Approval voting

© Bradley Lehman, 11/5/00. Send comments to

Suppose there is a carload of five people needing to choose a place to have dinner together. They agree that all will go to the same restaurant, and only need to decide which restaurant that will be. The entire group has to choose exactly one winner that best reflects the preferences of the people. They decide to vote.

Which method of voting is fairer?

  1. Give each person in the car exactly one vote: choose a restaurant from among the several choices. Those whose vote didn't win are at the mercy of the others. There was no opportunity to say anything about their alternate choices, if they have any.
  2. Let each person vote for as many choices as they would be content with. There is this way a better chance that most of the people will be pleased by the outcome, that they in fact endorsed the restaurant that won. It might not be their first choice, but it's a tolerable choice that was worth voting for. Or, if all their choices didn't win, at least they did the best they could, and their complex opinion was heard even though it didn't win.

An example: four, three, or two restaurants

The choices are McDonald's, Burger King, Wendy's, and Hardee's. The true preferences of the people (interviewing each person separately) are:
  • Alan - prefers Wendy's, can tolerate Burger King and Hardee's equally, can't stand McDonald's.
  • Beth - prefers Burger King, and the others are all the same to her. None are terrible.
  • Carl - prefers McDonald's or Hardee's equally above the other two.
  • Dan - prefers McDonald's, can tolerate Wendy's or Hardee's equally, can't stand Burger King.
  • Evy - prefers Hardee's, then Burger King, doesn't like the other two.

If each person gets exactly one vote, the whole group is at the mercy of Carl who (presumably) chooses randomly between McDonald's or Hardee's. If Carl votes McDonald's, then Alan and Evy (and possibly also Beth) are all unhappy that the group goes to McDonald's. And it's "Carl's fault"...Alan and Evy become upset with Carl. If instead Carl votes Hardee's, everybody is happy except possibly Beth. But it's a matter of luck, since Alan and Evy and Beth never got to express their dislike of McDonald's.

Which restaurant do you prefer?
Burger King      Hardee's      McDonald's      Wendy's     

Now instead, let each person have up to one vote per candidate rather than one per voter. This method is called approval voting: from the list of candidates, each voter simply crosses off any choices which are less acceptable. That is, the voter divides the candidates into two groups: "any of these would be better than any of those." The voter can draw that line wherever s/he wants to; it's even OK to vote for all of them or none of them if that truly expresses the voter's opinion. ("It's all the same to me" is a position.)

Which restaurant(s) do you prefer?
Burger King      Hardee's      McDonald's      Wendy's     
Vote for as many as you wish, until each restaurant you vote for
is preferable to each restaurant you do not vote for.
That is, uncheck any restaurant(s) that are less acceptable
than the remaining checked restaurant(s).

In this example Alan votes WBH; Beth votes B; Carl votes MH; Dan votes MWH; Evy votes HB. Hardee's clearly wins (4 votes out of 5), and also accurately reflects the group's preference that McDonald's is one of the two least popular (yet in the other count, it had a 50% chance of winning!).

Consider only Burger King - McDonald's - Wendy's: Suppose there isn't a Hardee's available, and the same people have the same preferences. With "one person one vote," the group comes up with a tie between McDonald's and Burger King, and has to do further negotiation. (They probably let Alan be the tie-breaker and go to Burger King.) With approval voting, it comes out WB, B, M, MW, B. The group clearly goes to Burger King, and there didn't have to be a second round of voting to determine what happens with Alan's "wasted" vote for Wendy's. And Carl and Dan at least have some consolation: they did the best they could in expressing their opinion against Burger King (even though they lost, they were heard).

Consider only Hardee's - McDonald's - Wendy's: Now instead strike Burger King from the list, so it's only M, W, or H. With "one person one vote" the group is at the mercy of Beth and Carl: Beth choosing randomly among all three, and Carl choosing randomly between McDonald's and Wendy's. (Inspection shows that Hardee's has the least chance overall of winning here. It first has to win Beth's vote, and then it has to win a runoff against whatever Carl chose.) With approval voting, it comes out WH, -, MH, MWH, H. Hardee's wins in a landslide, which accurately reflects the true preference of the individuals! (And Beth's indifference or randomness cannot swing the decision.)

Consider only Burger King - Wendy's - Hardee's: Now instead strike McDonald's. With "one person one vote" Hardee's clearly wins. With approval voting, it comes out W, B, H, WH, BH...Hardee's again wins. In this case the two voting methods deliver the same result.

Consider only McDonald's - Hardee's - Burger King: Now strike Wendy's. With "one person one vote" we're at the mercy of Alan and Carl, with Alan voting randomly between Burger King/Hardee's, and Carl between McDonald's/Hardee's. Hardee's might win, but there could also be a lot of ties. With approval voting, it's BH, B, MH, MH, HB...Hardee's clearly wins.

What about all the head-to-head decisions, two restaurants?

Which restaurant is better? Choose your preference among each pair.
Burger King      Hardee's      No preference     
Burger King      McDonald's      No preference     
Burger King      Wendy's      No preference     
Hardee's      McDonald's      No preference     
Hardee's      Wendy's      No preference     
McDonald's      Wendy's      No preference     

A simple table can be built from the interviews, assessing each restaurant against each other restaurant given only two choices:

Hardee'sBurger KingWendy's
Burger KingH 3-1
Wendy'sH 2-1tie 2-2
McDonald'sH 2-1B 3-2M 2-1
That is, with these five voters, Hardee's beats each of the other three restaurants head-to-head. Burger King beats McDonald's but ties Wendy's. McDonald's beats only Wendy's. Wendy's can do no better than tie Burger King.

Considering all these possible combinations (win-tie-loss), the group's overall ranking of the restaurants is evidently Hardee's (3-0-0), Burger King (1-1-1), McDonald's (1-0-2), Wendy's (0-1-2). And Hardee's is the "Condorcet" winner.

Problems with "one person one vote"

CONSISTENCY: Why should the deletion or addition of a single restaurant make all the rest of the outcomes shift around so much? Notice that all these results have shown us that the group's overall first choice is Hardee's, then Burger King. It seems like a total "duh" that if one of those two restaurants is deleted, the other should win. Yet with "one person one vote", notice what happens. Lacking a Hardee's there is a high probability that Burger King will lose to McDonald's. And lacking a Burger King, Hardee's has the least possible chance of winning! How screwy is that? You drive into a town that has no Burger King, and suddenly Hardee's (through no fault of its own) switches from the highest to the lowest choice!

Or start from the two-way decision of Hardee's vs Wendy's, Hardee's winning. If a McDonald's comes into play, under "one person one vote" observe that each Wendy's and McDonald's have a better chance of winning than Hardee's does. If Hardee's can beat each of the other restaurants individually, why should it lose a three-way decision? Especially when its own strongest competitor, Burger King, is not even in the race?

ABSTAINING: In the example MWH the group is at the mercy of B and C (voting randomly, or abstaining as a refusal to vote randomly). In the example MBH they're at the mercy of A and C (again either voting randomly or abstaining). Knowing that this is true, the people who "don't matter" will be inclined not to vote at all (feeling that their votes don't matter, being deadlocked)...which works only if all three of them don't vote. If only some of them abstain, the whole group's decision is thrown off. But if the method were approval voting, it doesn't encourage any of the people to abstain...except in MWH where person B's true opinion is that all three of the choices are indistinguishable. What sensible system encourages people to vote randomly or to abstain when they really do have an opinion that should count?

INSINCERITY: And what about the business whereby "one person one vote" goads people into insincere voting? In the example where there's no Hardee's available, the person A (afraid that Wendy's can't win anyway) is wise to abandon Wendy's and vote insincerely for Burger King just to make sure McDonald's doesn't win. Meanwhile D, through a misinformed opinion that McDonald's has no chance, throws her vote insincerely to Wendy's to make sure Burger King doesn't win. Again that's totally screwy. Why not have a sensible voting system that lets the people express their true opinions and feel that their vote influenced the election?

Improvements over "one person one vote"

A preference system such as the Single Transferable Vote (Hare method) would take care of some of the problems: it would discourage the insincerity and needless abstaining, and it would solve those problems of ties (eliminating runoffs). However, it would still encourage people such as A and D to vote randomly in writing their "2" and "3" preferences. Under STV, the voter marks as many choices as can be distinguished from the top, and then stops marking the ballot. It does not address the situation where the voter's true preference has a middle tier where choices are indistinguishable from one another, yet above a lower tier.

So, that particular preference system (Hare/STV) is better than a straight "one person one vote" plurality system, but I think it's still inferior to simple approval voting in determining the group's overall choice.

Approval voting itself can be refined in various ways. There could be weighting among the approval votes (unfortunately, this makes it more complicated for the voter). Or, keeping it simple, the group could stipulate that for the election to be conclusive, the winning candidate must have received approval of more than half the voters. (Therefore there would be a difference between voting "all are OK" and "none are OK".) That seems like an obvious goal in a democracy: the outcome pleases more than half the voters. Another obvious goal is that the voters should feel that their sincere preferences can be expressed honestly and without penalty.

And all those voting methods are inferior to decision by consensus, but consensus generally takes a lot more work and is sometimes impossible. If there is voting, at least let it be approval voting.

Consensus --> Approval voting --> Preference voting --> "one person one vote" Plurality


I hope that this restaurant example has shown the obvious inherent flaws in "one person one vote", and the superiority of approval voting. This has also been expressed elsewhere (and better) by political scientists and mathematicians. I particularly like the explanations and leads in Archimedes' Revenge: The Joys and Perils of Mathematics by Paul Hoffman.

"One person one opinion" (as complex as that opinion might be) - not "one person one vote"!

Bradley Lehman, 11/5/00. Send comments to


Addendum 11/09/00

Given the way the US presidential election has supposedly all come down to Florida's decision, with everything else evenly divided, "it's all Florida's fault" if the outcome was thought unsatisfactory. (Florida is like Carl in most of the examples above.) Or "it's all Nader's fault" if his votes changed the Bush/Gore balance and supposedly lost the election for Gore.

No, it's the voting system's fault for not letting the voters affirm more than one candidate. If it were simplified only to Bush-Gore-Nader, there are surely voters whose true opinions fall into each of these three camps:

  • Either Bush or Gore is better than Nader
  • Either Gore or Nader is better than Bush
  • Either Bush or Nader is better than Gore

But there is no way to express those opinions accurately if the voter gets only one vote. With approval voting, the voter can express eight possible opinions:

  • Bush is clearly the best - B
  • Gore is clearly the best - G
  • Nader is clearly the best - N
  • Nader is clearly the worst (i.e. either Bush or Gore is better than Nader) - BG
  • Bush is clearly the worst (i.e. either Gore or Nader is better than Bush) - GN
  • Gore is clearly the worst (i.e. either Bush or Nader is better than Gore) - BN
  • All of them are OK - BGN
  • None of them are OK - { }
And if (for example) Nader would drop out of the race, or if there would be any number of other "spoilers," it would not affect the two front-runners relative to each other. Either Bush gets more approval than Gore, or Gore gets more approval than Bush, or they are equally bad (both are clearly inferior to someone else), or they are equally good (they are indistinguishable from each other, but better than somebody else).

In practice Gore and Bush can individually clobber everybody else except each other, so the whole race comes down to counting the approval given to each of them. Whichever of them gets more approval votes wins. (And it's probably going to be more than 50% of the electorate affirming each of them!... it's nice to be able to say that more than half the voters in an election voted for the winner.)

Meanwhile, the percentages of support for Nader and all the other also-rans show up honestly. If x% of the voters affirm the Green Party (while possibly also affirming something else), approval voting delivers x% of the total, plain and simple. Nader doesn't lose his own support due to voters who decide to abandon him in favor of "making sure Bush doesn't win" or "making sure Gore doesn't win." Such a voter can vote for both Nader and somebody else. If the voter's honest preference is Nader over Gore, and Gore over Bush, and she wants to make sure her vote counts against Bush, it doesn't do any good for her to vote for only Gore and not also Nader.

See how elegant this is?

Approval voting allows eight possible opinions with three candidates; 16 with four; 32 with five; 64 with six; etc. There are 2^N possibilities, where N is the number of candidates. In effect, the voter simply separates the candidates into two groups (of any size) such that "any of these candidates in group A are better than any of these candidates in group B". The voter is still free to put only a single candidate into group A or group B, and everybody else into the other group, if that's the voter's true opinion (i.e. if this is a voter absolutely for or absolutely against one particular candidate). Or the voter can affirm several candidates while voting against several others. The point here is: the voter has the opportunity to express complex opinions.

The voter could hold a clearly hierarchical preference of three groups (or three candidates): "These in group A are clearly the best, those in C are clearly the worst, and there are others in the middle group B." That voter simply needs to decide whether it is more important to affirm only the best, or to treat the worst as a threat and therefore affirm everyone except them. That of course is part of the voter's personal opinion: is it more crucial to vote against the worst or vote for the best? Is it more important to me that A wins, or that C loses? Should I vote for B also as insurance, in case A is going to lose to C? Could my insurance vote for B cause my first choice A to lose to B? But how terrible would it be if B indeed won? This is the only place in approval voting where strategy comes into play, trying to guess what other voters will do and handling the middle layer accordingly...but even within this strategy, each voter is voting for his/her own sincere preferences.

And yet the balloting is simple for the voter: "from this list, cross out any candidates who in your opinion are not acceptable for this office". Or, wording that positively, "from this list, vote for as many candidates as you wish to affirm".

With "one person one vote", the only opinions allowed are "Candidate X is clearly the best" or "none of them". That is only N+1 opinions. All polarized voters are created equal. Every other voter has to compromise with an insincere strategy or a random vote:

  • If one's true opinion is "make sure my vote counts against candidate Y", the only way to express that is to find candidate Y's (perceived) closest opponent, the one most likely to need this vote to beat candidate Y, and vote for that person...even if that person is not your own first choice! That's just silly that the vote-counting method encourages such insincerity, while one's true first choice doesn't get a vote at all.
  • If one's true opinion is that "these m candidates are all better than those n candidates", the best strategy is to pick the (perceived) single most popular candidate in the m group to make sure the vote goes against everybody (and particularly the most dangerous candidates) in the n group. Again insincerity is encouraged: vote for the most likely popular candidate, the one most likely to "beat all the bad guys", not necessarily your own first choice.
  • If one's true opinion is "both these two are better than everybody else, but I can't distinguish between them" the best strategy is either to vote randomly between them or (again) to vote for whichever is perceived to have a better chance to win. What sensible voting system encourages randomness, or basing one's own decision on a projection of what other voters might do?

Shouldn't the voting method simply encourage voters to be true to their consciences, confident that the opinion will count, rather than adopting a game strategy "to make sure the vote counts"?

A good side effect of approval voting

Approval voting would force the candidates to run positive campaigns rather than smearing one another. And the candidates would have to adopt platforms that are centrist enough to please a sufficient number of voters. That is, the candidates would have to focus on serving the majority wishes of the citizens, rather than special interest groups. (Representation of the electorate, and being a public servant as elected: what a concept!)

Assume that every voter can separate the candidates into three groups (as above) as "best - A", "worst - C", and a "maybe - B" group in the middle. (The "maybe" group might be empty for some voters, but in a sense it still exists.) And the voter has to decide whether to vote for group A alone, or to vote also for group B as insurance against a victory from group C...allowing for the possibility that someone in B might then beat the A candidates. The voter has to weigh that danger. The voter then either draws the line of support above the B group (excluding it) or below the B group (including it).

From the candidate's perspective, it is obviously fatal to get into the group C of too many of the voters. If you alienate the voter, s/he will retaliate by voting for everybody except you...this is very bad for you. So it is attractive to get into at least the B group of as many voters as possible, if not the A group. If you make it into the A group, you certainly win a vote. If you make it into the B group, you might still win a vote from that voter. You want to be at least tolerable, to pick up the insurance vote, even if you're not in that voter's first rank.

If you're in a voter's B group, you want the voter to draw her line below the B group so you get a vote. How do you ensure that you pick up that insurance vote? By making it clear to that voter that the candidates in her C group are a major threat against those in her A group, and she needs insurance to be sure C doesn't win. (Her C candidates are not a threat against you, since she doesn't care whether you lose to them or not.) Yet you cannot campaign to smear those other candidates directly, or you will end up in someone else's C group! You will lose the votes of the people who can't stand negative campaigns. Therefore you have to align yourself with the things she likes about her A candidates, showing that your own platform and A's are both clearly better than C's. You have to show that you view C's position as respectable, reasonable, viable, and therefore dangerous, and show that you can do better.

If you're in the B group, you also have to make it clear that it won't be terrible for your voter if you happen to beat the candidates in her A group...otherwise she'll draw her line above the B group and not give you the chance.

So you have to win your A and B support on the merits of your own platform and character. You have to please the voters, showing that you will serve their needs as a good leader. Run a clean campaign, focus on the issues, and show yourself to be the best (rather than trying to show somebody else to be the worst). In their turn, the voters in good conscience will elect you. The approval system ensures that they can vote their consciences rather than adopting insincere strategies.


BPL, 11/9/00

9/9/01: See also my essay against Simple Plurality...