I presume you're saying that utility-based simulations are not credible. I don't think you're actually trying to say that they're not numerical estimates. So let me explain what I'm talking about, then say what parts I'm claiming are "credible".

I'm talking about monte-carlo simulations of voter satisfaction efficiency. You use some statistical model to generate thousands of electorates (that is, voters with numeric utilities for candidates); a media model to give the voters information about each other; and a strategy model to turn information, utilities, and choice of voting system into valid ballots for that voting system. Then, you see who wins each time, and calculate the average overall utility of that winners. Clearly, there are a lot of questionable assumptions in terms of the statistical, media, and strategy models, but the interesting thing is that exploring various assumptions in all of those cases shows that the (plurality-dictatorship)≈(good system-plurality) equation is pretty robust, with various systems such as approval, condorcet, majority judgment, score, or SODA in place of "good system".

There are certainly various ways to criticize the above.

• "Don't believe it": If you think that I've messed up my math or not done a good job with the sensitivity analysis, of course you'd question my conclusions. But if you want to play with my code to check it, it's here.

• "Utilitarianism is a bad metric": It may not be perfect, but as far as I can tell it's the only rational way to put numbers on things.

• "Democracy is a bad idea": In other words, if you think that the average voter's estimate of their utility for a candidate has 0 or negative correlation with their true utility of that candidate winning, then this simulation is garbage. I'd respond with the old saying about democracy being the worst system except all the others.

• "The advantages of democracy over dictatorship aren't in terms of who's in charge": if you think that democracy's clear superiority to dictatorship in terms of human welfare comes from something other than choosing better leaders (such as, for instance, reducing the prevalence of civil wars), then improving the voting system might not be able to have comparable payoff as instituting a voting system to begin with. I'd respond that this critique is probably partially right, but on the other hand, better leadership could credibly have better responses to crises (financial, environmental, and/or existential-risk) which could indeed be on the same order as the democracy dividend.

All in all, taking a more outside view, I see how the combination of the above objections would reduce your estimate of the expected "voting system dividend". Still, when I "shut up and multiply" I get: $80 trillion world GDP * plausible (conservative) effect size in a good year of 2% * .1 plausible portion of good years over time * .5 plausible portion of good years over space (some country's economies might already be immune to the kind of harm this could prevent) * .5 chance you trust my simulations * .1 correlation of voter preference with utility * .5 probability leadership makes any difference = about $2 billion/year potential payoff in expected value, even without compounding. That seems to me like (a) quite a conservative choice of factors, (b) not a totally implausible end result, and (c) still big enough to care about. Of course, it's incredibly back-of-the-envelope, but I invite you to try doing the estimation yourself.