the far east side, 114 northeast of Chattanooga). Here's the population breakdown by metro area (surrounding county): <br />
<div style="float:right; padding:2px; text-align:center"><br />
[[Image:CondorcetTennesee.png]]</div><br />
<br />
* Memphis (Shelby County): 826,330<br />
* Nashville (Davidson County): 510,784<br />
* Chattanooga (Hamilton County): 285,536<br />
* Knoxville (Knox County): 335,749<br />
<br />
Let's say that in the vote, the voters vote based on geographic proximity. Assuming that the population distribution of the rest of Tennessee follows from those population centers, one could easily envision an election where the percentages of votes would be as follows:<br />
<br />
<table border=1><br />
<tr><br />
<td><br />
'''42% of voters (close to Memphis)'''<br><br />
1. Memphis<br><br />
2. Nashville<br><br />
3. Chattanooga<br><br />
4. Knoxville<br />
</td><br />
<td valign="top"><br />
'''26% of voters (close to Nashville)'''<br><br />
1. Nashville<br><br />
2. Chattanooga<br><br />
3. Knoxville<br><br />
4. Memphis<br />
</td><br />
<td><br />
'''15% of voters (close to Chattanooga)'''<br><br />
1. Chattanooga<br><br />
2. Knoxville<br><br />
3. Nashville<br><br />
4. Memphis<br />
</td><br />
<td><br />
'''17% of voters (close to Knoxville)'''<br><br />
1. Knoxville<br><br />
2. Chattanooga<br><br />
3. Nashville<br><br />
4. Memphis<br />
</td><br />
</tr><br />
</table><br />
<br />
The results would be tabulated as follows:<br />
<table BORDER><caption>Pairwise Election Results</caption><br />
<tr><th colspan=2><th colspan=4 bgcolor="#c0c0ff">A</tr><br />
<br />
<tr><br />
<th colspan=2><th bgcolor="#c0c0ff">Memphis<br />
<th bgcolor="#c0c0ff">Nashville<br />
<th bgcolor="#c0c0ff">Chattanooga<br />
<th bgcolor="#c0c0ff">Knoxville<br />
</tr><br />
<tr><th bgcolor="#ffc0c0" rowspan=4>B<th bgcolor="#ffc0c0">Memphis<td><td nowrap bgcolor="#e0e0ff">[A] 58% <br>[B] 42% <br><td nowrap bgcolor="#e0e0ff">[A] 58% <br>[B] 42% <br><td nowrap bgcolor="#e0e0ff">[A] 58% <br>[B] 42% <br></tr><br />
<tr><th bgcolor="#ffc0c0">Nashville<td nowrap bgcolor="#ffe0e0">[A] 42% <br>[B] 58% <br><td><td nowrap bgcolor="#ffe0e0">[A] 32% <br>[B] 68% <br><td nowrap bgcolor="#ffe0e0">[A] 32% <br>[B] 68% <br></tr><br />
<br />
<tr><th bgcolor="#ffc0c0">Chattanooga<td nowrap bgcolor="#ffe0e0">[A] 42% <br>[B] 58% <br><td nowrap bgcolor="#e0e0ff">[A] 68% <br>[B] 32% <br><td><td nowrap bgcolor="#ffe0e0">[A] 17% <br>[B] 83% <br></tr><br />
<tr><th bgcolor="#ffc0c0">Knoxville<td nowrap bgcolor="#ffe0e0">[A] 42% <br>[B] 58% <br><td nowrap bgcolor="#e0e0ff">[A] 68% <br>[B] 32% <br><td nowrap bgcolor="#e0e0ff">[A] 83% <br>[B] 17% <br><td></tr><br />
<tr><th colspan=2 bgcolor="#c0c0ff">Ranking (by repeatedly removing Condorcet winner):<br />
<br />
<td bgcolor="#ffffff">4th<br />
<td bgcolor="#ffffff">1st<br />
<td bgcolor="#ffffff">2nd<br />
<td bgcolor="#ffffff">3rd<br />
</table><br />
<br />
* [A] indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption<br />
* [B] indicates voters who preferred the candidate listed in the row caption to the candidate listed in the column caption<br />
<br />
In this election, Nashville is the Condorcet winner and thus the winner under all possible Condorcet methods. <br />
<br />
== Use of Condorcet voting ==<br />
<br />
Condorcet voting is not currently used in government elections. However, it is starting to receive support in some public organizations. Organizations which currently use some variant of the Condorcet method are:<br />
<br />
# The Debian project uses Cloneproof Schwartz Sequential Dropping.<br />
# The Software in the Public Interest project uses Cloneproof Schwartz Sequential Dropping.<br />
# The UserLinux project uses Cloneproof Schwartz Sequential Dropping.<br />
# The Free State Project uses a Condorcet method for choosing its target state<br />
# The voting procedure for the uk.* hierarchy of Usenet<br />
#[http://www.rsabey.pwp.blueyonder.co.uk/rpc/fscc/ Five-Second Crossword Competition]<br />
{{fromwikipedia}}</div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Techniques_of_method_design&diff=386Techniques of method design2005-03-23T03:12:03Z<p>12.73.132.31: /* Special sets */</p>
<hr />
<div>This is a list of techniques and concepts which are more or less useful when designing a single winner election method, with some comments.<br />
<br />
== Scores ==<br />
Scores can be interpreted as measures of "goodness" of a candidate. Their usage can thus improve the "efficiency" of a method.<br />
* '''Direct support''' = ''no. of voters ranking the candidate first.'' Retains clone-proofness when used appropriately, requires ranked ballots, is not well defined when more than one candidate is ranked top.<br />
* (Total) '''approval score''' = ''no. of voters approving of the candidate.'' Retains clone-proofness when used appropriately, requires approval information.<br />
* '''Borda score''' = ''sum of ranks the candidate gets on all ballots.'' Destroys clone-proofness and most "independency"-properties, requires ranked ballots.<br />
* '''Copeland score''' = ''no. of alternatives the candidate beats pairwise.'' Destroys clone-proofness.<br />
* Laslier's '''minimal gain score''' = ''probability that an alternative beaten by the candidate wins divided by the probability that an alternative beating the candidate wins, when choosing from the bipartisan distribution on the set of all other candidates.'' Improves the monotonicity of the bipartisan distribution, but destroys clone-proofness and is complicated to compute.<br />
<br />
== Defeats and defeat strength ==<br />
Looking at pairwise defeats and their strength can help assessing the "immunity" of a candidate against certain types of complaints. Care has to be taken so that the definition of strength doesn't destroy monotonicity.<br />
* (Pairwise) '''defeat''' <=> ''more voters expressed to prefer A over B than expressed to prefer B over A'' <br />
* '''Winning votes''' (wv) = ''no. of voters preferring the winner of the defeat to the loser''<br />
* '''Margin(s)''' = ''winning votes minus no. of voters preferring the loser.'' Gives more strategic incentives and seems more arbitrary (why not take the quotient instead?) than wv<br />
* '''Pairwise opposition''' or '''Votes against''' = Certain methods behave differently when defeats are not considered; instead the number of votes for each side of a pairwise contest against the other are counted as "opposition."<br />
* Other similar definitions of strength are possible, especially when voters can distinguish between equivalence and undecidedness<br />
* '''Majority-strength defeat''' = ''pairwise defeat which has a wv-strength of more than half the no. of voters.'' Using only such defeats can reduce incentive to truncate by reducing the likelihood that additional preferences will harm earlier ones. Voters adding a preference can create a majority-strength win, but they can't reverse the direction of one.<br />
* '''Approval-consistent defeat''' (definitive defeat) = ''pairwise defeat in which the winner is more approved than the loser.'' These are acyclic and can thus all be respected without a contradiction<br />
* '''Winning approval''' = ''approval score of the winner of the defeat.'' Using this as defeat strength leaves only one immune candidate: the least approved of those who beat all more approved ones. Similar for other scores. <br />
* '''Approval-based support''' = ''no. of voters approving of the winner but not of the loser of the defeat.'' Gives special influence to preferences which cross the aproval cutoff and thus helps diminishing certain strategies. Useful when one assumes that only these voters will support the correspondig "majority complaint".<br />
* '''Cardinal rated''' strength = ''sum of difference in the candidates' cardinal ratings on all ballots which rate the winner over the loser of the defeat.'' Helps diminishing certain strategies even better, but requires interpersonally comparable cardinal ratings.<br />
<br />
== Steps ==<br />
Many methods work in more than one, perhaps similar, perhaps different steps.<br />
* '''Iteration''': Similar steps are performed until some break criterion applies.<br />
* '''Reduction to a subset''': All members outside some special set are removed.<br />
* '''Runoff''': Special case of reduction to a subset, for example by removing worst candidate according to some score. Can be iterated.<br />
* '''Ordering''': Many methods involve the construction of an ordering of the candidates or the defeats, not to be interpreted as a social order, but only as an intermediate tool.<br />
* '''Randomization''': Using a certain amount of randomness can help avoiding some strategies, especially when it leads to each candidate being beaten by a possible winner.<br />
<br />
== Special sets ==<br />
* '''Smith set''' (top cycle, top tier) = ''smallest non-empty set of candidates each of which beats all candidates outside the set.''<br />
* '''Schwartz set''' = ''smallest non-empty set of candidates so that each candidate beating a member of the set is also in the set.''<br />
* Woodall's '''[[CDTT]]''' ("Condorcet doubly-augmented top tier") = ''union of all minimal non-empty sets such that no candidate in each set has a majority-strength loss to any candidate outside the set''. Alternatively, ''the set containing every candidate ''A'' where, for each other candidate ''B'' who has a majority-strength beatpath to ''A'', ''A'' also has a majority-strength beatpath to ''B.<br />
* '''Uncovered set''' = ''set of candidates which have defeats or beatpaths of length two to all other candidates''. Not monotonic as a set.<br />
* '''Banks set''' = ''set of candidates which are top on some maximal sub-chain of the defeat graph.'' Not monotonic as a set.<br />
* Dutta's '''Minimal covering set''' = ''smallest non-empty set such that when any other candidate is added, that candidate is covered in the resulting set.<br />
* '''Bipartisan set''' = ''set of candidates getting non-zero probability in the bipartisan distribution.'' <br />
* Forest Simmons' set '''P''' = ''set of candidates which are not approval-consistently defeated.'' This set is totally ordered by the defeats and contains the most approved candidate and a possible Condorcet winner.<br />
<br />
== Orderings == <br />
* Ordering by some score<br />
* (Uniformly) '''random order''': ''Pick some order uniformly at random.'' Destroys clone-proofness.<br />
* '''Random ballot order''': ''A is above B when A is above B on the first ballot distinguishing between them, in a random sequence of ballots.'' This is an often used tie breaking rank order (TBRO). Retains clone-proofness when used appropriately.<br />
* '''Random ballot approval order''': ''A is above B when A is above B on the first ballot on which the approval cutoff is between them, in a random sequence of ballots.'' Retains clone-proofness when used appropriately.<br />
<br />
== Randomization ==<br />
* '''Randomized start''': Pick an initial candidate by some random process, then proceed with other steps.<br />
* '''Randomized order''': Use a random process to determine the order in which the candidates are processes in some way.<br />
* '''Final randomization''': Restrict to some subset, then use a random process to choose from that set.</div>12.73.132.31http://wiki.electorama.com/w/index.php?title=CDTT&diff=268CDTT2005-03-23T03:07:27Z<p>12.73.132.31: </p>
<hr />
<div>The '''Condorcet doubly-augmented top tier''' or '''CDTT''' is defined by Douglas Woodall as the union of all minimal nonempty sets of candidates such that no candidate in each set has a majority-strength pairwise loss to any candidate outside of the set.<br />
<br />
Equivalently it can be defined as the set containing each candidate ''A'' who has a majority-strength beatpath to every other candidate ''B'' who has a majority-strength beatpath to ''A''. That is, a candidate ''A'' is in the CDTT unless some candidate ''B'' has a majority-strength beatpath to ''A'' while ''A'' has no such beatpath to ''B''.<br />
<br />
== Uses ==<br />
<br />
Limiting an election method's selection to the CDTT members can permit it to satisfy the [[Strong Defensive Strategy criterion]] (or [[Minimal Defense]]) and [[Mutual majority criterion|Majority]], while coming close to satisfying the [[Later-no-harm criterion]]. Specifically, the CDTT completely satisfies [[Later-no-harm criterion|Later-no-harm]] in the three-candidate case, and failures can only occur in the general case when there are majority-strength cycles.<br />
<br />
In order to maximize Later-no-harm compliance, the CDTT should be paired with a method that itself fully satisfies Later-no-harm. In order to ensure that [[Monotonicity criterion|Mono-raise]] is not failed, the paired method should be used to generate a ranking of the candidates which is not influenced by which candidates make it into the CDTT. Then the CDTT member who appears first in this ranking is elected.<br />
<br />
Some methods which can be paired in this way with the CDTT:<br />
*'''[[Random Ballot]]''': This can be very indecisive, but it is conceptually simple, and it satisfies [[Monotonicity criterion|Mono-raise]] and Clone Independence.<br />
*'''[[Plurality voting|First-Preference Plurality]]''': This is decisive, simple, and [[Monotonicity criterion|monotone]], but fails Clone Independence.<br />
*'''[[Instant-runoff voting|Instant Runoff Voting]]''': This is more complicated. It satisfies Clone Independence but not [[Monotonicity criterion|monotonicity]].<br />
*'''[[Descending Solid Coalitions]]''': This is also somewhat complicated, but it's the only non-random option which satisfies Clone Independence and [[Monotonicity criterion|Mono-raise]].<br />
*'''[[Minmax|MinMax (Pairwise Opposition)]]''': This has the advantage that it is calculated based on the pairwise matrix, just as the CDTT itself is. However, it is somewhat indecisive and fails Clone Independence. It satisfies [[Monotonicity criterion|Mono-raise]].<br />
<br />
Regardless of the method paired with the CDTT, it should be noted that the combined method necessarily fails the [[Plurality criterion]] and [[Condorcet criterion]].</div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Descending_Solid_Coalitions&diff=264Descending Solid Coalitions2005-03-23T02:26:49Z<p>12.73.132.31: </p>
<hr />
<div>'''Descending Solid Coalitions''' (or '''DSC''') is a [[voting system]] devised by Douglas Woodall for use with ranked ballots.<br />
<br />
== Procedure ==<br />
<br />
Every possible set of candidates is given a score equal to the number of voters who are ''solidly committed'' to the candidates in that set. A voter is solidly committed to a set of candidates if he ranks every candidate in this set strictly above every candidate not in the set.<br />
<br />
Then the sets are considered in turn, from those with the greatest score to those with the least. When a set is considered, every candidate not in the set becomes ineligible to win, unless this would cause all candidates to be ineligible, in which case that set is ignored.<br />
<br />
When only one candidate is still eligible to win, that candidate is elected.<br />
<br />
== Properties ==<br />
<br />
DSC satisfies the [[Plurality criterion]], the [[Mutual majority criterion|Majority criterion]], [[Monotonicity criterion|Mono-raise]], [[Mono-add-top criterion|Mono-add-top]], the [[Participation criterion]], [[Later-no-harm criterion|Later-no-harm]], and Clone Independence.<br />
<br />
DSC fails the [[Condorcet criterion]] and [[Smith set|Smith criterion]].<br />
<br />
DSC can be considered a [[Plurality voting|First-Preference Plurality]] variant that satisfies Clone Independence.<br />
<br />
===Example===<br />
{{Tenn_voting_example}}<br />
<br />
The sets have the following strengths:<br />
100 {M,N,C,K},<br />
58 {N,C,K},<br />
42 {M,N,C},<br />
42 {M,N},<br />
42 {M},<br />
32 {C,K},<br />
26 {N,C},<br />
26 {N},<br />
17 {K},<br />
15 {C}.<br />
<br />
{N,C,K} is the strongest set that excludes a candidate. Memphis becomes ineligible.<br />
<br />
No matter in which order we consider the sets with 42% of the voters solidly committed to them, we will arrive at the same result, which is that Nashville will be the only candidate remaining. So Nashville is the winner.<br />
<br />
Notice that more than half of the votes held Memphis to be the worst alternative, yet the Memphis supporters' votes were still useful in securing their second choice, Nashville. If the Memphis voters had not listed any preferences after Memphis, the winner would have been Chattanooga.</div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Later-no-harm_criterion&diff=267Later-no-harm criterion2005-03-23T01:41:14Z<p>12.73.132.31: </p>
<hr />
<div><h4 class=left>Statement of Criterion</h4><br />
<br />
<p><em>Adding a preference to a ballot must not decrease the probability of election of any candidate ranked above the new preference.</em></p><br />
<br />
<h4 class=left>Complying Methods</h4><br />
<br />
<p>Later-no-harm is satisfied by [[Plurality voting|First-Preference Plurality]], [[IRV|Instant Runoff Voting]], [[Minmax|Minmax(pairwise opposition)]], Douglas Woodall's [[Descending Solid Coalitions]] method, and [[Random Ballot]]. It is failed by virtually everything else.</p></div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Plurality_criterion&diff=266Plurality criterion2005-03-23T01:33:26Z<p>12.73.132.31: </p>
<hr />
<div><h4 class=left>Statement of Criterion</h4><br />
<br />
<p><em>If the number of ballots ranking ''A'' as the first preference is greater than the number of ballots on which another candidate ''B'' is given any preference, then ''A''<nowiki>'</nowiki>s probability of election must be greater than ''B''<nowiki>'</nowiki>s.</em></p><br />
<br />
<h4 class=left>Complying Methods</h4><br />
<br />
<p>[[Plurality voting|First-Preference Plurality]], [[Approval voting]], [[IRV]], and many [[Condorcet method|Condorcet methods]] (using winning votes as defeat strength) satisfy the Plurality criterion. [[Condorcet method|Condorcet methods]] using margins as the measure of defeat strength fail it, as do [[Raynaud]] regardless of the measure of defeat strength, and also [[Minmax|Minmax(pairwise opposition)]].</p></div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Participation_criterion&diff=265Participation criterion2005-03-23T01:21:37Z<p>12.73.132.31: </p>
<hr />
<div><h4 class=left>Statement of Criterion</h4><br />
<br />
<p><em>Adding one or more ballots that vote X over Y should never change<br />
the winner from X to Y.</em></p><br />
<br />
<h4 class=left>Complying Methods</h4><br />
<br />
<p>[[Plurality voting]], [[Approval voting]], [[Cardinal Ratings]], [[Borda count]], and [http://www.mcdougall.org.uk/VM/ISSUE6/P4.HTM Woodall's DAC and DSC methods] all pass the Participation Criterion. [[Condorcet method | Condorcet methods]], [[Majority Choice Approval]], and [[IRV]] fail.</p><br />
<br />
''Some parts of this article are derived with permission from text at http://electionmethods.org''<br />
<br />
== See Also ==<br />
<br />
*[[Voting system]]<br />
*[[Monotonicity criterion]]<br />
*[[Condorcet Criterion]]<br />
*[[Generalized Condorcet criterion]]<br />
*[[Strategy-Free criterion]]<br />
*[[Generalized Strategy-Free criterion]]<br />
*[[Strong Defensive Strategy criterion]]<br />
*[[Weak Defensive Strategy criterion]]<br />
*[[Favorite Betrayal criterion]]<br />
*[[Summability criterion]]<br />
<br />
== External Links ==<br />
<br />
* [http://electionmethods.org/ Election Methods Education and Research Group]<br />
<br />
[[Category:Voting system criteria]]<br />
<br />
{{fromwikipedia}}</div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Minmax&diff=270Minmax2005-03-23T01:18:43Z<p>12.73.132.31: </p>
<hr />
<div>'''Minmax''' is the name of several election methods based on electing the candidate with the lowest score, based on votes received in pairwise contests with other candidates.<br />
<br />
'''Minmax(winning votes)''' elects the candidate whose greatest pairwise loss to another candidate is the least, when the strength of a pairwise loss is measured as the number of voters who voted for the winning side.<br />
<br />
'''Minmax(margins)''' is the same, except that the strength of a pairwise loss is measured as the number of votes for the winning side ''minus'' the number of votes for the losing side.<br />
<br />
Both of these methods satisfy the [[Condorcet criterion]], and both fail the [[Smith set|Smith criterion]]. Minmax(winning votes) also satisfies the [[Plurality criterion]]. In the three-candidate case, Minmax(margins) satisfies the [[Participation criterion]].<br />
<br />
'''Minmax(pairwise opposition)''' or '''MMPO''' elects the candidate whose greatest ''opposition'' from another candidate is minimal. Pairwise wins or losses are not considered; all that matters is the number of votes for one candidate over another.<br />
<br />
Minmax(pairwise opposition) does not strictly satisfy the [[Condorcet criterion]] or [[Smith set|Smith criterion]]. It also fails the [[Plurality criterion]], and is more indecisive than the other Minmax methods. However, it satisfies the [[Later-no-harm criterion]], and in the three-candidate case, the [[Participation criterion]].</div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Talk:Ranked_Approval_Voting&diff=5241Talk:Ranked Approval Voting2005-03-23T00:57:37Z<p>12.73.132.31: </p>
<hr />
<div>I put this page here, FYI. -Kevin Venzke</div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Ranked_Approval_Voting&diff=255Ranked Approval Voting2005-03-23T00:53:17Z<p>12.73.132.31: </p>
<hr />
<div>'''Ranked Approval Voting''' is an election method combining a ranked ballot with an approval measure. Possibly Kevin Venzke was the first to suggest it on the election methods mailing list, in 2003. It was given the name "Ranked Approval Voting" by Russ Paielli.<br />
<br />
== Ballot Format ==<br />
<br />
The voter ranks the candidates. Approval could be indicated by a cutoff placed by the voter, or it can be implicit that the approved candidates are all those that the voter chooses to rank.<br />
<br />
== Procedure ==<br />
<br />
Ranked Approval Voting is a '''[[Condorcet method]]''', which means it always elects a "Condorcet winner" if one exists. A Condorcet winner is a candidate who, given any other candidate ''Y'', more voters rank this candidate above ''Y'' than vice versa.<br />
<br />
If a Condorcet winner doesn't initially exist, then the candidate with the least approval is eliminated such that his pairwise contests are no longer considered. These eliminations continue until a Condorcet winner is created; that is, until some non-eliminated candidate has pairwise wins over every other non-eliminated candidate. Then this candidate is elected.<br />
<br />
== Advantages ==<br />
<br />
Ranked Approval Voting inherently satisfies the [[Smith set|Smith criterion]], without requiring an explicit step to reduce to the Smith set.</div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Talk:Techniques_of_method_design&diff=396Talk:Techniques of method design2005-03-23T00:37:02Z<p>12.73.132.31: </p>
<hr />
<div>Hi. I made additions regarding pairwise opposition as an alternative approach to defeat strength; majority-strength defeats; and also the definition of the CDTT. -Kevin Venzke</div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Techniques_of_method_design&diff=253Techniques of method design2005-03-23T00:31:51Z<p>12.73.132.31: /* Defeats and defeat strength */</p>
<hr />
<div>This is a list of techniques and concepts which are more or less useful when designing a single winner election method, with some comments.<br />
<br />
== Scores ==<br />
Scores can be interpreted as measures of "goodness" of a candidate. Their usage can thus improve the "efficiency" of a method.<br />
* '''Direct support''' = ''no. of voters ranking the candidate first.'' Retains clone-proofness when used appropriately, requires ranked ballots, is not well defined when more than one candidate is ranked top.<br />
* (Total) '''approval score''' = ''no. of voters approving of the candidate.'' Retains clone-proofness when used appropriately, requires approval information.<br />
* '''Borda score''' = ''sum of ranks the candidate gets on all ballots.'' Destroys clone-proofness and most "independency"-properties, requires ranked ballots.<br />
* '''Copeland score''' = ''no. of alternatives the candidate beats pairwise.'' Destroys clone-proofness.<br />
* Laslier's '''minimal gain score''' = ''probability that an alternative beaten by the candidate wins divided by the probability that an alternative beating the candidate wins, when choosing from the bipartisan distribution on the set of all other candidates.'' Improves the monotonicity of the bipartisan distribution, but destroys clone-proofness and is complicated to compute.<br />
<br />
== Defeats and defeat strength ==<br />
Looking at pairwise defeats and their strength can help assessing the "immunity" of a candidate against certain types of complaints. Care has to be taken so that the definition of strength doesn't destroy monotonicity.<br />
* (Pairwise) '''defeat''' <=> ''more voters expressed to prefer A over B than expressed to prefer B over A'' <br />
* '''Winning votes''' (wv) = ''no. of voters preferring the winner of the defeat to the loser''<br />
* '''Margin(s)''' = ''winning votes minus no. of voters preferring the loser.'' Gives more strategic incentives and seems more arbitrary (why not take the quotient instead?) than wv<br />
* '''Pairwise opposition''' or '''Votes against''' = Certain methods behave differently when defeats are not considered; instead the number of votes for each side of a pairwise contest against the other are counted as "opposition."<br />
* Other similar definitions of strength are possible, especially when voters can distinguish between equivalence and undecidedness<br />
* '''Majority-strength defeat''' = ''pairwise defeat which has a wv-strength of more than half the no. of voters.'' Using only such defeats can reduce incentive to truncate by reducing the likelihood that additional preferences will harm earlier ones. Voters adding a preference can create a majority-strength win, but they can't reverse the direction of one.<br />
* '''Approval-consistent defeat''' (definitive defeat) = ''pairwise defeat in which the winner is more approved than the loser.'' These are acyclic and can thus all be respected without a contradiction<br />
* '''Winning approval''' = ''approval score of the winner of the defeat.'' Using this as defeat strength leaves only one immune candidate: the least approved of those who beat all more approved ones. Similar for other scores. <br />
* '''Approval-based support''' = ''no. of voters approving of the winner but not of the loser of the defeat.'' Gives special influence to preferences which cross the aproval cutoff and thus helps diminishing certain strategies. Useful when one assumes that only these voters will support the correspondig "majority complaint".<br />
* '''Cardinal rated''' strength = ''sum of difference in the candidates' cardinal ratings on all ballots which rate the winner over the loser of the defeat.'' Helps diminishing certain strategies even better, but requires interpersonally comparable cardinal ratings.<br />
<br />
== Steps ==<br />
Many methods work in more than one, perhaps similar, perhaps different steps.<br />
* '''Iteration''': Similar steps are performed until some break criterion applies.<br />
* '''Reduction to a subset''': All members outside some special set are removed.<br />
* '''Runoff''': Special case of reduction to a subset, for example by removing worst candidate according to some score. Can be iterated.<br />
* '''Ordering''': Many methods involve the construction of an ordering of the candidates or the defeats, not to be interpreted as a social order, but only as an intermediate tool.<br />
* '''Randomization''': Using a certain amount of randomness can help avoiding some strategies, especially when it leads to each candidate being beaten by a possible winner.<br />
<br />
== Special sets ==<br />
* '''Smith set''' (top cycle, top tier) = ''smallest non-empty set of candidates each of which beats all candidates outside the set.''<br />
* '''Schwartz set''' = ''smallest non-empty set of candidates so that each candidate beating a member of the set is also in the set.''<br />
* Woodall's '''CDTT''' ("Condorcet doubly-augmented top tier") = ''union of all minimal non-empty sets such that no candidate in each set has a majority-strength loss to any candidate outside the set''. Alternatively, ''the set containing every candidate ''A'' where, for each other candidate ''B'' who has a majority-strength beatpath to ''A'', ''A'' also has a majority-strength beatpath to ''B.<br />
* '''Uncovered set''' = ''set of candidates which have defeats or beatpaths of length two to all other candidates''. Not monotonic as a set.<br />
* '''Banks set''' = ''set of candidates which are top on some maximal sub-chain of the defeat graph.'' Not monotonic as a set.<br />
* Dutta's '''Minimal covering set''' = ''smallest non-empty set such that when any other candidate is added, that candidate is covered in the resulting set.<br />
* '''Bipartisan set''' = ''set of candidates getting non-zero probability in the bipartisan distribution.'' <br />
* Forest Simmons' set '''P''' = ''set of candidates which are not approval-consistently defeated.'' This set is totally ordered by the defeats and contains the most approved candidate and a possible Condorcet winner.<br />
<br />
== Orderings == <br />
* Ordering by some score<br />
* (Uniformly) '''random order''': ''Pick some order uniformly at random.'' Destroys clone-proofness.<br />
* '''Random ballot order''': ''A is above B when A is above B on the first ballot distinguishing between them, in a random sequence of ballots.'' This is an often used tie breaking rank order (TBRO). Retains clone-proofness when used appropriately.<br />
* '''Random ballot approval order''': ''A is above B when A is above B on the first ballot on which the approval cutoff is between them, in a random sequence of ballots.'' Retains clone-proofness when used appropriately.<br />
<br />
== Randomization ==<br />
* '''Randomized start''': Pick an initial candidate by some random process, then proceed with other steps.<br />
* '''Randomized order''': Use a random process to determine the order in which the candidates are processes in some way.<br />
* '''Final randomization''': Restrict to some subset, then use a random process to choose from that set.</div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Techniques_of_method_design&diff=251Techniques of method design2005-03-23T00:23:20Z<p>12.73.132.31: /* Special sets */</p>
<hr />
<div>This is a list of techniques and concepts which are more or less useful when designing a single winner election method, with some comments.<br />
<br />
== Scores ==<br />
Scores can be interpreted as measures of "goodness" of a candidate. Their usage can thus improve the "efficiency" of a method.<br />
* '''Direct support''' = ''no. of voters ranking the candidate first.'' Retains clone-proofness when used appropriately, requires ranked ballots, is not well defined when more than one candidate is ranked top.<br />
* (Total) '''approval score''' = ''no. of voters approving of the candidate.'' Retains clone-proofness when used appropriately, requires approval information.<br />
* '''Borda score''' = ''sum of ranks the candidate gets on all ballots.'' Destroys clone-proofness and most "independency"-properties, requires ranked ballots.<br />
* '''Copeland score''' = ''no. of alternatives the candidate beats pairwise.'' Destroys clone-proofness.<br />
* Laslier's '''minimal gain score''' = ''probability that an alternative beaten by the candidate wins divided by the probability that an alternative beating the candidate wins, when choosing from the bipartisan distribution on the set of all other candidates.'' Improves the monotonicity of the bipartisan distribution, but destroys clone-proofness and is complicated to compute.<br />
<br />
== Defeats and defeat strength ==<br />
Looking at pairwise defeats and their strength can help assessing the "immunity" of a candidate against certain types of complaints. Care has to be taken so that the definition of strength doesn't destroy monotonicity.<br />
* (Pairwise) '''defeat''' <=> ''more voters expressed to prefer A over B than expressed to prefer B over A'' <br />
* '''Winning votes''' (wv) = ''no. of voters preferring the winner of the defeat to the loser''<br />
* '''Margin(s)''' = ''winning votes minus no. of voters preferring the loser.'' Gives more strategic incentives and seems more arbitrary (why not take the quotient instead?) than wv<br />
* Other similar definitions of strength are possible, especially when voters can distinguish between equivalence and undecidedness<br />
* '''Majority-strength defeat''' = ''pairwise defeat which has a wv-strength of more than half the no. of voters.'' Using only such defeats can help reduce incentive to truncate<br />
* '''Approval-consistent defeat''' (definitive defeat) = ''pairwise defeat in which the winner is more approved than the loser.'' These are acyclic and can thus all be respected without a contradiction<br />
* '''Winning approval''' = ''approval score of the winner of the defeat.'' Using this as defeat strength leaves only one immune candidate: the least approved of those who beat all more approved ones. Similar for other scores. <br />
* '''Approval-based support''' = ''no. of voters approving of the winner but not of the loser of the defeat.'' Gives special influence to preferences which cross the aproval cutoff and thus helps diminishing certain strategies. Useful when one assumes that only these voters will support the correspondig "majority complaint".<br />
* '''Cardinal rated''' strength = ''sum of difference in the candidates' cardinal ratings on all ballots which rate the winner over the loser of the defeat.'' Helps diminishing certain strategies even better, but requires interpersonally comparable cardinal ratings.<br />
<br />
== Steps ==<br />
Many methods work in more than one, perhaps similar, perhaps different steps.<br />
* '''Iteration''': Similar steps are performed until some break criterion applies.<br />
* '''Reduction to a subset''': All members outside some special set are removed.<br />
* '''Runoff''': Special case of reduction to a subset, for example by removing worst candidate according to some score. Can be iterated.<br />
* '''Ordering''': Many methods involve the construction of an ordering of the candidates or the defeats, not to be interpreted as a social order, but only as an intermediate tool.<br />
* '''Randomization''': Using a certain amount of randomness can help avoiding some strategies, especially when it leads to each candidate being beaten by a possible winner.<br />
<br />
== Special sets ==<br />
* '''Smith set''' (top cycle, top tier) = ''smallest non-empty set of candidates each of which beats all candidates outside the set.''<br />
* '''Schwartz set''' = ''smallest non-empty set of candidates so that each candidate beating a member of the set is also in the set.''<br />
* Woodall's '''CDTT''' ("Condorcet doubly-augmented top tier") = ''union of all minimal non-empty sets such that no candidate in each set has a majority-strength loss to any candidate outside the set''. Alternatively, ''the set containing every candidate ''A'' where, for each other candidate ''B'' who has a majority-strength beatpath to ''A'', ''A'' also has a majority-strength beatpath to ''B.<br />
* '''Uncovered set''' = ''set of candidates which have defeats or beatpaths of length two to all other candidates''. Not monotonic as a set.<br />
* '''Banks set''' = ''set of candidates which are top on some maximal sub-chain of the defeat graph.'' Not monotonic as a set.<br />
* Dutta's '''Minimal covering set''' = ''smallest non-empty set such that when any other candidate is added, that candidate is covered in the resulting set.<br />
* '''Bipartisan set''' = ''set of candidates getting non-zero probability in the bipartisan distribution.'' <br />
* Forest Simmons' set '''P''' = ''set of candidates which are not approval-consistently defeated.'' This set is totally ordered by the defeats and contains the most approved candidate and a possible Condorcet winner.<br />
<br />
== Orderings == <br />
* Ordering by some score<br />
* (Uniformly) '''random order''': ''Pick some order uniformly at random.'' Destroys clone-proofness.<br />
* '''Random ballot order''': ''A is above B when A is above B on the first ballot distinguishing between them, in a random sequence of ballots.'' This is an often used tie breaking rank order (TBRO). Retains clone-proofness when used appropriately.<br />
* '''Random ballot approval order''': ''A is above B when A is above B on the first ballot on which the approval cutoff is between them, in a random sequence of ballots.'' Retains clone-proofness when used appropriately.<br />
<br />
== Randomization ==<br />
* '''Randomized start''': Pick an initial candidate by some random process, then proceed with other steps.<br />
* '''Randomized order''': Use a random process to determine the order in which the candidates are processes in some way.<br />
* '''Final randomization''': Restrict to some subset, then use a random process to choose from that set.</div>12.73.132.31http://wiki.electorama.com/w/index.php?title=Techniques_of_method_design&diff=250Techniques of method design2005-03-23T00:12:23Z<p>12.73.132.31: /* Defeats and defeat strength */</p>
<hr />
<div>This is a list of techniques and concepts which are more or less useful when designing a single winner election method, with some comments.<br />
<br />
== Scores ==<br />
Scores can be interpreted as measures of "goodness" of a candidate. Their usage can thus improve the "efficiency" of a method.<br />
* '''Direct support''' = ''no. of voters ranking the candidate first.'' Retains clone-proofness when used appropriately, requires ranked ballots, is not well defined when more than one candidate is ranked top.<br />
* (Total) '''approval score''' = ''no. of voters approving of the candidate.'' Retains clone-proofness when used appropriately, requires approval information.<br />
* '''Borda score''' = ''sum of ranks the candidate gets on all ballots.'' Destroys clone-proofness and most "independency"-properties, requires ranked ballots.<br />
* '''Copeland score''' = ''no. of alternatives the candidate beats pairwise.'' Destroys clone-proofness.<br />
* Laslier's '''minimal gain score''' = ''probability that an alternative beaten by the candidate wins divided by the probability that an alternative beating the candidate wins, when choosing from the bipartisan distribution on the set of all other candidates.'' Improves the monotonicity of the bipartisan distribution, but destroys clone-proofness and is complicated to compute.<br />
<br />
== Defeats and defeat strength ==<br />
Looking at pairwise defeats and their strength can help assessing the "immunity" of a candidate against certain types of complaints. Care has to be taken so that the definition of strength doesn't destroy monotonicity.<br />
* (Pairwise) '''defeat''' <=> ''more voters expressed to prefer A over B than expressed to prefer B over A'' <br />
* '''Winning votes''' (wv) = ''no. of voters preferring the winner of the defeat to the loser''<br />
* '''Margin(s)''' = ''winning votes minus no. of voters preferring the loser.'' Gives more strategic incentives and seems more arbitrary (why not take the quotient instead?) than wv<br />
* Other similar definitions of strength are possible, especially when voters can distinguish between equivalence and undecidedness<br />
* '''Majority-strength defeat''' = ''pairwise defeat which has a wv-strength of more than half the no. of voters.'' Using only such defeats can help reduce incentive to truncate<br />
* '''Approval-consistent defeat''' (definitive defeat) = ''pairwise defeat in which the winner is more approved than the loser.'' These are acyclic and can thus all be respected without a contradiction<br />
* '''Winning approval''' = ''approval score of the winner of the defeat.'' Using this as defeat strength leaves only one immune candidate: the least approved of those who beat all more approved ones. Similar for other scores. <br />
* '''Approval-based support''' = ''no. of voters approving of the winner but not of the loser of the defeat.'' Gives special influence to preferences which cross the aproval cutoff and thus helps diminishing certain strategies. Useful when one assumes that only these voters will support the correspondig "majority complaint".<br />
* '''Cardinal rated''' strength = ''sum of difference in the candidates' cardinal ratings on all ballots which rate the winner over the loser of the defeat.'' Helps diminishing certain strategies even better, but requires interpersonally comparable cardinal ratings.<br />
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== Steps ==<br />
Many methods work in more than one, perhaps similar, perhaps different steps.<br />
* '''Iteration''': Similar steps are performed until some break criterion applies.<br />
* '''Reduction to a subset''': All members outside some special set are removed.<br />
* '''Runoff''': Special case of reduction to a subset, for example by removing worst candidate according to some score. Can be iterated.<br />
* '''Ordering''': Many methods involve the construction of an ordering of the candidates or the defeats, not to be interpreted as a social order, but only as an intermediate tool.<br />
* '''Randomization''': Using a certain amount of randomness can help avoiding some strategies, especially when it leads to each candidate being beaten by a possible winner.<br />
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== Special sets ==<br />
* '''Smith set''' (top cycle, top tier) = ''smallest non-empty set of candidates each of which beats all candidates outside the set.''<br />
* '''Schwartz set''' = ''smallest non-empty set of candidates so that each candidate beating a member of the set is also in the set.''<br />
* Woodall's '''CDTT''' = PLEASE ADD!<br />
* '''Uncovered set''' = ''set of candidates which have defeats or beatpaths of length two to all other candidates''. Not monotonic as a set.<br />
* '''Banks set''' = ''set of candidates which are top on some maximal sub-chain of the defeat graph.'' Not monotonic as a set.<br />
* Dutta's '''Minimal covering set''' = ''smallest non-empty set such that when any other candidate is added, that candidate is covered in the resulting set.<br />
* '''Bipartisan set''' = ''set of candidates getting non-zero probability in the bipartisan distribution.'' <br />
* Forest Simmons' set '''P''' = ''set of candidates which are not approval-consistently defeated.'' This set is totally ordered by the defeats and contains the most approved candidate and a possible Condorcet winner.<br />
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== Orderings == <br />
* Ordering by some score<br />
* (Uniformly) '''random order''': ''Pick some order uniformly at random.'' Destroys clone-proofness.<br />
* '''Random ballot order''': ''A is above B when A is above B on the first ballot distinguishing between them, in a random sequence of ballots.'' This is an often used tie breaking rank order (TBRO). Retains clone-proofness when used appropriately.<br />
* '''Random ballot approval order''': ''A is above B when A is above B on the first ballot on which the approval cutoff is between them, in a random sequence of ballots.'' Retains clone-proofness when used appropriately.<br />
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== Randomization ==<br />
* '''Randomized start''': Pick an initial candidate by some random process, then proceed with other steps.<br />
* '''Randomized order''': Use a random process to determine the order in which the candidates are processes in some way.<br />
* '''Final randomization''': Restrict to some subset, then use a random process to choose from that set.</div>12.73.132.31