Different uses for voting
need different types of voting.
Voting systems and election rules for Accurate Democracy

Teacher's Guide to
Tabletop Tallies

Introduction to voting systems, chapter contents

Tabletop voting is a hands-on experience for students who like to learn by doing.  The games are simple and memorable introductions to great voting rules.

The purpose is to see a fair-share tally organizing voters into a multi-winner decision.  Concepts to learn include a transferable vote and a winning threshold.

Voting rules are frequently presented as the most abstract of concepts: as mathematical proofs or ethical polemics.  Teachers usually connect the rules to current events but not to a student's physical activity and senses.  Making logic and fairness tangible helps many students gain a feel for these tally rules.

This workshop demonstrates three rules that use transferable votes.  It starts with the simple, one-winner rule which leads to multi-winner rules for electing a committee and for selecting projects.  Sections on setting budgets or enacting a policy may be included if time and attention permit.

Young students might do one voting rule each day for a week. 
Mature students can complete all four tallies in less than two hours.

The one-page handout on transferable votes is available in several styles.  The single sheets print on letter-size paper.  The booklets is made to print on legal-size paper then cut across, stack and fold.  The Primer includes the handout and much more.

Styles:.PDF .DOCOther
Dot Lists
One sheetIRV, STV…HandoutWeb
BookletIRV, STV…BookletSlide show
One sheetIRV, STV…Class Web

You may download a large-font edition in Acrobat IRV STV Big.pdf, or MS Word IRV STV Big.doc.

You also may download a full-choice ballot, free software, and voting cards formatted in Microsoft Word 2000 or Excel.)

{Preview: After lunch we’ll have a workshop on multi-winner democracy.  You will be voting for [home-made] cookies  — and you get to eat the winners! (Pepperidge Farm cookies  come in many chocolate flavors and in small bags so a class can eat several winners:-)

The Tally Board

This "vote transfer board" has a column for each candidate.  Each voter gets a "card" to cast in the column of his favorite candidate.  A candidate wins if her column of cards reaches a finish line.  (For simplicity and clarity, candidates have feminine pronouns while voters have masculine.)

Crowding around a vote transfer board may limit tabletop voting to groups of a few dozen voters, with each dozen taking a turn to place their initial votes.

Each card must have a colorful ID label so its owner can find it quickly and transfer it himself.  This helps students get a feel for transfers.  But it should not let one voter move another's card.
{The ballots or cards must be big enough for people in the back row to see which candidate's column has the fewest cards after all voters have placed their cards on the board. 

{Our tent-shaped paper cards fit the columns of an accordion-folding window shade.  /\ 1" square.  -or-

{You can use dowels for columns and alphabet blocks for cards.  For elections and projects, a voter slips his card over the top of a dowel and slides it down, securing it until that column wins or loses.  For agency budgets he puts each card between a pair of dowels so he can remove it quickly.

{The Equipment Guide suggests a variety of vote-transfer boards and has pages of cards to download.  Software from the Tool page can quickly tally these and other voting rules.

{Informed Choices

{Introduce the concepts of options and information as necessary foundations for democratic decisions.  An analogy to shopping is often used.

{What would you say if a convenience store offered this choice of cookies: Honey Grams or Cinnamon Grams? Not a very good store is it.  Would a choice of Republican Grams or Democrat Grams be more satisfying?  (It might also offer some expensive cookies in odd flavors, but that wouldn't give many people what they want.)

Electing One Winner by IRV

For Instant Runoff Voting (IRV) the board's finish line marks the height of half the cards plus one.  The winner is the candidate whose cards reach this quota or "winning threshold".

{Notice that this is the quota used in runoff elections.  The finish line works best when it is just a half vote above 50% for IRV or above 25% for 3-seat STV.

If no candidate wins half the cards plus one, the candidate with the fewest is eliminated.  (A dice or coin toss can break ties. )  A voter who supported that candidate now has to move his card to his next choice: he "transfers" his vote.  Cards are counted again and this step is repeated until...  The candidate whose cards reach the finish line wins.

IRV is used to elect the President of the Republic of Ireland, the Mayor of London and the Australian lower house.  It is used to elect student officers at over 20 top American universities including  Carlton, Cal Tech, Duke, Harvard, Johns Hopkins, MIT, Rice, Reed, Stanford, UC Berkeley, UC Davis, UCLA, Vassar, William & Mary, and the Universities of:  Illinois, Maryland, Washington, and Wisconsin.

By organizing voters, IRV avoids "spoiler" candidates and the lesser-of-two-evils choice, costly runoffs and winners-without-mandates.

{To show the affect of voting districts, tally the plurality winner for each dozen voters as they place their initial votes.  In homogeneous groups, students can help the demonstration if some vote the opposite of their true feelings.
{A better way to break ties:  If two candidates tie for last place, the one with the fewest votes in the previous round is eliminated.

{Discuss Transfers

{Not all of these candidates can win.  In fact, only one can.  But if someone wins by a plurality (not a majority), most voters will realize they did not vote for her.

{If we decide to eliminate the candidates one at a time, which one should we eliminate first?

{Say this candidate looses.  Do her voters have to lose? Not if we let them give their votes to their next choices.

IRV Questions

Can more than one candidate reach the 50% +1 threshold? {No.}

Can your second-choice vote hurt your first choice?

Is a transferred vote bigger than any other vote?  {No, they each count as one vote; the cards are all the same size.}  So does this voter have more votes than that voter?  {No.}  More power?  {No.}

Would you want your ballot to support your next choice or to be thrown out?  {Is it more fair to throw ballots out or transfer them to each voter's next choice?

How could your group use Instant Runoff Voting?

{Please write your answers on the back of your ballot.

Prove it makes a difference: ask, "Who won by plurality rule? "

A sample ballot is pictured below.  Its issues include dinner-party music, favorite videos and snack foods.  Optional topics include, group vacations, pizza toppings and ice creams.

{Other voting topics: past presidents, sports stars, books, movies and movie stars (academy awards).  (We may expect a multi-winner rule to elect both men and women, to reward both "guy" movies and "chick" movies.)  The print version of Tabletop Tallies has a ranked choice ballot.

{It is best to use voting on music [food] with the MMV rule because it is public if everyone must hear it.  Of course, the distraction of playing music [serving snacks] can wait until after showing the budget and Pairwise rules.
{Most of the issues above do not demand anonymity, so voters can place their cards openly.  But the goods are more private than public, therefore not strong examples of public issues.

For anonymity, students may put their ballots in a box, stir them, and pull out other ballots.  The box may contain "mail-in" ballots which may make the electorate more diverse.

{A class of 16 or less might give students their own cards plus cards to move according "mail-in" ballots.  This makes the electorate larger and more diverse so the transfers and winners are more true to life.  It also shows students how the paper ballot works toward helping a winner.

Electing a Council by STV

For a three-seat election by Single Transferable Vote (STV) the finish line marks the height of one quarter of the cards plus one.  {Droop quota}  To win a seat, a candidate's cards must reach this line.

{The Droop Quota to win a seat is:
Number of votes / (Number of seats + 1) + 1 vote.
Notice that this formula  applied to a one-seat election gives 50% +1.

{The history and logic of important quotas or  thresholds  for winning seats are explained in "The Single Transferable Vote" by Prof. Nicolaus Tideman in the Journal of Economic Perspectives. Winter 1995, 27-38.  This article is now available in pdf and text formats.

{Reusing the IRV ballots may show how STV3 rewards a larger percentage of voters than a one-winner rule used {3 times can.  (51% can win all {3 seats under Bloc Vote or in {3 plurality districts.  But {75% are needed to win {3 STV seats.

A voter may not give a card to a candidate who has crossed the finish line.  (So there are no "excess votes" to transfer.)
{If you want to transfer excess votes, you can use thin cards to represent the partial weights transferred from a winner to each voter's next choice.

The weakest candidates are eliminated one at a time and students move their votes until three candidates win!

STV is used in Australian {as Preference Voting} and Irish elections {as Single Transferable Vote}, in the Church of England, and at universi¬ties including Oxford, Cambridge, Berkeley, Harvard, Princeton, Vassar and Whitman.  It increases choices for voters turnout of voters, elects more women and gives each group their fair share of representatives.

STV Questions

Can 4 candidates reach the 25% +1 threshold? {No.}
What is the threshold for winning 1 of 5 seats? {20% + 1}
Can your second choice hurt your first choice? {No.}
Where could you use the Single Transferable Vote? {How could it help your life?

Prove it makes a difference: ask, "Which items would have been elected by a plurality? Which items were elected by a minority?"

Funding Projects by MMV

Movable Money Votes (MMV) give a costly item several columns to fill.  If each column represents $100, a project that costs $200 needs to fill 2 columns.

The bylaws might say an item needs moderate support from at least 8 voters to prove it has broad public support for public funding.  Columns are therefore 8 cards tall.

And because every 8 cards will fill a $100 column, we hand out 8 cards for every $100 in the fund.  Those cards can fill just enough columns to use up the fund.

Each voter receives a set of cards to represent his share of the funds.  You may place only one card in a column.  This prevents you from dumping all your cards in one column, so it keeps the threshold for public money at 8 moderate backers.

A voter might receive two single cards and one double card to represent his [$50] share of the overall budget for projects.  His usual strategy is to put the double card vertically in his favorite column to lift that project quickly.  This way, four enthusiastic supporters can fund one low-cost project.  Enthusiasm and number of voters both count.  {Enthusiasm is limited by the size of the largest card; a number of voters is required by the height of the columns.

{Arranging the voter’s cards by size defines a "utility curve".  The chapter about voting for projects graphically shows this.  The software ballots let voters create or choose their own curves which can be smooth continuous functions rather than stepped functions.  The width of an item always represents its cost (e.g. columns); the height is its number of votes (e.g. single or double cards), and the area under the curve, or integral, is the money offered to an item.

A project is funded if supporters fill all its columns.  The "banker" removes or hides those cards.

The teacher removes unpopular snacks one at a time.  First we drop snacks that cost more than all visible cards.  If none is so costly, we drop the one with the lowest fraction of its columns filled.

When a favorite is threatened, you may try to save it by briefly re¬moving your cards from lower-choice items.  {You may want to omit this step because it slows the tally and rarely changes the result.

If 1 of yours looses, move your card(s) from it to your next choice.

Voting ends when all remaining projects are funded fully.  (Any leftover cards would go to an emergency fund, an endowment or other politically neutral uses.)  {But leftovers may not go to tax refunds because that would lead groups advocating smaller government to try to manipulate the process to produce many leftover votes.}

Only a few snacks can win, but all voters can win something!

MMV Questions

1. Can your second choice hurt your first choice?

2. Should we drop the item that has 1) the lowest percentage of its columns filled, or 2) the greatest number of empty spaces?

[ Percentage is correct. What is fair to both expensive and low-cost items? (If number were used, it would cause sponsors to break big projects into many low-cost items.  And should a big project supported by many people be canceled if it needs 1 more card than a low-cost project?) ]

[ Students might ask, "Which is furthest from winning?" When choosing which candidate to eliminate, we would like to know each item's chance of getting more support in the next few rounds of transfers. The past rate of support is the simplest measure. A big item that has been getting offers a fast rate, relative to its cost, has a good chance of getting more in the next few rounds of transfers. ]

Prove it makes a difference: ask, "Which items would have been elected by a plurality?  Which items were elected by a minority?"

{How many moderate backers does a $2 item need? $4? $6? $8?
  {8, 8, 16, 16
{How many enthused backers does a $2 item need? $4? $6? $8?
  {4, 6, 6, 7
{These numbers result from the set of cards we give to each voter, not from the basic logic of MMV.

2b) Would a big contribution plus seven $0.01 contributions show real support to prove the item is a public good?

3. When paying for public goods, should a person's taxes equal his or her benefits? When paying to run a government, should a person's taxes equal his or her benefits? Should people who pay more taxes get more power to say how public money should be spent? to set public laws?

{Does paying taxes give a person moral superiority? Does the amount of tax paid measure moral judgment or worth? How do taxes relate to setting laws that rich and poor alike should obey.  Should industry leaders control regulations which protect the general public from industrial dangers?

{In the late 1800s, most German towns only let taxpayers vote.  They were divided into three groups, each of which paid one-third of the taxes.  The richest one-third elected one-third of the town council.  In one town, the factory owner appointed one-third of the council; his factory paid a third of the taxes so his handpicked board of directors selected another third of the council; and finally the other taxpayers elected the last third.

{Conservatives want to replace public welfare with private charity.  Does that give the rich more power?  Does it give the poor less power?

4. Do you belong to an organization that could use Movable Money Votes?  {Uses for MMV are discussed in MMV Cases.)  MMV is new, try it!  Make history by showing the world a more accurate democracy.  Lead by example!

The introduction to voting on projects describes some merits of Fair-share Spending by MMV.

Adjusting Agency Budgets with BRV

Budget Refill Voting (BRV) has several columns for a costly department to fill, as MMV did for a costly project.  But an agency cannot be eliminated.  A supporter’s cards raise its budget.

Each column of an agency begins the tally from a starting line $100... below a goal line — which may be last year's budget or one set by a Median Voter Process.  Supporters may push an agency above its goal line, but its gain will be another's loss.

Let's say 20 voters want to budget 4 small agencies with 1 column each plus 3 with 2 columns each.  They decide each column needs moderate support from at least 10 voters to restore its previous funding.  So a column needs 10 single cards from 10 voters to reach its goal line – or 5 double cards from enthusiastic voters can maintain fund¬ing for a small agency.

In this example, each voter gets 5 cards.
(That’s because 10 columns × 10 cards each = 100 cards needed to refill all columns.
And 100 cards / 20 voters = 5 cards for each voter.)
You may get one double and four single cards.

As a voter, you should set a target budget for each agency and rank your priorities.  As a budget nears your target, its priority likely goes down, and at some point you’ll want to move your cards from it to your next under-funded priority.  Reacting to other voters is the key to success!   {This makes us use cards designed slide down a column to fill in when someone removes cards below.  This is why BRV cannot use one-time paper ballots; voters must respond to each other’s moves.

Voting stops when a hidden timer sounds and voters lose cards that are not on the board.  This deters people from faking votes until a last moment switch.  A two-thirds majority may reopen voting.

{There are two basic and opposite strategies.  First, delay your vote, waiting to cast the last votes; they can't decide winners and losers but they can decide relative winners and losers.  On the other hand, elected reps will want to tell voters they cast the first votes for popular budgets — for instance, the tax-cut and deficit-reduction columns.

{A more hard-headed rule lets voters push an agency down with negative cards, and push overall spending up or down.  (If the rules do not let a voter change overall spending, he must use an equal number of positive and negative cards.)  Red columns for negative cards are paired with black columns for positive cards.  An agency starts at its goal line, not below it.  If the agency’s black columns are filled higher than its red columns, its funding has increased by that difference; it is "in the black".

{The tabletop BRV rule is not in the family of elimination rules that includes IRV, STV and MMV.  Instead, it is simplified from an Influence Point rule developed from Hylland-Zeckhauser's rule and implemented in software for this web site.

BRV Questions

Does each voter have transferable votes? {Yes.}
Do agencies have a winning threshold? {No.}
Can your second choice hurt your first choice? {No.}
Where could you use Budget Refill Voting? It's new, try it!

{Does BRV motivate sponsors to merge unpopular little agencies, each of which has less than [10] supporters?  That would let each supporter dump all his cards in the super agency’s columns, in effect giving all his money to his favorite sub-agency.  "Omnibus" spending bills already are routine in American legislatures.  BRV makes each supporter more visible and therefore accountable to voters.

{Prove it makes a difference: ask, "Were any items increased by a majority? Which items were raised by a minority?"

The cards and columns can be on spreadsheets.  On-line "cards" are easier to move than real cards and any number of voters fit at the "board".  Tally software also makes the STV or MMV transfers easy and quick.  A voter just ranks the options once and the computer does the rest.

Pairwise Demonstration

This seating chart of seven students can produce the numbers in the handout's Pairwise table.  This small group of voters is slightly and evenly polarized.  But there is one voter with a proposal near the middle.  Condorcet's Pairwise rule will enact that policy.  The swing voter is in seat 33.  That student's position will center the final decision.

The teacher can tell selected students, “You are a voter.” or “You are a voter with a proposal.” Hand each voter a piece of yellow paper to raise when voting.  Each proposal could be a piece of paper, folded as a tent, with a large letter on each side.

Compared with letting everyone vote, this may seem contrived and therefore unconvincing.  But it is faster.  It lets most students focus on the tally, not on their personal votes or who's on whose team.  And yes, it is certain to show the differences between voting rules.

Seat IDs are column and row numbers such as: 11, 21, and 55.  The gray seats are unoccupied.  The median student sits at 33.
Proposal IDs or flags are upper case letters: A, B, C, and D.

Seating Chart 1a and Proposal Flags
  11   21 31 41   51  
12    22 A 32 42 52
13 23 33 C D 43    53
14    24 B 34 44 54
15 25 35 45 55

Voters Ranks
1st 2nd 3rd 4th
13 B A C D
22 A C B D
24 B C A D
33 C D B A
42 D C A B
43 D C B A
54 D C B A

“Are you closer to flag A than B?
If so, please raise your hand.” 
Then test A against C, etc.
Put each total in the table.

Pairwise Table
against  A B C D
for A 2 2 3
for B 5 2 3
for C 5 5 4
for D 4 4 3

Option C wins its three pair-wise tests.  So it is the Condorcet pair-wise winner.  Three of the options win at least one majority.  But only one can win over-all majorities.
The plurality winner is D with 3 first-choice votes.  The IRV winner is D.  (Whether we drop B or C after the first count, the other loses the next count.)  Thus the Condorcet winner is squeezed out in an early round.

Here is a seating chart to show a centrist proposal with narrow appeal losing by Condorcet's rule.  For this example, the electorate is more polarized.  Proposal R has a narrow appeal show by a short red ribbon. (4' or 5' depending on seating)  The wide appeal of proposal B is show by a long blue ribbon.

Seating Chart 2 and Proposal Flags
  11   21 31 41   51  
   12 A 22 32 42 52
13 23 33RB 43 D 53   
14    24 B 34 44 54
15 25 35 45 55

Voters Ranks
1st 2nd
12 B R
13 B R
24 R B
33 R B
44 R B
52 B R
53 B R

The wide appeal of B wins this single pair-wise test by four votes to three.
Adding proposals further from the center would not affect this result.

 If the flags are places for a heater in an icy cold room. 
A) Do we put it at our center or in the biggest group?
    {Center it.}
B) Do we turn on its fan to spread the heat wide?
    {Yes, because most voters are not next to the heater.}

Questions on Pairwise Voting

1. Can the middle voter(s) enact any policy alone?
    {No, he alone is not a majority.}
2. Can fringe voters affect the Pairwise result?
    {Yes, they can.}
3. Does it favor narrowly-centrist or widely-popular policies?
    {Broad-majority policies are favored.}
4. Does it favor balanced or one-sided policies?
    {Balanced policies are favored.}
5. Does it eliminate the weakest option and move its ballots?
    {No, Pairwise does not use sequential elimination.}
6. Should a first-choice vote count more?
    {No, not for finding the one option that tops each rival.}

{About Ballots

We want this workshop to teach the differences between voting rules.  So the ballot should produce different results by each tally rule.  Plurality rules are erratic; usually they do not elect a minority representative, but sometimes they do.

The ballot should elect only reps of the majority under IRV.  And IRV should elect another majority rep if the first winner is removed before the tally.  STV should elect one or two reps for the majority and one for the other group(s).  The same is true for MMV.

To make this likely, we choose candidates in two or three categories that will polarize the voters into two or three groups with high party loyalty and little mixing of favorites.

We instruct each voter to give top ranks 1, 2, 3... to options in his favorite category.

Fun ballot issues for IRV or Pairwise include best actress or player and homecoming royalty.  For STV try video and music play lists.  For MMV try field trips, snack and soft drinks, pizza toppings and deserts...

Those who want anonymity may put their ballot in a ballot box and pull out another student’s or a “mailed-in” ballot.  The latter let a teacher add some variety to the electorate for a more surprising tally.

{About Workshops

Some suggestions for your first time leading a workshop:
What does a workshop need in order to change behaviors?
Practice recall: Most people "get it" with movable votes very quickly.  But they don’t use it and so they can’t recall it a month later.  Several techniques increase retention and recall.
Review sections and the whole (using fill in the blank sentences).

Moment of silence
Recall a positive learning experience.

Help them see and fix misconceptions: Before the presentation, ask students what they know, what voting rules they have heard about.

Involve emotions:  How do you feel when you think about ___ .
Motivate, energize, awaken: 
The voting rules in this workshop increase voter turnout and representation by women which increases funding for social issues such as education, child care, heath care, ...
Inspire:  If we want better democracy, we will have to work for it.  Is it worth it?  Think of the value you inherited from the work of Sam Adams, Elizabeth Caddy Stanton, or Dr. Martin Luther King.  Or consider the benefits to India, South Africa and Asia from Gandhi, Mandela, Dae joong Kim, Ang San Suu Kyi, None of those heroes succeeded alone.  They all inspired and needed the creative work of other egalitarian people.
Envision practice:  How could the U.S. use this?  How could your club or congregation use transferable votes ?  Involve imagination, internalize.  Move from classroom to application.

Involve the Senses: Tabletop tallies audio, visual, tactile, kinesthetic, olfactory (treats); Desires: social, altruistic, hunger,
Reward changes with food, optimism, money (opportunity prosperity) friends, neighbors...
Give souvenirs: handouts, business cards, cards, sets of cards, software,

Before a workshop, try to gather a survey.  Ask students to rank categories of music.  "Your favorite is number one, of course.  The second best is number two and so on." Then design a ballot to polarize voters.  The multi-winner rules should select winners for all large groups.

This material is designed to hold adults’ attention through a two-hour workshop.  Classroom teachers might try one voting rule a day for a week.  A fifth day might be a review, discussion or quiz.
Previous weeks might introduce the history and problems of plurality rules including minorities, gerrymander and apportionment, and then the ABC majority rules: Approval, Borda and Condorcet or Pairwise.  The latter conducts an instant, round-robin tournament.


Transferable votes quickly organize powerful groups supporting popular choices.  They are a great improvement over primitive rules when selecting a person, a whole committee, a set of projects or renewed budgets.

These rules strengthen a democracy.  They give “effective votes” to more voters.  So they give stronger mandates, legitimate authority, to the winners.  That means the rules organize voters, and expand the base of power by increasing the number of voters who select and support:
Center  a Chairperson from a plurality to a majority;
Full Rep  a Council from a plurality to over three quarters;
Budgets  a Budget from a few power blocs to all members;
Center  a Policy from a one-sided to an over-all majority.

Movable-vote rules are usually the best way to distribute winners to interest groups.  But this hints that they often are not the best rules for picking the one central policy with the broadest support from all groups.  That goal is the topic of the next workshop on Pairwise Tournament Tallies. Pairwise Condorcet Voting Workshop

Each rule has other web pages detailing its logic, uses and effects.

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