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The monotonicity criterion is a voting system criterion used to evaluate both single and multiple winner ranked voting systems. A ranked voting system is monotonic if it is neither possible to prevent the election of a candidate by ranking them higher on some of the ballots, nor possible to elect an otherwise unelected candidate by ranking them lower on some of the ballots (while nothing else is altered on any ballot). 〔D R Woodall, ("Monotonicity and Single-Seat Election Rules" ), ''Voting matters'', Issue 6, 1996〕 In single winner elections that is to say no winner is harmed by up-ranking and no loser can win by down-ranking. Douglas R. Woodall called the criterion mono-raise. Raising a candidate on some ballots ''while changing'' the orders of other candidates does ''not'' constitute a failure of monotonicity. E.g. harming candidate by changing some ballots from to isn't a violation of the monotonicity criterion. The monotonicity criterion renders the intuition, that there should be neither need to worry about harming a candidate by (nothing else than) up-ranking nor it should be possible to support a candidate by (nothing else than) counter-intuitively down-ranking. There are several variations of that criterion, e.g. what Douglas R. Woodall called ''mono-add-plump'': A candidate should not be harmed if further ballots are added that have top with no second choice. Agreement with such rather special properties is the best any ranked voting system may fulfill: The Gibbard–Satterthwaite theorem shows, that any meaningful ranked voting system is susceptible to some kind of tactical voting, and Arrow's impossibility theorem shows, that individual rankings can't be meaningfully translated into a community-wide ranking where the order of candidates and is always independent of irrelevant alternatives . Noncompliance with the monotonicity criterion doesn't tell anything about the likelihood of monotonicity violations, failing in one of a million possible elections would be as well a violation as missing the criterion in any possible election. Of the single-winner ranked voting systems Borda count, Schulze method, ranked pairs / maximize affirmed majorities, descending solid coalitions〔(Electorama:Descending Solid Coalitions ).〕 and descending acquiescing coalitions〔〔(Electorama:Descending Acquiescing Coalitions ).〕 are monotonic, while Coombs' method, runoff voting and instant-runoff voting (IRV) are not. Most variants of the single transferable vote (STV) proportional representations are not monotonic, especially all that are currently in use for public elections (which simplify to IRV when there is only one winner). All plurality voting systems are monotonic if the ballots are treated as rankings where using ''more than two ranks is forbidden''. In this setting first past the post and approval voting as well as the multiple-winner systems single non-transferable vote, plurality-at-large voting (multiple non-transferable vote, bloc voting) and cumulative voting are monotonic. Party-list proportional representation using D'Hondt, Sainte-Laguë or the largest remainder method is monotonic in the same sense. In elections via the single-winner methods range voting and majority judgment nobody can help a candidate by reducing or removing support for them, but as they are not ''ranked'' voting systems, they are out of the monotonicity criterion's scope. ==Instant-runoff voting and the two-round system are not monotonic== Using an example that applies to instant-runoff voting (IRV) and to the two-round system, it is shown that these voting systems violate the mono-raise criterion. Suppose a president were being elected among three candidates, a left, a right, and a center candidate, and 100 votes cast. The number of votes for an absolute majority is therefore 51. Suppose the votes are cast as follows: : According to the 1st preferences, Left finishes first with 35 votes, Right gets 33 votes, and Center 32 votes, thus all candidates lack an absolute majority of first preferences. In an actual runoff between the top two candidates, Left would win against Right with 30+5+16=51 votes. The same happens (in this example) under IRV, Center gets eliminated, and Left wins against Right with 51 to 49 votes. But if at least two of the five voters who ranked Right first, and Left second, would raise Left, and vote 1st Left, 2nd Right; then Left would be defeated by these votes in favor of Center. Let's assume that two voters change their preferences in that way, which changes two rows of the table: : Now Left gets 37 first preferences, Right only 31 first preferences, and Center still 32 first preferences, and there is again no candidate with an absolute majority of first preferences. But now Right gets eliminated, and Center remains in round 2 of IRV (or the actual runoff in the Two-round system). And Center beats its opponent Left with a remarkable majority of 60 to 40 votes. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「monotonicity criterion」の詳細全文を読む スポンサード リンク
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