[EM] Condorcet strategy spreadsheet (was, ...maybe it should be about Condorcet...)
Jameson Quinn
jameson.quinn at gmail.com
Fri Jan 29 08:17:19 PST 2010
2010/1/28 Juho Laatu <juho.laatu at gmail.com>
>
> To be exact, one could also break an already existing cycle for strategic
> reasons (compromise to elect a better winner). And yes, the strategies are
> in most cases difficult to master (due to risk of backfiring, no 100%
> control of the voters, no 100% accurate information of the opinions,
> changing opinions, other strategic voters, counterstrategies, losing second
> preferences of the targets of the strategy).
>
Yes. Some months ago, when I proposed "Score DSV" voting, I did some playing
with a spreadsheet to see the true individual benefit and social cost of
various types of strategy in various 3-way condorcet tie scenarios. A link
to the spreadsheet is
here<https://spreadsheets.google.com/ccc?key=0Am3BsUGKovVvdEVsSld1VjlTbVJoRTVsR0FGVXlmcEE&hl=en>.
There's a lot more black magic there than I care to explain fully - that's
why I didn't share this earlier - but I think that something like this is
useful in exploring the nature of strategies. So, I'm putting it out there
for any geeks like me who are interested. Here's a "quick" (that is,
incomplete) explanation of how it works. If you want to skip the technical
details, there's a couple paragraphs about what I learned from it at the end
of this message.
...
The voting system used, in all cases, is Score DSV. This is a system which
uses Range ballots and meets the Condorcet criterion. As a Condorcet
tiebreaker, it is intended to give the win to the candidate whose opposing
voters would be, overall, least motivated to use strategy to defeat her. (Of
course, this "least" is after the normalization step. This is inevitable
since normalization is the only mathematical means of comparing preference
strength across voters.) Still, while the mechanics of Score DSV are unusual
for a Condorcet system, its results are not so much. A typical Condorcet
system would give results which are broadly comparable. (Actually, since
only the 3 candidate, no-honest-equalities case is considered, the winner
and all non-equal-ranking-based strategies are mutually identical for a
large set of Condorcet systems, including, IIANM, Schultz, Tideman, Least
Margin, and others, but not Score DSV).
The spreadsheet works by first creating a 3-way Condorcet tie scenario. To
do so, you set 7 parameters, the red numbers in the blue area. Feel free to
change the red numbers, but please, if you want to change the spreadsheet in
another way, use a copy. The basic parameters are:
-In the column "num voters", the size of the three pro-cyclical voting
groups - ABC, BCA, and CAB. Without loss of generality, the first group is
the largest.
-To the right of each voter number is the average vote within that group.
All groups vote 1 for their favorite of the three candidates and 0 for their
least favorite, but you can change their honest utility for the middle
candidate to any number between 0 and 1.
-The voter population is assumed to have some anticyclical voters (ACB, CBA,
and BAC). However, you do not set these numbers directly. The anticyclical
voters are assumed to be a "bleed over" of the cyclical voters. For
instance, if the ABC voters assign a relatively high utility to B, then some
fraction of them will actually become BAC voters. To change the overall size
of the anticyclical vote, change the value in cell B1 ("cohesion power"). A
higher value there will give a smaller anticyclical vote. Values should be 1
or greater. Lower values are probably more "realistic" but lead to weaker
(or even broken) condorcet cycles. Values over 3-4 lead to essentially
negligible anticyclical voters.
Once your scenario is created, the spreadsheet will calculate the utility of
various strategy options for the different voter groups. Each strategy is
placed to the right of the group to which it applies, and continues through
the row. Each strategy has intrinsic values and calculated values. The
intrinsic values include the strategy name, the candidate it is "for"
(intended to favor), the candidate it is "against" (intended to disfavor),
and the strategy (if any) it is intended to respond to or defend against.
The values calculated for each strategy include:
-Works: this is true (green) if the strategy has any hope of working, and
false (red) if not. If this value is false, the rest of the row for this
strategy consists of GARBAGE values, and should not be considered.
-Undefensible: true if there is no rational strategy which could defend
against or change the results of this strategy.
-Payoff/voter: if the strategy works, how much "utility per vot" would be
gained for this voter group?
-Semi-dishonesty/risk: by how much would the voters in question have to
change their ballots in order for this strategy to work? Or, equivalently:
if the strategy ends up backfiring for some reason, how much utility would
this voter group lose? It is reasonable to assume that the higher this
number is, the more difficult it will be to organize this strategy. This is
expressed as a total, not a per-voter number, since a strategy which
requires the cooperation of a lot of voters will be harder, just as a
strategy which requires voters to "hold their nose" more strongly and vote a
seriously dishonest ballot (rather than just a minor change from their true
utilities).
There are also "probabilistic" values calculated for each strategy. The
probabilities are run using the assumption that there will be some random
noise in the results. The quantity of this noise is set by the "effective
uncorrelated electorate size" (EUES, cell Z24). A lower number here means
that the noise will be more significant. If the EUES is 30, then the actual
"election day turnout" will be a poisson distribution around 30, and each
voting bloc will turnout in a poisson distribution of the appropriate
fraction of 30. This "noise" could simulate polling error (that is, voter
uncertainty of the true makeup of the electorate due to statistical weakness
of polls), voting-day error (that is, turnout fluctuations due to random
chance events), or true error (last-minute swings in the electorate, polling
bias, etc.)
Thus, each voting bloc has an "expected value" for the election, and each
strategy has an expected payoff. This payoff can be negative because it
includes the probability that the strategy will backfire. In order to
calculate these expected payoffs, there are two more parameters for
"strategic cohesion" of offensive and defensive strategies (cells Z27, Z28);
this is the portion of the group in question which may be expected to use
the strategy (since there will always be some fraction of nonstrategic
holdovers).
....
The spreadsheet overall is quite slow in Google Docs. If you want to play
with it more than a small amount, it's probably worth downloading a copy and
opening it in your favorite desktop spreadsheet application (ie OpenOffice,
Excel, etc.)
What I learned from this spreadsheet is that, in a Condorcet tie situation,
there are always some strategies which are rational. As far as I can tell,
while it is possible for a good system to minimize the strategic incentives,
it is not possible to create a system without at least some scenarios where
the expected payoff of a strategy is significant. This holds even in the
face of a fair amount of "noise", and even with a system designed to
minimize strategic payoff. Before making this spreadsheet, I had hoped that
Score DSV would be good enough that, with some noise, the risk of any
strategy would be enough to discourage its use, but that is not the case.
Still, to find scenarios where a strategy clearly pays off takes some work.
I have not done any systematic statistical sampling, but I'd say that with
Score DSV, such scenarios represent around 1/3 of condorcet ties. Given that
condorcet ties should probably occur in somewhere between 1% and 15% of
real-world elections, and that the group of voters for whom strategy is
rational is typically around 25% of the electorate, that means the average
voter will have a rational strategy less than 2% of the time (perhaps far
less). I'd say that that's negligible enough to hope that some kind(s) of
honest normalized voting would be a dominant strategy. Certainly, it seems
to me that this shows that it's unwarranted to imagine 100% strategy in
Condorcet, or to compare the results of Bayesian Regret simulations from N%
strategy in Condorcet systems against the same N% strategic voters in Range
systems.
Jameson Quinn
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