Expected Strategic Influence Factors

A simple tool for evaluating strategic voting

Suppose there’s an election coming up. You’ve decided how good each candidate is and you’ve looked at some polls, so you have a decent idea of how likely each of them is to win. Given this information, how should you fill out your ballot to get the best results according to your preferences?

This is the fundamental question of strategic voting. It’s often difficult to answer, and many people disagree. It’s of particular interest to people who worry about whether a voting method is vulnerable to a particular form of strategic voting, since, for such a vulnerability to exist, the proposed strategy must actually advance the strategists’ interests.

The Definition of ESIF

To assess whether a strategy S is effective, I’ve created a metric I call its Expected Strategic Influence Factor (ESIF). Here’s how you calculate it:

• Decide on some background strategy B that everyone will use by default.
• Create a (small) simulated electorate, where each voter has randomly generated preferences about each simulated candidate. The preferences must be in the form of utilities, i.e. numbers that show how much better/worse one candidate is than any other candidate, not just whether the candidate is better or worse.
• See who wins if every voter uses the background strategy.
• For each voter, see what happens if that voter uses the strategy S while every other voter uses the background strategy B. Record the utility the voter gives to the candidate who wins when the voter uses S, the utility given to the candidate who wins when using B like everyone else, and the utility given to the winner if the voter abstains. (These numbers will often be the same since a single vote won’t always change the outcome.)
• Repeat this for thousands of simulations, and sum to the utilities for S, B, and abstaining across all simulations and across all voters. Call these sums U(S), U(B), and U(A)
• The strategy S’s Expected Strategic Influence Factor is (U(S)-U(A))/(U(B)-U(A)).

This means that using the same strategy as everyone else (B) has an ESIF of 1 and abstaining has an ESIF of 0. A strategy that is more effective than the background strategy will have an ESIF greater than 1, and if a strategy’s ESIF is less than 1 then voters are better off not using it.

Note that a strategy’s ESIF isn’t determined solely by the strategy. It’s also determined by the context of the simulations: the background strategy, the number of voters and candidates, the quality of the polling data used by the strategy, and the voter model used to randomly generate electorates.

Advantages and Limitations of ESIF

ESIF has a straightforward interpretation: “By what factor does using this strategy increase the power of my ballot?” It judges a strategy entirely based on the preferences of a voter who’s thinking of using it, and it sidesteps the question of “How do you aggregate preferences across multiple voters who are using this strategy?” since it only considers a single voter at a time. It implicitly accounts for the possibility a strategy will backfire in a very natural manner; cases where a strategy backfires are simply ones in which the voter gets less utility from the attempted strategy than from using the background strategy, and this just gets added to the total.

ESIF realistically models voters as acting under uncertainty. They don’t know how everyone else will vote, and a strategy has to do a good job of balancing risks vs. rewards in order to have a high ESIF. But my favorite part about ESIF is that it doesn’t overemphasize coordinated strategies that can only work if they’re used by a large fraction of the electorate. (The effective level of coordination allowed can actually be adjusted by altering the number of voters; coordinated strategies will work far better with 9 voters than with 90.)

That said, ESIF is a tool for evaluating strategies, not voting methods. It can tell you how much better or worse one strategy is than another strategy in a given context, but not whether there could exist some other strategy that significantly outperforms everything you’ve thought of. I wrote earlier about a strategic voting metric used by Green-Armytage, Tideman, and Cosman:

It should be noted their R metric does not necessarily model realistic choices for voter behavior. It instead asks whether there is some group, who, if given perfect knowledge of how everyone else was voting, could change their votes in response to this knowledge, while the remainder of the electorate remains utterly ignorant to their preferences and strategies. A more precise name for R might be “post hoc cooperation resistance.” What R does do is give an upper bound for how often strategic voting can sway elections; it catches every case in which strategic voting can determine the outcome of an election at the cost of “catching” cases in which it would be utterly preposterous for strategic voting to affect elections given a realistic electorate. R can exonerate, but it cannot convict.

ESIF is largely the opposite of their R metric. ESIF asks whether a single predetermined strategy would be advantageous if used by a single voter; R asks if there is any strategy that would benefit a group of voters. Coordinated strategies are always rejected by ESIF with a large enough electorate, but they aren’t penalized at all by R. And while ESIF can show that a dishonest strategy is effective, it, unlike R, can never show that there isn’t an effective dishonest strategy. ESIF, unlike R, can’t exonerate. (ESIF also isn’t perfect at convicting a voting method for being vulnerable to strategy; even if a problematic strategy has an ESIF well over 1, there could be an unproblematic strategy you haven’t thought of that has an even higher ESIF.)

Example: Evaluating strategies in Approval Voting

Let’s use ESIF to investigate strategic voting under Approval Voting. In particular, should voters actually vote for multiple candidates very often, or are they better off “bullet voting”, i.e. only voting for their favorite?

Let’s start with the background strategy of voting for every candidate who looks better to you than the average candidate. Running 3,000 simulations with this background strategy, 21 voters, and 5 candidates (I won’t bother getting into the voter model) shows that bullet voting has an ESIF of 1.42. That is to say, if you bullet vote, your ballot will be 1.42 times as impactful. Does this mean you should always bullet vote in an Approval election?

No. It just means that bullet voting does better than this particular background strategy. Let’s try something a little less extreme, and start by normalizing the utilities you ascribe to the candidates by requiring that the average candidate has a utility of 0 and the best candidate has a utility of 1. Consider the following three strategies.

• U > 0.25: Vote for a candidate if their utility is at least 0.25
• U > 0.5: Vote for a candidate if their utility is at least 0.5
• U > 0.75: Vote for a candidate if their utility is at least 0.75

These strategies (with the same background strategy as earlier, which can be written as U > 0) have ESIFs of 1.30, 1.47, and 1.50, respectively. Let’s try using U > 0.75 as a background strategy. This yields the following ESIFs:

• U > 0: 0.98
• U > 0.25: 1.05
• U > 0.5: 1.07
• U > 0.75: 1 (The background strategy always has an ESIF of 1)
• Bullet voting: 0.91

Now bullet voting does this worst out of all these strategies! We also notice that U > 0.75 outperformed U > 0.25 and U > 0.5 with U > 0 as the background strategy, but that both of these strategies outperform U > 0.75 when U > 0.75 is the background strategy. Also, U > 0.75 only slightly outperforms U > 0 here, but the difference was massive with U > 0 as the background strategy. The general pattern is that, the fewer candidates other voters vote for, the more candidates you should vote for.

Out of these strategies, U > 0.5 turns out to be the Nash equilibrium; if everyone else is using it, you’re still best off using it yourself. You can still do better if you use polling data though; I was able to get an ESIF of 1.19 with my preferred viability-aware strategy with U > 0.5 as the background strategy.

Generalizing ESIF to multi-winner elections

In principle, generalizing ESIF to multi-winner elections is trivial: just replace the utilities for individual candidates with utilities for every possible set of winning candidates. This begs the question: What is a realistic voter model for assigning utilities to sets of possible winners? When I look at my own preferences for a city council or Congress, I find that I want:

1. A majority that agrees with me on as many issues as possible
2. Someone with very similar opinions to me on issues where I have strong preferences and most people don’t care much; such a representative will be well positioned to pull policy ropes sideways.
3. Everyone to be competent, ethical, willing to compromise, etc.
4. Ideological diversity; I know I’m wrong in plenty of ways, and I want people who are well-positioned to correct the representatives who agree with me.
5. Racial/ethnic/gender diversity

The first three points are easy to approximate from utilities for individual candidates. The first corresponds to caring about the utility of the median winner, the second to caring about the utility of the best winner, and the third to caring about the utility of the average winner. Capturing the last two parts sounds really hard and would require devising a new and very complicated voter model. This suggests that we can use the median, maximum, and mean of the utilities of individual candidates to get the utility of electing a given set of candidates, and the choice of which option to go with leads to three metrics: ESIF-median, ESIF-max, and ESIF-mean. I will make ample use of these metrics in a future post on strategic voting under proportional cardinal voting methods.