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.
| 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.
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.