The Mathematics of Elections Part II: Voting

The Mathematics of ElectionsPart II: Voting

Mark Rogers

Last time…

• We discussed systems of apportionment.– how we determine how many representatives

there should be in an elective body and how those representatives are to be distributed among various subgroups of the population as a whole

• Several methods can be used.– Founding Fathers: Hamilton, Jefferson,

Adams, Webster– Huntington-Hill method: used since WWII

Last time…• Problems can result from any type of apportionment.

– Alabama paradox: in an even larger group of representatives, one group could end up losing one, even with the same percentage of the population

– Population paradox: one group can lose a representative to another group, even though they are growing at a faster rate

– New states paradox: one group can lose a representative to another group if a new group is added, with their own delegation considered separately

– Violation of the apportionment criterion: each representative should represent (approximately) the same number of voters

– Violation of the quota rule: using some methods, a group can end up with a number of representatives very different than their “deserved” proportion of the entire body (too many or too few)

Last time…

• Gerrymandering: redrawing district lines to favor or hinder one particular group– Can be done to “pack” similar voters into the same

district, or to “crack” or split similar voters among several districts, thus diluting them

– Done by state legislatures to maximize their party’s chances of winning the Congressional seats

– This works because most of our elections follow a simple “majority wins” method, in which candidates with a minority of votes (and their supporters) will have no say in that district.

1992 & 2000 Presidential Elections

• The lopsided 1992 results and the upside-down 2000 results are due to the fact that our Electoral College system is based on a subdivided measure of the national popular vote, and one which does not necessarily require a majority of it.

2000 Candidate Popular Vote Electoral Vote

George W. Bush 50,456,002 (47.9%) 271 (50.4%)

Al Gore 50,999,897 (48.4%) 267 (49.6%)

1992 Candidate Popular Vote Electoral Vote

Bill Clinton 44,908,254 (43.0%) 370 (68.8%)

George W. Bush 39,102,343 (37.7%) 168 (31.2%)

Majorities and supermajorities

• “majority”: more than 50% of the votes (“50% plus 1”)• “supermajority”: a higher-than-majority percentage of

votes– The specific definition can vary according to the situation.– In the U.S. Senate, a 60% supermajority is needed to end a

filibuster.– In both houses of Congress, a two-thirds supermajority is

needed to override a Presidential veto.

• The majority criterion: if a majority of voters rank candidate X as their first choice, then X deserves to win the election.– This seems like common sense, but often a majority cannot be

attained.

What if a majority is not achieved?

• Many elections are run using the plurality method: the candidate chosen by the most voters as their first choice wins, regardless of whether they attain a majority or not.

• Many others use the plurality-with-elimination method: if no candidate attains a majority, then a second round of voting (or possibly several) takes place, with some candidates (possibly as many as all but the top two from the first round) eliminated from the later ballot(s).

• The plurality-with-elimination method can often result in the second-place candidate winning in the second round of voting, as supporters of the dropped candidates flock to his or her side.– Often, a coalition of voters forms to defeat the candidate who was in first

place after the first round.

2007 French presidential elections

• 12 candidates, including four major ones in an extremely tight race:– Conservative Nicolas Sarkozy, favored in early polls by 26% of voters– Socialist Ségolène Royal, favored by 25%– Centrist François Bayrou, favored by 24%– Far-right nationalist Jean-Marie Le Pen, favored by 16%

• The top two candidates would advance to a runoff.• However, Bayrou’s support was more widespread; not being as

despised by members of the other extreme as either Sarkozy or Royal, he would have been able to attract far more voters in a runoff than either of them…if he could make the runoff.

• Unfortunately for him, he did indeed finish in 3rd place, leaving him unable to surpass the two frontrunners in this manner.

• In the end, Sarkozy won the runoff against Royal 53% to 47%.• While this is not surprising or even unexpected, curious paradoxes

can occur in certain other situations.

The problem with a “revote”• The consistency criterion: if the electorate is split into two or more

divisions, and candidate X is the winner in each portion, then X deserves to win the overall election. (This idea is somewhat analogous to Simpson’s paradox in statistics.)

• The independence-of-irrelevant-alternatives criterion: if candidate X wins an election, and a recount is taken after some non-winner(s) are eliminated from the ballot, then candidate X should still win the election.

• The monotonicity criterion: if candidate X wins an election, and if in a revote the only voters who change their preference do so in favor of X, then X should still win the revote.– Again, these both sound like the expected result, but in certain

circumstances candidate X will actually lose the second round despite winning the first round and being the only person to gain supporters!

1992 & 2000 Presidential Elections• Stronger-than-usual showings by third-party candidates are thought

to have tilted the balance in recent elections.

• Experts theorize that if even “most” (say, 2/3) of the third-party voters had stayed with the main candidate of their (former) party, both elections would have swung the other way.

1992 Candidates National Vote

Bill Clinton 44,908,254

George H.W. Bush 39,102,343

Ross Perot 8,085,294

2000 Candidates Florida Vote

George W. Bush 2,912,790

Al Gore 2,912,253

Ralph Nader 97,488

Electoral Vote

With “2/3” support

Clinton 370 267

Bush Sr. 168 271

Electoral Vote

With “2/3” support

Bush Jr. 271 246

Gore 267 292

Splitting your vote

• Many voters see this sort of “splitting” of the vote of one end of the political spectrum to be self-defeating.– By opting for their favorite candidate instead of a slightly less-

favored alternative whose higher support levels might have given them a more realistic chance of winning, they “hand the election” to the close-running candidate of the opposite persuasion.

• Cumulative voting: Voters can choose to split their vote into fractional portions divided among several candidates, or concentrate it for a single candidate.– Voters could give small-party candidates enough fractional votes

to help make them viable, but reserve the bulk of their vote for mainstream candidates who might otherwise lose a close race.

An unexpected election result• First-round results:

• Since Bonnie was the first choice of the fewest number of voters, she is eliminated and a runoff held.

• Given each voter’s runner-up preferences, the following results are likely in the second round:

• Thus, Charles wins the second-round runoff.• However, suppose that due to irregularities, the election is ordered

to be started from scratch. Charles should still win, right?

Preference 12 voters 9 voters 3 voters 8 voters

1st Charles Bonnie Adam Adam

2nd Adam Charles Charles Bonnie

3rd Bonnie Adam Bonnie Charles

Preference 12 voters 9 voters 3 voters 8 voters

1st Charles Charles Adam Adam

2nd Adam Adam Charles Charles

An unexpected election result• This time, the group of three voters in the third column decide that they were

disappointed by Adam’s behavior in the runoff, and decide to switch their first-round vote from him to Charles (whom they just watched “win” anyway).

• New first-round results:

• This time, Adam was the first choice of the fewest number of voters, so he is eliminated from the runoff. The revised likely runoff preferences are:

• Thus, Bonnie wins the second-round runoff, despite no changes to her votes.– She wasn’t even in the original runoff!

Preference 12 voters 9 voters 3 voters 8 voters

1st Charles Bonnie Charles Adam

2nd Adam Charles Adam Bonnie

3rd Bonnie Adam Bonnie Charles

Preference 12 voters 9 voters 3 voters 8 voters

1st Charles Bonnie Charles Bonnie

2nd Bonnie Charles Bonnie Charles

1991 Louisiana Gubernatorial Race

• Former KKK member David Duke ran a close second to Edwin Edwards, a candidate long suspected of corruption.

• In Louisiana’s open-party, multiple-candidate, plurality-with-elimination system, the top two candidates advanced to a runoff.

• In a case of “better the devil you know,” most voters flocked to Edwards (a candidate they distrusted) in the runoff in order to block the more embarrassing Duke.

Candidate 1st-round votes 2nd-round votesEdwin Edwards 523,096 1,057,031

David Duke 491,342 671,009

Buddy Roemer 410,690

Others 124,127

1991 Louisiana Gubernatorial Race• Similarly, many Duke supporters distrusted

Edwards, but could not gather enough support to defeat him directly. Suppose instead that 81,000 of them had stayed away from the polls.

• With Duke out of the runoff, voters would have felt comfortable voting for the safer Roemer; polls indicated that this would have occurred by a wide margin.

• Thus, David Duke’s supporters could have done more to defeat their chief opponent by not voting!

• A similar result occurred in France in 2002.– Far-right nationalist candidate Jean-Marie

Le Pen upset the Socialist challenger to the conservative president Jacques Chirac to force a runoff, in which the 3rd-place Socialists were compelled to vote for their archrival Chirac.

– Le Pen voters could have defeated Chirac by being willing to finish in third place.

Candidate 1st-round votes (rev.)

Edwin Edwards 523,096

Buddy Roemer 410,690

David Duke 410,342

Others 124,127

A reusable ballot

• One way to avoid the time and expense of a runoff round of voting is to use ballots that allow voters to express their levels of preference for each candidate, by ranking them in order.– These “preference ballots” can then be reused as needed,

since they are presumed to express the widest possible intention of the voters in a variety of scenarios.

– The ballots could be used as a preemptive runoff ballot, with voters getting either their original 1st choice (if still in the running) or their highest-ranked runner-up still in the running. This is also known as instant-runoff voting.

Winner take all?

• Unfortunately, any plurality-based voting system rewards only the winner, with the runners-up receiving no credit for being a “close second.”– It also makes little distinction between a runner-up in 2nd place

and a runner-up in 10th place.• Both were simply “less favored” than the 1st choice, who typically

was the only one who got that person’s vote.

• A weighted system that awarded each candidate points based on how high they placed in the preference ballot would acknowledge how close the 2nd-place candidate was to the 1st-place candidate (and how not close either of them was to the 10th-place candidate).

The Borda count method

• Jean-Charles de Borda (1733-1799), French mathematician and mariner

• If there are n candidates in an election, each voter ranks their preferences in order from 1st place through nth place.– The candidate in 1st place receives n points.

– The candidate in 2nd place receives (n – 1) points.

– The candidate in 3rd place receives (n – 2) points.

– And so on, until the candidate in nth (last) place receives 1 point.

• Then all such points are totaled for each candidate, and the one with the highest point total is the winner.

Bigger than an election…

• Most sports polls use the Borda count method.– The “top 25” teams are ranked based on ballots from

sportswriters, coaches, etc., ranking their personal top 25 choices.

– Each 1st-place team receives 25 points.– Each 2nd-place team receives 24 points.– (And so on…) Each 25th-place team receives 1 point.– The weighted averages are then used to rank all

teams in order.– Typically, as many as 40 or 50 teams may receive

votes (points) from at least a few voters.

NCAA Women’s Preseason PollESPN/USA Today (31 writers)

Team/Rank Points Team/Rank Points

1. Connecticut 773 (30 1st-place votes, one 3rd-place)

18. Arizona State 215

2. Stanford 676 19. Ohio State 188

3. Rutgers 639 20. Old Dominion 163

4. North Carolina 629 21. Auburn 154

5. Maryland 624 22. LSU 133

6. Tennessee 602 (one 1st-place vote) 23. Pittsburgh 104

7. Oklahoma 600 24. Florida State 101

8. Duke 508 25. Georgia 94

9. California 495 26. Xavier 87

10. Texas A&M 493 27. Purdue 85

11. Louisville 481 28. Kansas State 72

12. Vanderbilt 419 29. Iowa State 63

13. Texas 315 30. Marist 48

14. Notre Dame 294 … …

15. Virginia 279 41. (3 teams) 3

15. Oklahoma State 279 44. (4 teams) 2

17. Baylor 272 48. (5 teams) 1

A method from very close to home• The Bucklin method: a multi-stage variation on

the Borda count method– Voters rank their choices on a preference ballot.– The preferences are considered one level at a time.

• Initially, only the 1st choices are considered. If one candidate receives a majority of them, that candidate is the winner.

• If not, the 2nd choices are added to each candidate’s total.– There are now twice as many “votes,” cast by the same

number of voters.– If any candidate now has the support of a majority of voters,

that candidate is the winner.

• If multiple candidates surpass a majority in the same round, the one with the highest level of support is the winner.

• This process repeats through as many stages as needed.

– Invented by James W. Bucklin (1856-1919), co-founder of Grand Junction.

The pairwise-comparison method• Also known as Copeland’s method• Voters first rank all candidates with a preference ballot.• Then, for each combination of two candidates, we consider their

head-to-head record solely against each other.• If candidate A was ranked higher than candidate B on a majority of

the ballots, regardless of by how much, then A receives 1 point.– If each one was ranked higher than the other by the same number of

voters, then each one receives ½ point.

• After all possible combinations are compared, the total points awarded to each candidate are summed up.– The total of all points awarded will be the total number of matchups

(which will be nC2, a fact we can use to make sure we haven’t missed any matchups).

• This is also the round-robin method used in many competitions, in which the team(s) with the most head-to-head victories advance.

Another pairwise-comparison method• One alternative version is the Kemeny-Young method.• Voters again rank all candidates with a preference ballot and tally

their head-to-head record for each combination of two candidates.• This time, we keep track of how many voters ranked candidate A

higher than candidate B on the ballots, not just “who was higher.” • For each of the possible n! permutations of the rankings of the n

candidates, we tally the percent of voters who preferred each of the n candidates to each of the ones below them.– Example: For ranking ACB (1 of 6 A/B/C ones), find how many

preferred A over C, A over B, and C over B. (no other comparisons)

• After all of those matchups are compared, the sum of all the percentages calculated gives the Kemeny sequence score.– Whichever specific sequence of the n candidates has the highest

Kemeny sequence score is decreed to be the “winning sequence.”• Its rankings are considered the aggregate voters’ 1st choice, 2nd choice, etc.

Bad news from France• In the 2007 French elections, François Bayrou

would have likely defeated all of his opponents using either the Borda count or pairwise-comparison methods.

• Condorcet’s criterion: if candidate X can defeat all other candidates in a head-to-head match, then X deserves to win the election.

– Marie Jean Antoine Nicolas Caritat, the Marquis de Condorcet (1743-1794), French mathematician

• Clearly, anyone who is ranked first on all ballots will win the election.

– They will automatically win every head-to-head matchup, for an unsurpassable total of (n - 1) points.

• However, it is possible for a candidate who would win using the plurality method to lose using the Borda count method, if they are ranked low on many other ballots.

Using the plurality method

• For each column above, the heading tells us how many voters submitted an identical ballot of that type.

• Using this method, Helen would be declared the winner.

Preference 15 voters 7 voters 13 voters 5 voters 2 voters

1st Helen Eddie Grover Donna Eddie

2nd Donna Donna Flora Grover Flora

3rd Eddie Grover Eddie Flora Grover

4th Flora Flora Donna Eddie Donna

5th Grover Helen Helen Helen Helen

Using the plurality-with-elimination method

• Suppose that only Helen and Grover, the two candidates with the most number of “1st choice” votes, advance to a runoff.

– Barring any unforeseen change of heart, the likely result for the runoff would be:

Preference 15 voters 7 voters 13 voters 5 voters 2 voters

1st Helen Eddie Grover Donna Eddie

2nd Donna Donna Flora Grover Flora

3rd Eddie Grover Eddie Flora Grover

4th Flora Flora Donna Eddie Donna

5th Grover Helen Helen Helen Helen

Preference 15 voters 7 voters 13 voters 5 voters 2 voters

1st Helen Grover Grover Grover Grover

2nd Grover Helen Helen Helen Helen

• Helen is “hated” by most voters, but her supporters are numerous enough to get her into the runoff, where all other voters flock to Grover, who would be declared the winner using this method.

Using the Borda count method

• Donna would be declared the winner using this method.

– She was the top choice for very few voters, but was an acceptable alternative for many others, whose combined support increased her total.

• Helen suffers the most, a victim of the low “weight” she is given by most voters.

Preference 15 voters 7 voters 13 voters 5 voters 2 voters

1st (5 pts.) Helen (75) Eddie (35) Grover (65) Donna (25) Eddie (10)

2nd (4 pts.) Donna (60) Donna (28) Flora (52) Grover (20) Flora (8)

3rd (3 pts.) Eddie (45) Grover (21) Eddie (39) Flora (15) Grover (6)

4th (2 pts.) Flora (30) Flora (14) Donna (26) Eddie (10) Donna (4)

5th (1 pt.) Grover (15) Helen (7) Helen (13) Helen (5) Helen (2)

Candidate Total Points

Donna 143

Eddie 139

Flora 119

Grover 127

Helen 82

Using the pairwise-comparison method

• Consider the ten possible head-to-head matchups:

Preference 15 voters 7 voters 13 voters 5 voters 2 voters

1st Helen Eddie Grover Donna Eddie

2nd Donna Donna Flora Grover Flora

3rd Eddie Grover Eddie Flora Grover

4th Flora Flora Donna Eddie Donna

5th Grover Helen Helen Helen Helen

Cand. Votes Cand. Votes Cand. Votes Cand. Votes

Donna 20 Eddie 22 Eddie 24 Grover 18

Donna 27 Flora 15 Eddie 27 Helen 15

Donna 27 Grover 15 Flora 17 Grover 25

Donna 27 Helen 15 Flora 27 Helen 15

Eddie 24 Flora 18 Grover 27 Helen 15

Using the pairwise-comparison method

• Eddie would be declared the winner using this method.– He was perfectly balanced, finishing

ahead of Donna and Helen on roughly half of the ballots and ahead of Flora and Grover on the other half.

– With the “margin of victory” irrelevant, he had enough narrow head-to-head victories to win all four matchups.

• Helen again suffered from being the bottom choice for most voters, insuring she would lose all possible matchups.

Candidate Total Points

Donna 3

Eddie 4

Flora 1

Grover 2

Helen 0

Recap of the “Four-Method Faceoff”

• majority method: no immediate winner• plurality method: Helen wins• plurality-with-elimination method: Grover wins• Borda count method: Donna wins• pairwise-comparison method: Eddie wins• Only Flora, a mediocre candidate in all voters’

minds, fails to win under any of these methods.

Recap of the “Four-Method Faceoff”• Each method yielded different results because each was

violating a different “fairness criterion.”

• So...which voting method should we use?

Violates majority criterion?

Violates Concorcet’s criterion?

Violates indep.-of-irrelevant-alt. criterion?

Violates monotonicity criterion?

Plurality method No Yes Yes No

Plurality-with-elimination

No Yes Yes Yes

Borda count method

Yes Yes Yes No

Pairwise-comp./ Kemeny-Young

No No Yes No

More bad news…• Once again, we ask the question, “Is

there a best way to do this?”• Once again, we receive the unfortunate

answer, “No.”• Arrow’s Impossibility Theorem: In

any election involving more than two candidates, there is no voting system that will satisfy all four of the “fairness criteria.”– Kenneth Arrow (1921- ), RAND Corporation– youngest-ever winner of the Nobel Prize in

Economic Sciences, at age 51• won for his work in equilibrium theory

More Equal Than Others• “All animals are created equal…but some animals are

more equal than others.”– George Orwell, Animal Farm

• Under weighted voting systems, some voters receive “supervotes” that count more than others.– U.N. Security Council: 15 members, 5 of which have a veto

• All 5 founding members must vote yes (or abstain) for any measure to pass.

– Juries: in criminal cases, a single holdout can prevent a decision

• Each of the jurors thus has veto power, able to prevent a verdict either way.

– Stockholder meetings: preferred-stock shares, which grant heavier voting power…for a premium share price

– Dictators: their single “vote” outweighs all others combined• Roman Empire: the tribune’s cry of “Veto!” (“I forbid it!”) could overturn any

decision of the Senate

Building a Coalition

• quota: the number of votes necessary to pass a measure

• Whether the voting system is equally weighted or not, the voters can attempt to form coalitions to consolidate their block of voting power.– If their combined voting power is greater than or equal

to the quota, then it is a winning coalition.– In systems where a supermajority quota is needed,

minority blocs can still form a blocking coalition capable of preventing a measure from passing.

• U.S. Senate: 41 members can sustain a filibuster

Keeping a Coalition

• A voter or group whose participation in a coalition is necessary for it to remain a winning coalition is called a critical voter.

• A voter or group could thus proclaim its importance by calculating how often it is the critical component of a coalition.

• John F. Banzhaf III (1940- ): GWU law professor

coalitions winningin critical is

is pvoter/grou times of #

coalitions winningin critical

is pvoter/grou a times of #

=index power Banzhafanytotal

Canadian House of Commons• With just short of a majority, the

Conservatives can theoretically choose any of the other three major parties to ally with for a ruling coalition.

• They are thus critical in every winning coalition except the triad coalitions of the other three (163 MPs total).

• However, so are any of the three when only one works with the Conservatives, or when the three band together.

• Winning coalitions: {CL}, {CQ}, {CN}, {CLQ}, {CLN}, {CQN}, {CLQN}, {LQN}, and each with either/both of the other 2– (i.e., 4 versions of each of these 7)

Party MPs

Conservative 143

Liberal 77

Québécois 49

NDP 37

Others 2

Total 308

Party BPI

Conservative

Liberal

Québécois

NDP

%50=4824

%.616=488

%.616=488

%.616=488

The power of electoral votes

• Theoretically, there are 251 possible Electoral College outcomes if we only include two parties as potential winners.– Within these, there are more than 9 trillion possible

winning coalitions to consider.– With computer assistance, we can calculate the

Banzhaf power index of each state and D.C.• California: critical in 11% of winning electoral-vote totals• Texas and New York: critical in 6% of winning combinations• Colorado: critical in 1.5% of winning combinations• 7 least-populated states and D.C., each with only 3 electoral

votes: critical in 0.55% of winning combinations

A new Electoral College?• The Constitution mandates the use of the Electoral College, so

amending it would require a ⅔ vote in the House and Senate, plus the consent of ¾ of the state legislatures (38 of 50!).– However, the manner in which each state names its electors is up to the

legislatures.

• Many proposals and referendums have been made for states to award their electoral votes by Congressional district, with the winner of the state as a whole receiving the 2 additional electoral votes.– Maine and Nebraska are currently the only states to do so.

– This would mean battleground states were split almost evenly.• Would candidates visit if they knew they would receive half of the votes either way?

– It would also give minority-party voters in lopsided states a voice.• Candidates with a minority of support would still have a reason to campaign there.

• California wouldn’t be “quite so blue” of a state, Texas not “quite so red.”

• Thus, for now, same-party state legislatures in “very red” or “very blue” states have little incentive to end their winner-take-all advantage.

– Under this system, Bush would have beaten Gore 289 to 249.

Other voting dilemmas

• Presidential primary season– Statewide party votes staggered throughout the winter and

spring of election years– Intense focus on early primaries (NH, IA, SC), with very little on

states late in the primary season (winners chosen before then)– 2008: historical exception, with 10 candidates from each party

• Democratic candidate Obama: did not achieve majority of convention delegates until final primaries in early June

• Republican candidate McCain: achieved majority of delegates relatively early

• Candidates dropped out quickly if they found little early support– Lower turnout in Republican primaries once the contest was “over”– Later ballots would include only 2 or 3 of the 10 original candidates– Republican voters in late-primary states felt robbed of their chance to

vote for their preferred candidate

A method to please everyone

• Approval voting: candidates are not ranked; instead, voters can cast a vote for as many of the candidates as they like.– The winner is the candidate receiving the most votes of

approval.– Encourages the election of consensus-building candidates

acceptable to all, rather than the ones who can build the largest core group of supporters

– Can reduce infighting, since candidates do not have to take support away from others to gain it for themselves

– Used to select leaders in many “collegial” organizations• Mathematical Association of America• National Academy of Sciences• Many universities• U.N. Secretary-General• Boy Scouts of America

What have we learned?• Much like allocating a group of representatives, deciding how they

will be elected is not easy.• In yet another world of paradoxes, contradictory results, and

unhappy candidates, no method of voting is perfect.• Depending on the voting system used, a candidate can end up with

a narrow victory, a crushing landslide, or a stunning defeat, all with the same vote tallies.

• Even in our “two-party” system, smaller third-party candidates can have a major mathematical impact.

• Under our current Electoral College system, large states get more attention because there are more ways they can end up on the “winning side.”– If their electoral votes were split up, the “balance of power” might

change. (Of course, that’s why it probably won’t be changed.)• And don’t even get me started about Parliamentary systems.

References• Most liberal-arts college-mathematics course (ex.: MATH 110) textbooks

– Including ours, Thomas L. Pirnot’s Mathematics All Around, 3rd edition

• The Alfred Nobel Foundation: www.nobelprize.org• Banzhaf Power Index analysis of the Electoral College (2 studies):

– www.everything2.com/node/1678149

– www.cs.uiowa.edu/~dsidran/2000%20Power%20Indices.htm

• Banzhaf’s own website: www.banzhaf.net• National Archive 2000 Election Returns:

– www.archives.gov/federal_register/electoral_college/votes/2000_2005.html#2000

• Analysis of 2000 Presidential election given different House sizes:– www.thirty-thousand.org/pages/Neubauer-Zeitlin.htm

• Interactive electoral maps, both historic and modern: www.270towin.com• The story of GJ’s own James W. Bucklin: www.gjhistory.org/cat/main.htm

– And look for his Grand Junction charter and voting system debut at books.google.com!

• And yes, of course, Google and Wikipedia.• My Mesa State homepage, at www.mesastate.edu/~mcrogers, will have this

presentation plus the previous one.