Why range voting is "uniquely best" among all common proposals for single winner voting systems

For the Bayesian-regret measurement study via massive computer election simulations using many election systems, see paper #56 here.

Here is some typical data from that study.

Regret measurements. Column A: 5 candidates, 20 voters, random utilities; Each entry averages the regrets from 4000000 simulated elections. Column B: 5 candidates, 50 voters, utilities based on 2 issues, each entry averages the regrets from 2222222 simulated elections.
Voting system Regret A Regret B
Magically elect optimum winner 0 0
Range (honest voters) 0.04941 0.05368
Borda (honest voters) 0.13055 0.10079
Approval (honest voters) 0.20575 0.16549
Condorcet-LR (honest voters) 0.22247 0.14640
IRV (honest voters) 0.32314 0.23786
Plurality (honest voters) 0.48628 0.37884
Range & Approval (strategic exaggerating voters) 0.31554 0.23101
Borda (strategic exaggerating voters) 0.70219 0.48438
Condorcet-LR (strategic exaggerating voters) 0.86287 0.58958
IRV (strategic exaggerating voters) 0.91522 0.61072
Plurality (strategic voters) 0.91522 0.61072
Elect random winner 1.50218 1.00462

As you can see, Range voting has a lot smaller Bayesian regret than the other systems. Note: this table makes it appear that Borda is the second-best system after range. But in fact the full study considers about 100 tables of this kind, and in many of them, Borda is not second best, in fact in many of them it is way down in the rankings. The question of which system is second best has no clear answer - some of them are better in some kinds of election situations, others in others. But range voting always came out best (or at least tied for best) in all the tables.

How big an improvement is this?

Huge. In this (and other) tables, reduction in regret you get from switching to range is instead of plurality is a factor of 7-10 for honest voters and 2-3 for strategic ones.

That was larger than the improvement in regret we got by switching from non-democratic systems such as monarchy (we assume monarchs, on average, were as least as good rulers as a random candidate, since, e.g. they were trained from birth to rule), to plurality: only a factor of 2-4 for honest voters and about 1.6 for strategic voters.

Also, if you do not like multiplicative factors - you like additive differences - then the improvement we got from switching from random winner to (our current) strategic plurality is comparable to the improvement we will get from the future switch to range voting with a mixture of honest & strategic voters.

In other words you get comparable or more improvement in democracy by switching from plurality to range voting, than you get from the very invention of democracy in the first place.