Approval Strategy: Summary and Mathematical Basis

In Approval elections, we'll often know how we want to vote, without considering complicated strategies, and that's a perfectly valid way to vote. The strategies discussed here are intended for those elections in which it isn't otherwise obvious how we want to vote.

Though voting in Approval involves strategy, it isn't more difficult than the strategy that we're already familiar with in Plurality -- except that in Approval no one ever has incentive to bury their favorite by voting someone else over him/her. In Approval, for the first time, everyone would always feel free to fully support their favorite.

If you know how to vote in Plurality, then you know how to vote in Approval.

In Plurality, many people vote for whichever of the two expected frontrunners they like more than the other. I'll call that the Best-Frontrunner strategy. For someone who votes that way in Plurality, a good Approval strategy would be to do the same thing: vote for whichever of the two expected frontrunners you like more than the other, and also vote for everyone whom you like more than that. That includes your favorite. That's the big difference between Approval and Plurality.

When there are two clear expected frontrunners, Best-Frontrunner will probably be the most popular Approval strategy. That will be the case immediately after a switchover from Plurality to Approval.

In this and subsequent Approval strategy articles, I'll discuss some mathematical approaches to Approval strategy. But I emphasize that these approaches are no more necessary in Approval than they are in Plurality. The Best-Frontrunner strategy that I've described, for instance, doesn't require mathematics. For some strategies, some algebra will be used, for those who don't object to that. Approval has something for everyone. Your Approval strategy (like your Plurality strategy) can be as fancy as you like. If you like mathematics, or if you're willing to read some first year algebra, mathematical Approval strategy best brings out the beauty and merit of Approval Voting.

Here are a few more Approval strategies:

As you know, there are many who don't vote, because they feel that there are no winnable candidates who deserve their vote. What if that's true? Suppose there are candidates in the election who are completely unacceptable but who could win.

In that case, Approval strategy is greatly simplified: Vote for all of the acceptables, but for none of the unacceptables. I'll call that the acceptable/unacceptable strategy. It's the one that I (Mike Ossipoff) would use if Approval were used in our public elections.

For Plurality, under those conditions, your best strategy is to vote for the acceptable candidate who is most likely, with your help, to be able to take victory away from an unacceptable candidate.

Notice, under those conditions, how much simpler Approval is than Plurality. And notice how much simpler Approval strategy becomes if the election has candidates who are unacceptable but capable of winning.

All the other strategies described in these articles assume that there are no unacceptable candidates who could win.

Which of these strategies is best depends on what is easiest to estimate or judge. If you have a good estimate of who the top two will be, then use Best-Frontrunner. If not, then there are other approaches that ideally should result in optimal voting strategy.

For instance, in Approval Strategy II, I'll show why, given some reasonable approximations, you can maximize your utility expectation by merely voting for each candidate who is so good that you would prefer having him/her in office instead of holding the election. That will be demonstrated in that article in terms of the mathematics in this article. I'll call that the "Better-Than-Expectation" strategy because if you'd rather have a particular candidate in office instead of holding the election, then you probably don't expect anyone as good as him/her to win.

Better-Than-Expectation strategy is a good idea when there's no clear perception of which two candidates are going to be the top two, but when you have a feel for how good an outcome you expect. For instance, for candidate Smith, you could ask yourself: Would I rather have Smith in office, instead of holding this election? If he's that good, vote for him.

Now I've already named the three Approval strategies that one would most likely choose from in our public political elections:

  • Best-Frontrunner
  • Acceptable/Unacceptable
  • Better-than-expectation

Best-Frontrunner and Better-Than-Expectation are for elections when there aren't completely unacceptable candidates who could win. Most people vote as if there aren't -- as if there's instead a gradation of lesser and greater evils. As I said, Best-Frontrunner is the favorite strategy now with Plurality, and will probably be the favorite with Approval too.

In the rest of this article, I'll describe the mathematics that will be used in the subsequent articles that elaborate on what's been said here, for those who like fancier strategy.

In the next paragraph I'm going to use two words that I haven't defined yet, but I'll define them immediately afterward.

In Plurality, people vote as if their goal is to maximize their utility expectation, based on their utilities for the candidates, and their beliefs about frontrunners. For that reason, utility expectation maximization is the goal of the strategies described in these articles.

Utility simply means value to you, an effort to assign a number to that value. Obviously that number assignment will often be approximate, or a guess. The expectation for an outcome is the probability of that outcome multiplied by its value (utility, money, etc.). The expectation for the overall event is the sum of the expectations for all of its possible outcomes.

Suppose someone says they'll flip a coin, and give you $5 if it's heads, and $3 if it's tails. Your expectation is (1/2)5 + (1/2)3 = 2.5 + 1.5 = $4. The probabilities involved in calculating your expectation in an election are obviously guesses too, for the most part. Based on your estimates of the candidates' utilities, and your estimates of certain probabilities, you can easily calculate your strategy to maximize your utility expectation in Approval or Plurality. I re-emphasize that none of the 3 Approval strategies that I've defined here require mathematics. The mathematics is optional, for more elaborate strategies, and for demonstrating why certain strategies are optimal.

All these articles on Approval strategy shouldn't be taken to mean that Approval needs this strategy more than Plurality or Instant Runoff, or that Approval's strategy is more complicated than that of those methods. The need for strategy and the complexity of the mathematical strategy are about the same in Plurality and Approval, though those two methods differ greatly in the degree of the defensive strategy that they require. Though the promoters of IRV claim otherwise, IRV requires strategy too, in the form of burying one's favorite to help a compromise -- something that Approval never gives incentive for. And IRV's mathematical strategy is much more complicated than that of Approval and Plurality.

Your ballot can change an election result if, when we count all the ballots except your own, there's a tie or a near-tie (the top two vote getters have vote totals differing by one vote), and you vote for one of those top two but not the other. In that way, you can make or break a tie.

If you change the winner from candidate j to candidate i, that accomplishment's utility is Ui-Uj. But, if a tie between candidates i and j is solved randomly, by flipping a coin, then the value for you of that tie is halfway between the utilities of i and j. So if you change a j win into an ij tie, or if you change an ij tie into an i win, then the utility of doing that is (Ui-Uj)/2. Half as much as if you changed it from a j win to an i win.

As I said the expectation for an outcome is its utility for you multiplied by its probability. What is the probability that you'll accomplish what's described in the previous paragraph?

In public elections with many voters, it's possible to ignore ties and near-ties between more than two candidates, because they're so much more unlikely. And even in a small committee, of course, a three-way, though not out of the question, is still significantly less likely than a two-way tie, and so ignoring three-way ties and near-ties is still a reasonable approximation even in small committees, especially since the uncertainties in the estimates of probabilities and utilities (the inputs for these methods) are such guesses that there isn't really much precision to be lost by ignoring three-way ties and near-ties.

So consider the probabilities of ties and near ties:

Pij is the probability that, when we've counted all the ballots except yours, either i and j have the same vote total, or j has one morevote than i. In other words, Pij is the probability that you can make orbreak a tie between i and j by voting for i and not for j.

Pji is of course the same, except instead of "j has one more vote than i," it's "i has one more vote than j." The probability that by voting for j and not for i, you can make or break a tie between i and j. It's reasonable to assume that Pij and Pji are the same, and I make that assumption.

As I said, the expectation for an outcome is its probability multiplied by its utility value for you. Let's define "ballot-expectation" as your expectation for what your ballot will do for you, compared to not voting. Let's consider the ballot-expectation for the outcome that your vote for i and not for j will make or break an ij tie. After that, as described above, we sum those outcome expectations to get your overall ballot expectation for the election.

From the definition of the expectation for an outcome, applied to the outcome in which you make or break an ij tie by voting for i and not for j, your ballot expectation with respect to that outcome is:

The probability that you'll make or break an ij tie, in i's favor, multiplied by the utility of doing so. Pij multiplied by (Ui-Uj)/2. Or, Pij(Ui-Uj)/2. (This formula could evaluate negative if you prefer j to i).

Actually, since the factor of 1/2 is present in every term of that type in this calculation, it makes no difference if we leave it off. Let's leave it off for simplicity.

Talking about your ballot expectation with respect to the outcome in which could you make or break an ij tie in i's favor:

Let's say that it's already decided that you aren't voting for j. Pij(Ui-Uj), then, is your ballot-expectation if you vote for i.

But what if it's been decided that you're voting for j? Then what's the ballot expectation of voting also for i? Well, the ballot expectation from voting for j and not for i, by the above formula, is: Pji(Uj-Ui). Since Pji = Pij, and since (Uj-Ui) = -(Ui-Uj), Pji(Uj-Ui) = -Pij(Ui-Uj)

So, when it's decided that you're voting for j, the ballot expectation of voting also for i is Pij(Ui-Uj), because you're getting rid of -Pij(Ui-Uj), by no longer voting for j and not for i. In other words, whether you vote for j or not, the ballot-expectation of voting for i is Pij(Ui-Uj).

That means that we can calculate the ballot expectation for voting for i, with regards to ij ties, without considering whether or not we're voting for j, since I've shown that it's the same whether or not we're voting for j.

To determine the total ballot expectation of voting for i, we evaluate Pij(Ui-Uj) repeatedly, letting each of the candidates other than i take their turn as j. We sum the results of those calculations. That gives the total ballot expectation of voting for i. That's the sum, over all j, of Pij(Ui-Uj), where j is different from i.

Well, we should vote for i only if the event ballot expectation of doing so is positive.

That ballot expectation is also called i's strategic value. If i's strategic value is greater than 0, then we should vote for i. If i's strategic value is less than 0, then we shouldn't vote for i.

The earliest description of that strategy was written by Robert Weber, the mathematician who first proposed Approval. So I'll call it Weber's strategy.

That's for Approval. By the way, if the method is Plurality, then we should vote for the candidate with highest strategic value. That appears to be what voters are doing in our Plurality elections. They feel, for example that P(Gore,Bush) is virtually 100%, and that P(Nader,Bush) and P(Nader,Gore) are essentially zero, assuring that Gore's strategic value is positive, and giving Nader a lower strategic value than Gore's high strategic value, resulting from his high probability of being frontrunners with Bush.

Nothing said here is intended to endorse the judgments of utilities or probabilities that lead those who prefer Nader to vote Democrat.

What that voter will say is "It's between Gore and Bush. We should make our votes count by helping Gore beat Bush."

Of course, with Approval in use, and everyone therefore able to vote for their favorite, true preferences would be better revealed, and the probability and viability beliefs that voters now have might turn out to be inaccurate. We might find that the Republicans and Democrats aren't the only ones with a chance of being among the top two vote getters.

As I said, the subsequent articles use Weber's strategy for the candidates whose strategic value is positive.

Of course it can be difficult to estimate the Pij. Especially if there are a lot of candidates, since it would have to be estimated for each pair of candidates. It would be easier if an estimate only had to be made for each candidate. Hence, some of the strategies in the subsequent articles are based on the following simplification:

Estimate the probability that, if there's a tie for first place, or a near tie for first place (two top candidates differing by one vote), i will be one of those candidates.

Call that probability Pi. We can estimate Pij as Pi*Pj. For some of the strategy methods described in later articles, the Pi are easier to use than the Pij.

Approval Strategy IV will describe some ways of estimating those numbers, for Weber's strategy, though the three strategies that I've already described seem the most useful.

But say we don't have any information about the other voters' preferences: no information about how big the factions are, no information on how popular the candidates are. In that case, we have no way to estimate Pi or Pij. We'll call that a zero-info election.

I emphasize that our public political elections aren't zero-info. And neither is a committee vote on something that's been well-discussed. So zero-info elections will probably be limited to a few rare instances in which the alternatives haven't been discussed. But they're still interesting to discuss, and are useful when we're comparing methods without having viability assumptions. That's their main use.

In that case, all the Pij are equal. Therefore we can leave them out. Replace Pij(Ui-Uj) with (Ui-Uj).

Let's calculate that sum again, with that simpler formula.

The sum, over all j, of (Ui-Uj) is the same as the sum of Ui over all j, minus the sum of Uj over all j.

What does it mean to say "The sum of Ui, over all j"? Once for each of the other candidates, while each of them takes a turn as j, we add Ui.Obviously Ui doesn't change. So we add up all those identical Ui terms. In other words, that sum is Ui(N-1), where N is the number of candidates.

What's the sum of Uj, over all J. The sum of the utility of everyone but i. I'll call that Uallj. So Ui(N-l) - Uallj > 0 if we're to vote for i. That can be rearranged to Ui > Uallj/(N-1).

In other words, vote for i if i's utility is greater than the average of the other candidates' utilities.

It can be shown, but is intuitively clear, that that condition is also true if i's utility is greater than the average utility of all the candidates (including i). Say i's utility is greater than the average of the others. The average utility of all the candidates must be somewhere between i's utility and the average for all those but i. So if i's utility is greater than the average of all the other candidates, then i's utility is also greater than the average of the utilities of all the candidates, including i. The argument is similar if we say that i's utility is less than the average of the others.

Thus we get the zero-info strategy: Vote for all the above-mean candidates.

Subsequent Approval Strategy articles will elaborate on this article. In particular, I'll show why Better-Than-Expectation is optimal, and will describe some refinements of Best-Frontrunner.