Preface: In a March 2004 Scientific American article entitled, "The Fairest Vote of All," Partha Dasgupta and Eric Maskin (now a Nobel Laureate) argued persuasively for their conception of a "True Majority Winner" of a single winner election based on ranked choice ballots. Taking for granted the Majority Criterion that mandates electing the candidate that outranks all of the other candidates on more than half of the ballots (when there is such a candidate), they propose that when there is no such candidate, when possible they at least keep this less demanding but crucial property of a Majority top ranked candidate: such a candidate outranks any competitor on more ballots than not. Why not say, "on more than half of the ballots" instead of "more ballots than not"? Because voters are not required to rank all of the candidates. Indeed, some voters may simply "bullet vote" for their favorite, while leaving the other candidates unranked or "truncated." The dictionary definition of "majority" is flexible enough to include this usage of "more than not," so Dasgupta and Maskin's "True Majority Winner" terminology is perfectly acceptable to Webster, Cambridge, OED, etc. Their Scientific American article briefly alluded to the rare public election possibility where a ballot set might yield neither a "more than half" first place majority winner nor a (less demanding) True Majority Winner. It was not the purpose of their article to prescribe a course of action to cover that rare case, since they were not making a proposal for a specific election method to be adopted and written into law for some specific democratic electorate. Their purpose was to expound and publicize to the broader scientific community and other interested citizens a principle that has been respected among social choice thinkers at least since the time of Ramón Llull of twelfth century Spain. We now pick up where they left off with a proposal for how to decide the winner in the case of no True Majority Winner (TMW). For ease of reference we repeat (my wording of) the Dasgupta/Maskin definition of True Majority Winner, namely a candidate that outranks every competitor on more ballots than not. Also, "bullet ballot" ... a ballot that truncates after its top choice. We also need the concept of a "ballot superset:" In the current context it is a ballot set augmented with a number of bullet ballots to gauge how far away a ballot set is from having a True Majority Winner. Our idea is to complete the quest for a True Majority Winner by augmenting the given ballot set with the bare minimum of bullet ballots to ensure the existence of a True Majority Winner for the augmented ballot set. In other words, we elect the candidate closest to being a TMW when there is no TMW. So here it is: If the submitted set of marked ballots does not have a True Majority Winner (i.e. a candidate that outranks each opponent on more ballots than not), then elect the True Majority Winner of the smallest ballot superset that does have a True Majority Winner. The above description completely and decisively defines the winner without recommending one procedure over another for tallying the submitted ballots. There are many possible counting procedures, (some more efficient than others) but any that require multiple passes through the ballot set (as do elimination methods like Instant Runoff) are inefficient, hence to be avoided. One efficient procedure is to immediately (at the precinct level) summarize each ballot in the form of a table with K rows and K columns, where K is the number of candidates. The i_th entry in the j_th row of the table is a one or zero depending on whether or not candidate j outranks candidate i on the ballot being tabulated. Once a ballot is converted to this K by K tabular format it can be added in to the precinct total. In turn the precinct totals are added together at some central location to arrive at a grand total table T. Apparently the i_th entry in the j_th row of table T is the number of ballots on which candidate j outranks candidate i. Similarly, the i_th entry in the j_th column of table T is the number of ballots on which candidate i outranks candidate j. Therefore, when we subtract the corresponding elements of the j_th column from the j_th row we get a new table D in which the i_th entry of the j_th row is the difference between the number of ballots on which j out ranks i and the number of ballots on which i outranks j. If this difference is positive, then candidate j outranks candidate i on more ballots than not. Therefore, if the j_th row of D has all positive differences, then candidate j is the True Majority Candidate. If there is no such row j with all positive entries, find the row j that needs the least multiple of the "bullet row" added to it in order to wipe out all of its (row j's) negative entries. A bullet ballot row consists entirely of ones: [1, 1, ..., 1]. This row j identifies the True Majority Winner of the ballot set that has been augmented with the minimum number of bullet ballots to achieve a TMW ... in ther words the candidate closest to being a TMW of the original ballot set. Don't worry about the details of this tally procedure .. that's for trained election officials to learn. But do take note that a methodical method involving mostly copying and adding of table entries (derived from ranked ballots) with a subtraction of table columns from corresponding table rows is all there is before determining how many bullet ballot rows are need to wipe out all of the negatives from one row ... just methodical use of arithmetic ... for someone equipped with an adding machine to worry about. Questions? Suggestions for improved exposition? Gripes? Thanks! -Forest

I’m sorry to say that I don't like Dasgupta and Maskin’s terminology at all - I think it's a trick to persuade readers that they've already signed up for the Condorcet Principle when they signed up for the Majority Criterion. I agree that "Condorcet winner" sounds forbiddingly esoteric to the layman - I sometimes consider "outright winner" as a more down-to-earth term.    Forest's definition is concise and avoids reference fo matrices of defeat margins and similar machinery, but I'm not sure if that makes it any easier to understand since the conceptual level is quite high.    Colin On 31/05/2022 05:23, Forest Simmons wrote: > Preface: > > In a March 2004 Scientific American article entitled, "The Fairest > Vote of All," Partha Dasgupta and Eric Maskin (now a Nobel Laureate) > argued persuasively for their conception of a "True Majority Winner" > of a single winner election based on ranked choice ballots. > > Taking for granted the Majority Criterion that mandates electing the > candidate that outranks all of the other candidates on more than half > of the ballots (when there is such a candidate), they propose that > when there is no such candidate, when possible they at least keep this > less demanding but crucial property of a Majority top ranked > candidate: such a candidate outranks any competitor on more ballots > than not. > > Why not say, "on more than half of the ballots" instead of "more > ballots than not"? > > Because voters are not required to rank all of the candidates. Indeed, > some voters may simply "bullet vote" for their favorite, while leaving > the other candidates unranked or "truncated." > > The dictionary definition of "majority" is flexible enough to include > this usage of "more than not," so Dasgupta and Maskin's "True Majority > Winner" terminology is perfectly acceptable to Webster, Cambridge, > OED, etc. > > Their Scientific American article briefly alluded to the rare public > election possibility where a ballot set might yield neither a "more > than half" first place majority winner nor a (less demanding) True > Majority Winner. > > It was not the purpose of their article to prescribe a course of > action to cover that rare case, since they were not making a proposal > for a specific election method to be adopted and written into law for > some specific democratic electorate. > > Their purpose was to expound and publicize to the broader scientific > community and other interested citizens a principle that has been > respected among social choice thinkers at least since the time of > Ramón Llull of twelfth century Spain. > > We now pick up where they left off with a proposal for how to decide > the winner in the case of no True Majority Winner (TMW). > > For ease of reference we repeat (my wording of) the Dasgupta/Maskin > definition of True Majority Winner, namely a candidate that outranks > every competitor on more ballots than not. > > Also, "bullet ballot" ... a ballot that truncates after its top choice. > > We also need the concept of a "ballot superset:" In the current > context it is a ballot set augmented with a number of bullet ballots > to gauge how far away a ballot set is from having a True Majority Winner. > > Our idea is to complete the quest for a True Majority Winner by > augmenting the given ballot set with the bare minimum of bullet > ballots to ensure the existence of a True Majority Winner for the > augmented ballot set. In other words, we elect the candidate closest > to being a TMW when there is no TMW. > > So here it is: > > If the submitted set of marked ballots does not have a True Majority > Winner (i.e. a candidate that outranks each opponent on more ballots > than not), then elect the True Majority Winner of the smallest ballot > superset that does have a True Majority Winner. > > The above description completely and decisively defines the winner > without recommending one procedure over another for tallying the > submitted ballots. > > There are many possible counting procedures, (some more efficient than > others) but any that require multiple passes through the ballot set > (as do elimination methods like Instant Runoff) are inefficient, hence > to be avoided. > > One efficient procedure is to immediately (at the precinct level) > summarize each ballot in the form of a table with K rows and K > columns, where K is the number of candidates.  The i_th entry in the > j_th row of the table is a one or zero depending on whether or not > candidate j outranks candidate i on the ballot being tabulated. > > Once a ballot is converted to this K by K tabular format it can be > added in to the precinct total. In turn the precinct totals are added > together at some central location to arrive at a grand total table T. > > Apparently the i_th entry in the j_th row of table T is the number of > ballots on which candidate j outranks candidate i. > > Similarly, the i_th entry in the j_th column of table T is the number > of ballots on which candidate i outranks candidate j. > > Therefore, when we subtract the corresponding elements of the j_th > column from the j_th row we get a new table D in which the i_th entry > of the j_th row is the difference between the number of ballots on > which j out ranks i and the number of ballots on which i outranks j. > > If this difference is positive, then candidate j outranks candidate i  > on more ballots than not. > > Therefore, if the j_th row of D has all positive differences, then > candidate j is the True Majority Candidate. > > If there is no such row j with all positive entries, find the row j > that needs the least multiple of the "bullet row" added to it in order > to wipe out all of its (row j's) negative entries. > > A bullet ballot row consists entirely of ones: [1, 1, ..., 1]. > > This row j identifies the True Majority Winner of the ballot set that > has been augmented with the minimum number of bullet ballots to > achieve a TMW ... in ther words the candidate closest to being a TMW > of the original ballot set. > > Don't worry about the details of this tally procedure  .. that's for > trained election officials to learn. But do take note that a > methodical method involving mostly copying and adding of table entries > (derived from ranked ballots) with a subtraction of table columns from > corresponding table rows is all there is before determining how many > bullet ballot rows are need to wipe out all of the negatives from one > row ... just methodical use of arithmetic ... for someone equipped > with an adding machine to worry about. > > Questions? > > Suggestions for improved exposition? > > Gripes? > > Thanks! > > -Forest > > > > > > ---- > Election-Methods mailing list - see https://electorama.com/em for list info