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Line 1: '''Maximum Constrained Approval Bucklin''' (MCAB) is a multiwinner method devised by [[Kristofer Munsterhjelm]], based on [[Bucklin voting]]. It uses linear optimization to calculate candidate support by assuming earlier surplus transfers were maximally favorable to the candidate in question, and thus reduces the strategic impact of lowering or raising a winning candidate on a ballot. MCAB works in multiple rounds, each of which sets an implicit approval cutoff for every ballot. The first round considers first preferences as approved, the second round considers first and second preferences, and so on. For each round, the method evaluates every remaining unelected candidate. The unelected candidate with the greatest support is elected if his support is greater than a Droop quota, and a round may elect more than one candidate. Line 7: Determining which voters to eliminate to maximize the support of a candidate subject to earlier constraints is relatively simple to do by linear programming, but hard to do by hand; MCAB can't be counted entirely by hand. What ended up as MCAB was initially proposed in 2017<ref>{{cite web\|url=http://lists.electorama.com/pipermail/election-methods-electorama.com/2017-January/001276.html\|title=Bucklin multiwinner method\|website=Election-methods mailing list archives\|date=2017-01-06\|last=Munsterhjelm\|first=K.}}</ref> and simplified in later that year<ref>{{cite web\|url=http://lists.electorama.com/pipermail/election-methods-electorama.com/2017-September/001584.html\|title=A simpler vote management-resistant Bucklin LP\|website=Election-methods mailing list archives\|date=2017-09-15\|last=Munsterhjelm\|first=K.}}</ref>. The method detailed here has been further modified from the EM posts to resist Woodall [[free riding]]. == Determining the support for X == Let <math>r</math> be the rank and round number, <math>n</math> the number of candidates so far, and <math>c_k</math> the kth elected candidate. Consider unranked candidates to be ranked equal below every explicitly ranked candidate, i.e. never approved in any round. The linear program for determining the support of candidate X as (n+1)th candidate is: maximize: sum over all voters v: support[v][n+1] Line 28: (sum over i = 1 ... n+1: support[v][i]) <= v's initial weight where <math>c_({n+1)}</math> is provisionally defined as X for the purpose of determining X's support. The three clauses do the following: # imposes the Droop constraint: that any elected candidate <math>c_i</math> must have more than a Droop quota's worth of approvals according to the implicit approval cutoff for round r. # defines support: <math>c_i</math>'s support is the number of voters who rank <math>c_i</math> at or above rank <math>r</math>. # defines each voter's budget: no voter can spread more support across the candidates than his ballot's initial weight. The initial weight is 1 per voter for ordinary elections, or some other value in case of a weighted vote. Line 107: Since A and B were elected in both cases, the vote-management failed. == ~~Criteria~~Criterion compliances == MCAB passes the following criteria: Line 150: [[Category:Multi-winner voting methods]] [[Category:Proportional voting methods]] [[Category:~~Preferential~~Ranked voting methods]] [[Category:PSC-compliant voting methods]]