Range voting produces the highest voter satisfaction index

By Clay Shentrup

Utility measurements: Group A: 5 candidates, 20 voters, random utilities; Each entry averages the results from 4,000,000 simulated elections. Group B: 5 candidates, 50 voters, utilities based on 2 issues, each entry averages the results from 2,222,222 simulated elections.
Voting system VSI A VSI B
Magically elect optimum winner 100.00% 100.00%
Range (honest voters) 96.71% 94.66%
Borda (honest voters) 91.31% 89.97%
Approval (honest voters) 86.30% 83.53%
Condorcet-LR (honest voters) 85.19% 85.43%
Range & Approval (strategic exaggerating voters) 78.99% 77.01%
IRV (honest voters) 78.49% 76.32%
Plurality (honest voters) 67.63% 62.29%
Borda (strategic exaggerating voters) 53.26% 51.78%
Condorcet-LR (strategic exaggerating voters) 42.56% 41.31%
IRV (strategic exaggerating voters) 39.07% 39.21%
Plurality (strategic voters) 39.07% 39.21%
Elect random winner 0.00% 0.00%
These results are from 2 of 720 different models, with millions of elections simulated for each model. Note that range voting is approximately as great an improvement over plurality voting, as plurality is over random selection; range voting effectively doubles the benefit brought about by the invention of democracy. These experimental results also strongly suggest that range voting is the least susceptible to strategic voting, of these common methods.

Voter satisfaction index, or "VSI" for short (also called "social utility efficiency" by some), is an objective measure of how utilitarian a voting method is. That is, the most utilitarian voting method is the one that does the most good for the most people. Utility has been defined by various thinkers as satisfaction, happiness, pleasure, etc. For instance, if you like apples twice as much as oranges, we say that you would derive twice as much utility from an apple as from an orange.

So let us imagine a scenario in which three brothers at a farmer's market have been given enough money by their mother for one piece of fruit to split. We describe their utilities for each potential choice in the following table, using an arbitrary "happiness unit". Note that these numbers have no bounds, because in reality, you can always like something a little bit more than something else. You can always be a little more satisfied than you are.

boy 1
boy 2
boy 3

Here we see that an orange clearly gives the highest overall utility. Some would argue from a majoritarian standpoint, that "banana" should win, since a majority of voters prefer it. That is how I myself thought, quite adamantly, until taking considerable time to debate and discuss the matter. But as we see, choosing the orange gives boy 1 and boy 2 just a little less satisfaction, whereas it makes a world of difference to boy 3, since he doesn't like bananas at all. It is my belief that majoritarianism is outdated, and stems from our traditional experiences choosing between just two options, or using plurality voting, where the candidate who gets the most first-place votes wins. It has been done this way for so long, that we fail to even consider any alternative.

Also, if you know much about economics and/or statistics, you should be interested in having the highest "expected value" from a decision. If you had to use a voting method to choose something with a group of other random people, you would want the most utilitarian voting method, because your personal expected satisfaction with the outcome of the election would be maximized. After all, you never know when you may not be in the majority. This is true whether you were voting for a piece of fruit, or a politician.

Testing Utility

In order to get an idea of how utilitarian various voting systems are, there are two fundamental methods we can employ. If we could somehow precisely read minds, to get honest utility values, and if we had the money to scientifically study and interview millions of voters over millions of real elections, using all different kinds of election methods, we would certainly prefer to do that. But we can't read minds, nor can we hold millions of large-scale elections. We can't even use results from real elections, because you cannot infer utilities based on votes. For example, in a plurality election for the fruit in the above scenario, boy 1 and boy 2 would have voted for banana, and boy 3 would have voted for orange. But we would have no way to determine their honest utility values based on that information. Hence the second method.

While we can't infer utilities from votes, we can infer votes from utilities, making certain assumptions about how honest/strategic, or informed/ignorant the voters are. Thus, when Princeton Ph.D. Warren D. Smith set out to calculate the utility produced by various voting methods, he used five parameters, or "knobs", to specify these things, and then used a computer program he wrote, to perform millions of simulated elections for each of 720 different parametrizations, or knob settings. The number of candidates were varied from two, to several. Some elections used 100% honest voters, while others used 100% strategic voters. The effect of voter ignorance was simulated, such that uninformed voters might behave as though they liked a candidate more or less than they really would have, had they researched him further. The goal was to produce enough simulations with enough different knob settings, that at least some of them would very closely model reality. This page explains in greater detail why we use computer simulations to measure utility.

Expressing Utility

Say we were to use the values from the scenario above, in which three brothers buy a piece of fruit at the market. We would create millions of scenarios like this, using different fruits, and different buyers, and of course, different voting methods. But what system should we use to describe the results? Warren D. Smith prefers a method called "Bayesian regret", which actually measures how much utility was wasted because a voting method chose the wrong winner. In our fruit scenario, the regret is simply the utility produced by the chosen fruit, subtracted from the utility produced by the ideal fruit, orange. So picking the orange would produce a regret of 0, whereas picking the apple would produce a regret of 6. Bayesian regret is the average of this value, over as many hypothetical scenarios as we choose to execute. An ideal voting system that always picked the ideal winner would have a Bayesian regret of 0, by definition.

While doctor Smith cites some academic reasons for preferring to express utility in the form of Bayesian regret, some find this system problematic. One of the chief criticisms of this method is that a lower number is actually better, and this can confuse people who are new to the concept. Another problem that some cite is that the utility units have an arbitrary magnitude, making it difficult to compare Bayesian regret figures from two different simulations.

A proposed solution to these problems is to use voter satisfaction indexes. VSI is calculated as

(U - R) ÷ (O - R)

where U is the utility produced by the voting method in question, R is the expected utility produced by random selection (in other words, the average utility of all candidates), and O is the utility of the optimum, or ideal, candidate.

In our above scenario, the VSI for a system which selected "orange", would be 100%. That is, (9 - 6) ÷ (9 - 6), or 3/3 = 1. Expressing this as a percentage simply helps to make it clearer that it is a ratio. The VSI for a system that picked banana would be 0, because a utility of 6 is exactly equal to the expected utility produced by selecting a random winner. That means that VSI can actually be negative; this would happen if a voting system selected less satisfying leaders than what would be produced by simply drawing a name out of a hat. The following table attempts to clearly illustrate the method used to derive the voter satisfaction index and the Bayesian regret from utility calculations.

Voting method Average voter utility
(in "happiness units")
= U
Voter satisfaction index
(U - R) ÷ (O - R)
Bayesian regret
(O - U)
Magically elect optimum winner
1.31348 = O
Range voting
1.26407 96.71%
Borda count
1.18293 91.31%
Approval voting
1.10773 86.30%
Condorcet (LR method)
1.09101 85.19%
Instant runoff voting
0.99034 78.49%
Plurality voting
0.8272 67.63%
Elect random winner
-0.1887 = R


It turns out that my previous mention of drawing names out of a hat is actually quite appropriate for explaining just how great range voting is. If you were to draw names out of a hat, the VSI would be 0%, by definition. If you use plurality voting, the VSI will be somewhere around 50%, depending on variables like how strategic the voters are. Yet range voting produces very close to 100% VSI, nearly doubling the benefit brought about by the original invention of democracy, as opposed to random selection, say by accident of birth (monarchy).

Range voting outperforms the other voting methods listed here, by a substantial margin. Since the entire purpose of an election is to give voters the greatest satisfaction possible — just like when you make a personal choice, you try to select the option that will bring about the greatest overall benefit to you, however you determine that — I argue that range voting is objectively the best out of all these voting methods.

Improve These Simulations

Okay, so maybe you don't like our results. You're a skeptic. You don't trust people who advocate a given voting method to make unbiased simulations saying that it is "the best". In that case, we encourage peer review. And by review, I mean that you can actually extend the code that was used to perform these simulations. If you know the C programming language, you are invited to dive in and add your own utility generators and strategy algorithms, by simply adding functions. Get the code here. We would particularly like for someone to translate the code to the D programming language.

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