By the way, my first message mistakenly said MCA fails the Summability criterion; I meant the Consistency criterion.<div><br></div><div>Here's the latest version of the criteria compliance, which is the same as before except for the above change and some editing and reformatting:</div>

<div><br></div><div><meta http-equiv="content-type" content="text/html; charset=utf-8"><span class="Apple-style-span" style="font-family: sans-serif; font-size: 13px; line-height: 19px; "><h2 style="color: rgb(204, 102, 0); background-image: none; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; font-weight: normal; margin-top: 0px; margin-right: 0px; margin-bottom: 0.6em; margin-left: 0px; padding-top: 0.5em; padding-bottom: 0.17em; border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: rgb(170, 170, 170); font-size: 19px; background-position: initial initial; background-repeat: initial initial; ">

<span class="editsection" style="float: right; margin-left: 5px; font-size: 13px; "><br class="Apple-interchange-newline">[<a href="http://wiki.electorama.com/wiki/index.php?title=Majority_Choice_Approval&action=edit&section=3" title="Edit section: Criteria compliance" style="text-decoration: none; color: rgb(0, 43, 184); background-image: none; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: initial initial; background-repeat: initial initial; ">edit</a>]</span><span class="mw-headline" id="Criteria_compliance">Criteria compliance</span></h2>

<p style="margin-top: 0.4em; margin-right: 0px; margin-bottom: 0.5em; margin-left: 0px; line-height: 1.5em; ">All MCA variants satisfy the <a href="http://wiki.electorama.com/wiki/Plurality_criterion" title="Plurality criterion" style="text-decoration: none; color: rgb(0, 43, 184); background-image: none; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: initial initial; background-repeat: initial initial; ">Plurality criterion</a>, the <a href="http://wiki.electorama.com/wiki/Majority_criterion_for_solid_coalitions" title="Majority criterion for solid coalitions" class="mw-redirect" style="text-decoration: none; color: rgb(0, 43, 184); background-image: none; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: initial initial; background-repeat: initial initial; ">Majority criterion for solid coalitions</a>, <a href="http://wiki.electorama.com/wiki/Monotonicity_criterion" title="Monotonicity criterion" style="text-decoration: none; color: rgb(0, 43, 184); background-image: none; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: initial initial; background-repeat: initial initial; ">Monotonicity</a> (for MCA-AR, assuming first- and second- round votes are consistent), and <a href="http://wiki.electorama.com/wiki/Minimal_Defense_criterion" title="Minimal Defense criterion" style="text-decoration: none; color: rgb(0, 43, 184); background-image: none; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: initial initial; background-repeat: initial initial; ">Minimal Defense</a> (which implies satisfaction of the <a href="http://wiki.electorama.com/wiki/Strong_Defensive_Strategy_criterion" title="Strong Defensive Strategy criterion" style="text-decoration: none; color: rgb(0, 43, 184); background-image: none; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: initial initial; background-repeat: initial initial; ">Strong Defensive Strategy criterion</a>).</p>

<p style="margin-top: 0.4em; margin-right: 0px; margin-bottom: 0.5em; margin-left: 0px; line-height: 1.5em; ">All of the methods are <a href="http://wiki.electorama.com/wiki/Summability_criterion" title="Summability criterion" style="text-decoration: none; color: rgb(0, 43, 184); background-image: none; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: initial initial; background-repeat: initial initial; ">matrix-summable</a> for counting at the precinct level. Only MCA-IR actually requires a matrix (or, possibly two counting rounds), and is thus "<a href="http://wiki.electorama.com/wiki/Summability_criterion" title="Summability criterion" style="text-decoration: none; color: rgb(0, 43, 184); background-image: none; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: initial initial; background-repeat: initial initial; ">summable for k=2</a>" ; the others require only O(N) tallies, and are thus "<a href="http://wiki.electorama.com/wiki/Summability_criterion" title="Summability criterion" style="text-decoration: none; color: rgb(0, 43, 184); background-image: none; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: initial initial; background-repeat: initial initial; ">summable for k=1</a>".</p>

<p style="margin-top: 0.4em; margin-right: 0px; margin-bottom: 0.5em; margin-left: 0px; line-height: 1.5em; ">The <a href="http://wiki.electorama.com/wiki/Participation_criterion" title="Participation criterion" style="text-decoration: none; color: rgb(0, 43, 184); background-image: none; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: initial initial; background-repeat: initial initial; ">Participation criterion</a> and its stronger cousin the <a href="http://wiki.electorama.com/wiki/Consistency_criterion" title="Consistency criterion" style="text-decoration: none; color: rgb(0, 43, 184); background-image: none; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: initial initial; background-repeat: initial initial; ">Consistency criterion</a>, as well as the <a href="http://wiki.electorama.com/wiki/Later-no-harm_criterion" title="Later-no-harm criterion" style="text-decoration: none; color: rgb(0, 43, 184); background-image: none; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: initial initial; background-repeat: initial initial; ">Later-no-harm criterion</a> are not satisfied by any MCA variant, although MCA-P only fails Participation if the additional vote causes an approval majority.</p>

<p style="margin-top: 0.4em; margin-right: 0px; margin-bottom: 0.5em; margin-left: 0px; line-height: 1.5em; ">Other criteria are satisfied by some, but not all, MCA variants. To wit:</p><ul style="line-height: 1.5em; list-style-type: square; margin-top: 0.3em; margin-right: 0px; margin-bottom: 0px; margin-left: 1.5em; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; list-style-image: url(http://wiki.electorama.com/wiki/skins/monobook/bullet.gif); ">

<li style="margin-bottom: 0.1em; "><a href="http://wiki.electorama.com/wiki/Strategic_nomination" title="Strategic nomination" style="text-decoration: none; color: rgb(0, 43, 184); background-image: none; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: initial initial; background-repeat: initial initial; ">Clone Independence</a> is satisfied by most MCA versions. In fact, even the stronger <a href="http://wiki.electorama.com/wiki/Independence_of_irrelevant_alternatives" title="Independence of irrelevant alternatives" style="text-decoration: none; color: rgb(0, 43, 184); background-image: none; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: initial initial; background-repeat: initial initial; ">Independence of irrelevant alternatives</a> is satisfied by MCA-A, MCA-P, MCA-M, and MCA-S. Clone independence is satisfied along with the weaker and related <a href="http://wiki.electorama.com/wiki/ISDA" title="ISDA" class="mw-redirect" style="text-decoration: none; color: rgb(0, 43, 184); background-image: none; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: initial initial; background-repeat: initial initial; ">ISDA</a> by MCA-IR and MCA-AR, if ISDA-compliant Condorcet methods (ie, <a href="http://wiki.electorama.com/wiki/Schulze" title="Schulze" class="mw-redirect" style="text-decoration: none; color: rgb(0, 43, 184); background-image: none; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: initial initial; background-repeat: initial initial; ">Schulze</a>) are used to choose the two "finalists". Using simpler methods (such as MCA itself) to decide the finalists, MCA-IR and MCA-AR are not strictly clone independent.</li>

</ul><ul style="line-height: 1.5em; list-style-type: square; margin-top: 0.3em; margin-right: 0px; margin-bottom: 0px; margin-left: 1.5em; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; list-style-image: url(http://wiki.electorama.com/wiki/skins/monobook/bullet.gif); ">

<li style="margin-bottom: 0.1em; ">The <a href="http://wiki.electorama.com/wiki/Condorcet_criterion" title="Condorcet criterion" class="mw-redirect" style="text-decoration: none; color: rgb(0, 43, 184); background-image: none; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: initial initial; background-repeat: initial initial; ">Condorcet criterion</a> is satisfied by MCA-IR if the <a href="http://wiki.electorama.com/wiki/Pairwise_champion" title="Pairwise champion" class="mw-redirect" style="text-decoration: none; color: rgb(0, 43, 184); background-image: none; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: initial initial; background-repeat: initial initial; ">pairwise champion</a> (aka CW) is visible on the ballots and would beat at least one other candidate by an absolute majority. It is satisfied by MCA-AR if at least half the voters at least approve the PC in the first round of voting. These methods also satisfy the <a href="http://wiki.electorama.com/wiki/Strategy-Free_criterion" title="Strategy-Free criterion" style="text-decoration: none; color: rgb(0, 43, 184); background-image: none; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: initial initial; background-repeat: initial initial; ">Strategy-Free criterion</a> if an SFC-compliant method such as <a href="http://wiki.electorama.com/wiki/Schulze" title="Schulze" class="mw-redirect" style="text-decoration: none; color: rgb(0, 43, 184); background-image: none; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: initial initial; background-repeat: initial initial; ">Schulze</a> is used to pick at least one of the finalists. All other MCA versions, however, fail the Condorcet and strategy-free criteria.</li>

</ul><ul style="line-height: 1.5em; list-style-type: square; margin-top: 0.3em; margin-right: 0px; margin-bottom: 0px; margin-left: 1.5em; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; list-style-image: url(http://wiki.electorama.com/wiki/skins/monobook/bullet.gif); ">

<li style="margin-bottom: 0.1em; ">The <a href="http://wiki.electorama.com/wiki/Later-no-help_criterion" title="Later-no-help criterion" style="text-decoration: none; color: rgb(0, 43, 184); background-image: none; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: initial initial; background-repeat: initial initial; ">Later-no-help criterion</a> and the <a href="http://wiki.electorama.com/wiki/Favorite_Betrayal_criterion" title="Favorite Betrayal criterion" style="text-decoration: none; color: rgb(0, 43, 184); background-image: none; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: initial initial; background-repeat: initial initial; ">Favorite Betrayal criterion</a> are satisfied by MCA-P. They're also satisfied by MCA-AR if MCA-P is used to pick the two finalists.</li>

</ul><ul style="line-height: 1.5em; list-style-type: square; margin-top: 0.3em; margin-right: 0px; margin-bottom: 0px; margin-left: 1.5em; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; list-style-image: url(http://wiki.electorama.com/wiki/skins/monobook/bullet.gif); ">

<li style="margin-bottom: 0.1em; ">MCA-AR satisfies the <a href="http://wiki.electorama.com/wiki/Guaranteed_majority_criterion" title="Guaranteed majority criterion" style="text-decoration: none; color: rgb(0, 43, 184); background-image: none; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: initial initial; background-repeat: initial initial; ">Guaranteed majority criterion</a>, a criterion which can only be satisfied by a multi-round (runoff-based) method.</li>

</ul><p style="margin-top: 0.4em; margin-right: 0px; margin-bottom: 0.5em; margin-left: 0px; line-height: 1.5em; ">Thus, the MCA method which satisfies the most criteria is MCA-AR, using <a href="http://wiki.electorama.com/wiki/Schulze" title="Schulze" class="mw-redirect" style="text-decoration: none; color: rgb(0, 43, 184); background-image: none; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: initial initial; background-repeat: initial initial; ">Schulze</a> over the ballots to select one finalist and MCA-P to select the other. Also notable are MCA-M and MCA-P, which, as <i>rated</i> methods (and thus ones which fail Arrow's <i>ranking</i>-based <a href="http://wiki.electorama.com/wiki/Universality_criterion" title="Universality criterion" style="text-decoration: none; color: rgb(0, 43, 184); background-image: none; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: initial initial; background-repeat: initial initial; ">Universality criterion</a>), are able to seem to "violate <a href="http://wiki.electorama.com/wiki/Arrow%27s_Theorem" title="Arrow's Theorem" class="mw-redirect" style="text-decoration: none; color: rgb(0, 43, 184); background-image: none; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: initial initial; background-repeat: initial initial; ">Arrow's Theorem</a>" by simultaneously satisfying monotonicity and <a href="http://wiki.electorama.com/wiki/Independence_of_irrelevant_alternatives" title="Independence of irrelevant alternatives" style="text-decoration: none; color: rgb(0, 43, 184); background-image: none; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: initial; background-position: initial initial; background-repeat: initial initial; ">independence of irrelevant alternatives</a> (as well as of course sovereignty and non-dictatorship).</p>

</span></div>