[EM] New BR numbers (preliminary)

[Jameson Quinn]

I've added to my BR simulator, and fixed a bug, so I have some further
numbers.

First, the bug. My strategy code was sorting in the wrong order, so
strategic voters were strategizing between the bottom two candidates, not
the top two. Oops. Fixed now.

What I've added since yesterday is the possibility of one-sided strategy.
That is, I run each election, with the same voter utilities, 3 times:
-with honest votes
-with strategic votes, that exaggerate the difference between the two
frontrunners from the honest election
-with one-sided strategy; honest if the voter prefers the honest winner
over the honest runner-up, but strategic if the voter prefers the honest
runner-up over the honest winner.

I measured the average social utility versus best winner for each case, and
the proportion of cases for which the one-sided result differed from the
honest result (this is a sort of "strategic incentive" measure). Here's a
run of 5000 elections, with 4 candidates and 100 voters each.

Score [-0.01351796752462983, -0.074143581160690683, -0.066539696670647963]
0.2128
Mav [-0.049848219124324029, -0.048864684230964794, -0.051033067381167654]
0.1386

So one-sided strategy shifts the score result 21% of the time, and only 14%
of the time for MAV. I expected such a difference, but frankly I would have
guessed it would be larger. So in that sense, I have to admit that my
intuition was exaggerated. This is not a night-and-day difference, though
it remains to be seen how things work out with partial one-sided strategy.

For score, one-sided strategy actually gives better utility than two-sided
strategy, because under one-sided strategy the honest runner-up, who has
better than average utility, will more often win; whereas under two-sided
strategy, anybody can win. For MAV, (straddle) strategy barely shifts the
result at all. If you used 2-sided approval strategy, results would be the
same as 2-sided strategic Score.

Note that the advantage for MAV over Score in strategic cases has grown
slightly now that I fixed the bug.

....

I have several other improvements just about finished.

   - Extra systems: I'm doing MJ and Grand-Junction (ie, the original US
   version of Bucklin) at Aaron Hamlin's request, and Ranked Pairs at my own
   curiosity; any other system requests?
   - Extra "styles": partial strategy and partial one-sided strategy.
   Should I just set that to 25%, 50%, and 75%?

There are some other things I plan to get to:

   - Extra election models: 1D voters with 2D candidates (that is, ideology
   plus quality); 1.5D voters (higher variance in one dimension) with 2.5D
   candidates; both using Euclidean distance (L2 norm); impartial
   culture/random voters.
   - Extra media environments: "poll" only the first N candidates and M
   voters, so that strategy is based on flawed info on frontrunners.
   - More data: characterize winning margins; utility payoff for just the
   strategic voters in the 1-sided cases; how frequent are Condorcet cycles;
   proper statistical analysis, starting with standard deviations and standard
   errors.

Any other suggestions?

Jameson


2014/1/9 Jameson Quinn <jameson.quinn at gmail.com>

> I've written a new Bayesian Regret simulator in python. I'll open-source
> the code, put it on github, and explain exactly how it works later. For
> now, I only have a few methods done, but I'll have more tomorrow.
>
> I just wanted to share the first numbers I've gotten. This is an average
> of 1000 runs with 100 voters and 4 candidates. Each voter has a standard
> normal utility for each candidate. Voters are created in clusters in a
> Polya/Dirchlet/Husse process, with mutations - that is, a new voter has a
> correlation of 0.81 on all utilities with a randomly-selected existing
> member of the electorate (or, to be more precise, the electorate plus one
> wildcard which, when picked, creates a voter in a new cluster uncorrelated
> with any of the existing ones). The initial electorate consists of two
> "opposite" voters with a correlation of -0.5 on all utilities. For each
> system, honest results are given first, then results in which all voters
> use first-order strategy based on the honest results. Instead of straight
> BR, I give the average social utility compared to the best (that is, the
> inverse of BR); thus, higher numbers (less negative) are better. This was
> my first sizable run; it took about 40 seconds to complete.
>
> ('Score', -0.020281932241917173, -0.073726013277532365)
> ('MAV', -0.045652626758973726, -0.052529728966832126)
>
> In other words: this shows that a median method can have better BR than
> score voting in some cases. This is because in the median method (here,
> MAV), the voters strategize to ensure that their grades for the two
> frontrunners straddle the honest top two medians, but they do not
> necessarily go all the way to an approval-style ballot. In exceptional
> cases, this even leads to a better BR, something that almost never happens
> with score.
>
> Note: Adding more candidates makes honest Score look better, because the
> variability of the normalization goes down. So 4 candidates is probably the
> worst number for honest Score. This effect has very little impact on any of
> the other three numbers above, though.
>
> While I find these assumptions reasonable, I realize that others will
> disagree. There will be plenty more numbers from where these came from, and
> plenty of time to adjust assumptions and argue about which ones are the
> most realistic. My main point here is that median methods can, in at least
> some circumstances, be better than score voting, even with the same
> proportion of strategic voters. I hope to explore the boundaries of those
> circumstances further in the days to come.
>
> Jameson
>
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