[EM] Preliminary simulation of fpA-{sum, max} fpC and compromise vs burial incentive

[Forest Simmons]

Kristofer,

It seems to me that it might be worth the cost of an extra pass through the
ballots (after the pairwise wins have been ascertained) to fix the
"no-first-preference" problem: just use, in place of the number (or
percentage) of the ballots on which candidate X is ranked in first place,
the number (or percentage) of the ballots on which X covers every candidate
ranked ahead of it.

We could call such a candidate X the minimal prudent compromise (MPC)
candidate for the ballot, since if X covers all of the candidates ranked
ahead of it, none of those stand a snowflake's proverbial chance in
comparison.

In any case, for me it would be interesting to see if this MPC substitution
for First Place Votes preserves the burial resistance of fpA-SumfpC, while
lessening the need for actual voter favorite betrayal on the ballots ...
i.e. alleviating the tension between compromise and burial resistance.

-Forest

On Tue, Nov 22, 2022, 4:02 PM Kristofer Munsterhjelm <km_elmet at t-online.de>
wrote:

> So I finally wrote something quick (and hopefully less messy than my
> previous manipulability checker) to determine strategy susceptibility
> for a few voting methods. It's still very unoptimized and thus quite
> slow, but I've attached the Python source to my post. PDFs showing
> results can be found here:
> https://munsterhjelm.no/km/python_vote_strats/ and the Python source is
> available there as well.
>
> This should both give some ideas as to where to take fpA-fpC, as well as
> the relative proportion of compromise vs burial for some Condorcet
> methods. (However, as my figures seem somewhat different from Daniel
> Carrera's, there may still be bugs. So take this with a pinch.)
>
> I'll first note some observations, then explain how to use the Python
> code, and then how to read the bar charts.
>
> For impartial culture, the observations seem to be:
>
> - The two simple fpA-fpC generalizations are not robustly strategy
> resistant the way IRV is. While they do well with three candidates, as
> the number of candidates increases, the susceptibility increases; and at
> 10 candidates, it's up at Minmax level.
>
> - That said, even Contingent vote (which is a quick proxy for IRV) gets
> more susceptible to strategy as the number of candidates increases. I
> suspect that for any voting method in impartial culture, and any
> susceptibility level, there exists a sufficiently large number of voters
> and candidates where that method is vulnerable to strategy at least that
> often.
>
> - The simple generalizations also lose strategy resistance when the
> number of voters is increased for more than four candidates, whereas
> Contingent vote and its Smith,Plurality version both seem to level off
> for a fixed number of candidates. See e.g. the contingent 99 voters 5
> candidates graph.
>
> - fpA-max fpC is more resistant than fpA-sum fpC, both in terms of total
> susceptibility and in how much can be accomplished by just burial or
> compromise vs having to do both at once.
>
> - There seems to be a baseline compromise level that all the depicted
> Condorcet methods follow -- see the sum of the blue plus purple bars for
> Copeland, Minmax, and the two fpA-fpC methods. If this really is
> universal, it might be possible to derive results about what that level
> is for *any* Condorcet method as number of voters approaches infinity.
> But I'd need to implement something known to be strategy resistant and
> Condorcet, e.g. Smith//IRV, to know whether the pattern is real or just
> a fluke.
>
> - On some of the graphs, Contingent has a lower susceptibility than
> Smith-Contingent, even for three candidates. On the face of it, this
> shouldn't be possible since there's a proof that "Condorcet else X" is
> never less resistant than X alone, assuming certain majority-based
> criteria hold for X. I think the answer is that my simulation doesn't
> take more complex strategies into account, which means that Contingent
> has quite a few strategies that are neither full compromise nor full
> burial nor both at the same time.
>
> - Smith-Contingent vote is both summable and robust to strategy with the
> number of candidates fixed in IC. This shows that summability and
> robustness is compatible. Furthermore, the method is vulnerable to my
> "nobodies ranked first" problem: if every voter ranked himself first, it
> would be unable to call a winner. So neither ISDA nor (the weaker)
> immunity to the nobodies ranked first problem is required for a method
> to be strategically robust in IC.
>
> ("Strategically robust" here would probably mean something close to: 1.
> the method starts at a low susceptibility level for modest number of
> candidates in IC. 2. With not too many candidates, the limit of
> strategic susceptibility as number of voters goes to infinity is not one.)
>
> Note that this is all under impartial culture, which in a sense is the
> most tie-prone of them all. Spatial models might show much less
> potential for strategy.
>
> ---
>
> Usage:
>
> The Python code can be used by starting an interactive interpreter (e.g.
> python3 or ipython3), importing the strategy and generators modules, and
> then calling plot_MC_proportions, for instance like this:
>
> strategy.plot_MC_proportions(election_space=generators.strict_ordinal_elections(99,
>
> 5))
>
> Note that burial doesn't yet work with the truncated ballot generators,
> as the burial function doesn't check if the candidate is unranked. And,
> as mentioned, it is very slow. Implementing a cache (for things like the
> current election's Condorcet matrix) would help a lot, as would reusing
> elections, e.g. drawing just 1000 and testing the same 1000 on each
> method instead of drawing 1000 * num methods elections in the
> Monte-Carlo simulator.
>
> The code may understate nonmonotone methods' strategic susceptibility
> because it's possible that e.g. upranking a compromise all the way to
> first may not work but upranking it to second place does, as a
> consequence of pushover. But I don't have any nonmonotone methods
> implemented yet, so that shouldn't be a problem :-)
>
> ----
>
> Reading the bar charts:
>
> These are stacked charts, so the total height of a method's bar gives
> how often simple (compromise, burial, or minmaxing/two-sided
> maipulation) strategies work. It doesn't check for more complex
> strategies or coordinated arbitrary-ballot ones, so the total bar may
> still understate susceptibility slightly. But as Daniel Carrera noted
> back in January, most methods' strategy vulnerability can be exploited
> with simple strategies.
>
> The "burial only" bar corresponds to elections where burial worked, but
> compromising didn't. The "compromise only" bar similarly corresponds to
> elections where compromising worked but burial didn't. The "burial and
> compromise" bar is for elections where either strategy worked, and
> "two-sided only" is for elections where two-sided strategy worked but
> neither burial nor compromising worked.
>
> Two-sided strategy is simply putting the candidate you're trying to win
> first and the candidate who used to win last, i.e. doing both burial and
> compromising at once. Burial is putting the winner last, and
> compromising is putting a favored candidate first.
>
> When an election is tested, it's tested for the possibility that the
> group that prefers A to the current winner W can get A elected by
> employing strategy, for any other candidate A. If there exists any such
> A, then the strategy is considered to be successful. In the case of
> ties, what is counted is whether the A>W voters are able to insert A
> into the set of winners.
>
> -km----
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>
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