It has been suggested to us (we do not agree, but the case can be made) that approval voting is a superior voting system to range voting. Well first of all, Approval is simpler. However, we are here focusing not on simplicity but on inherent voting system quality. Why? Because honest range voters can be "victimized" by strategic ones who "exaggerate" their opinions to get more power, ranking candidates artificially 0 or 99 rather than honest scores like 23. In contrast, with approval, to a rough approximation (but only approximately since honest and strategic approval voting are not exactly the same thing), all voters are "forced" to be strategic, approximately getting rid of this inequality. So that is why approval is supposed to be better.
There are many responses to that criticism of range voting. We won't discuss them fully here. We'll only focus on a few. The "moral" response is the honest range voters were not "victimized" because they voluntarily chose not to exaggerate in order to help society, or because they did not consider it worth it, or for whatever reason. (The exaggeration strategy is so obvious that few if any voters would be too stupid to conceive of it, so it was a full-knowledge, voluntary choice.) Or (another flavor) those voters shouldn't complain because evidently their feelings that candidate A was better than B were not very intense if they voted A=51, B=50, so they didn't exactly suffer catastrophically from B's victory.
But some critics are unhappy with that response. E.g. maybe it isn't as "obvious" as I think.The "happier society" response is that this "victimization" does not matter, because, society-wide, everybody, on average, ends up happier with range voting, even though the strategic-voter group can end up relatively happier than the honest-voting group. That's because the good effect on decision quality that range voting gets from having better-quality information, is enough to outweigh any bad effects of this sort. We will demonstrate that is true, by computer simulation below.
But some attackers feel that response is inadequate. They vaguely feel that society being happier is not good enough. They also want "fairness." They feel a system which allows the tactically stupid, or more honest and less strategic voters, to suffer a relative disadvantage, is bad. Of course, such voters suffer such a relative disadvantage in essentially every voting system, but our critics are vaguely unsatisfied with that response too, feeling that somehow range voting is worse in this respect than some other systems.
That leads us to the "not really victimized" response. Again we fire up our computer election simulators. This time, we evaluate the social utility of the election results we get from range and approval voting not for all society, but instead only for the honest-voter subset of society. The point is to see whether and when the honest voters really are victimized by the combination of range-voting rules plus nasty exaggerator voters. They are victimized if their average utility is made worse by range voting rules than it would have been under approval voting rules.
Our computer sims find that it depends how many strategic voters there are and how many honest ones there are. (Our simulation simplistically assumes that all the voters are either 100% honest or 100% maximally-exaggerating.) "Victimization" in this relative sense indeed does occur, but only if the fraction of strategic voters is sufficiently large. (You can see from our tables below just how large.) In other words, the "not really victimized" defense is often, but not always, valid. But remember, even in the region where victimization does occur, society overall is still better off.
(Also you can see our other computer sims.)
Below table shows summed utility difference (util for range minus util for approval) summed for all the voters when the first t voters strategize (rest honest), for 100-voter elections with 2 to 10 candidates. All utilities of candidates for voters are independent standard normal random deviates (the simplest possible utility generator, not claimed to be especially politically realistic, but we doubt that matters much for the present purposes). Each data point is based on 350000 elections, which is not infinity, so we get some noise. Note that every table entry is positive, indicating range is a superior voting system to approval from the point of view of society-wide utility (i.e. averaged over all voters, i.e. both the honest and strategic ones) at all simulation parameters here. The exceptions to that statement are the 2-candidate column and 100% strategic row. In the 2-candidate case honest normalized range votes and approval votes are the same thing ("normalized" means rescaled so each voter's favorite candidate is scored MAX possible while her most-hated candidate scored MIN possible), and in the 100% strategic case range and approval also are the same thing at least as far as this simulation is concerned – so the first column and last row, shown pink, consist entirely of zero-centered random noise. They are only useful as a "noise-level gauge."
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In summary, up to a small amount of statistical noise that should go away if we devote enough computer time to this, the "happier society" response seems completely valid.
Below table shows summed utility difference (util for approval minus util for range) for the 100-t honest normalized-range-voters only (the other t are strategic), for 100-voter elections with 2 to 10 candidates. The "strategic" voters are using mean-candidate-utility as an approval-theshhold, because that is the best strategy we know of for 100-voter elections. Each data point is based on 350000 elections, which is not infinity so we get some noise. Nevertheless the noise is small enough that the boundary (shown bold) between the honest-voters-prefer-range and honest-voters-prefer-approval regions is now becoming visible. (The former region has negative entries, the latter positive.) There isn't any boundary in the 2-candidate case, though. That's because in the 2-candidate case, honest normalized range voters and approval voters are the same thing. So the 2-candidate column, shown pink, is just entirely random noise equally likely to be positive or negative. The limit as we approach 100% strategic voters (last row) would also be random noise.
We find that with mostly honest voters, range is better than approval (for the honest voters) utility-wise. That's not surprising since range seems obviously a better voting system than approval if there is a lot of voter honesty. Then as we increase the strategic-voter percentage, at some point this flips, and now the honest voters prefer approval over range so that they do not get "victimized" by the strategic voters. That also makes sense.
Note that a substantial region of this table is negative (i.e. range better in everybody's view). That fact would have to be the basis for any "not really victimized" defense. It appears this defense has some, albeit not a huge amount of, validity; and it has more validity if there are 3 or 4 candidates, than if there are 10.
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Below table shows difference in average utility (AvgUtil for strategists minus AvgUtil for honest voters) when the first t voters strategize (remaining 100-t voters honest), for 100-voter elections with 2 to 10 candidates. Each data point based on 1500000 elections. The "strategic" voters are using mean-candidate-utility as an approval-theshold. The more positive the table entry, the more incentive an honest voter has to strategize. The 2-candidate column again is pure noise since strategic and honest voting then are the same thing, so is only useful as a "noise-level gauge."
These results indicate that you have the most incentive to strategize if there are few strategic voters and many candidates. In practice we expect there will be effectively fewer candidates than there actually are (since many will be "no hopers") in which case the incentive to strategize will be reduced.
We thank raphfrk both for suggesting this experiment and for donating the computer time.
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You can download the entire code.
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