[EM] Condorcet strategy spreadsheet (was, ...maybe it should be about Condorcet...)

[Juho Laatu]

On Jan 31, 2010, at 9:18 PM, Jameson Quinn wrote:

> .
>
> 2010/1/31 Juho Laatu <juho.laatu at gmail.com>
> Yes. It is not easy to give exact numbers on how often strategies  
> are possible in Condorcet. ...Even the second challenge of election  
> specific commendations based on already available information is  
> hard to meet.
>
> The spreadsheet is about trying to find such strategies.
>
>
> There have been quite a number of (non-political) Condorcet  
> elections but I have not seen anyone point out any obvious strategic  
> opportunities even after the elections. Maybe this also says  
> something about how common the vulnerabilities are (more experiments  
> needed though).
>
> So, even when there is a theoretical vulnerability that some set of  
> voters could use to improve the end result from their point of view  
> by altering their votes, that may still be quite far from practical  
> implementation of the strategy. Have you maybe generated some rules  
> that the voters or parties/candidates could recommend to implement  
> some of the strategies in real life?
>
> The fundamental thing you need for any strategy to work is a  
> (potentially) "cyclical" election. That is, you need to know that  
> (at least aside from your own group of voters) ABC + BCA + CAB > CBA  
> + BAC + ACB. Let's take real cycles first, and potential cycles in  
> the next paragraph. Real cycles very hard to find in real world  
> elections; if A voters hate C, then C voters tend to hate A  
> symmetrically. Thus, IMO all the scenarios that the spreadsheet  
> explores are more-or-less unrealistic; I can't make up good stories  
> about how they could happen in real life.

In principle "natural" cycles could occur in real life. Let's say  
there are three voter groups (33%, 33%, 33%). Each group is mainly  
interested in one topic (A, B, C). We have three candidates (C1, C2,  
C3). The time and effort that candidate C1 allocates to different  
topics is A=>80% B=>20%. C2 allocates B=80% C=20%. C3 allocates C=80%  
A=20%. As a result the voters are likely to vote so that candidates  
C1, C2 and C3 form a preference cycle. Everything is quite normal  
(although simplified, but more complex sets of opinions and candidates  
will not spoil this example). The three candidates just happened to  
plan their campaigns so that they generate a cycle. That could well  
happen.

> However, I have found situations which, if they did occur, would  
> have an obvious strategy for some voter groups. Basically, if you  
> know there'll be an A>B>C>A cycle, and you can guess A will probably  
> win the tiebreaker

One of the polls before the election might indicate so.

> , then B>C>A (by definition, under 1/3 of voters under most  
> tiebreakers) should vote C>B>A to elect C, and B>A>C voters (by  
> definition, a small minority - under 1/6 of voters) should vote  
> B>C>A to help elect B. Note that these strategies contradict each  
> other - but either one is somewhat robustly rational on its own.

I believe we are by default talking about large public elections. Now  
we have the problem of informing the voters on what to do. Candidate  
B's campaign office can not propose B>C>A voters to vote C>B>A in  
order to elect C since B>A>C voters would not like that. Such  
recommendations might also sound like B will give up. It is maybe  
better (and more typical) for B to continue campaigning, claim to a  
potential winner and try to get the missing votes somewhere (so that B  
would win already with sincere votes). Candidate C's campaign office  
might recommend B>C>A voters to vote C>B>A, but candidate B and  
candidate B supporters might think this is just a campaign trick. C  
office might give this kind of recommendations (propaganda) even if  
they would not believe that the poll is accurate and that voter  
behaviour will be similar at the election day.

I just picked some points above to show that it is not easy to control  
and achieve the intended results in this kind of strategy campaigns.  
Do we have some nice set of poll answers that some campaign office or  
voters themselves (one by one, maybe getting assistance from the media  
on how strategic voting works in Condorcet) could study and use as a  
basis for strategic voting recommendations, and then implement this  
strategy successfully? (= working election specific strategy) Can the  
campaign offices trust on some particular poll enough to make their  
move based on that? Public strategic recommendations may also cause  
reactions among the voters (sincere changes of opinions and other  
strategic moves). (And how abut continuing positive campaigning to  
change the opinions of the voters before the election day?)

>
> But all of that assumes a knowable honest cycle - implausible to me  
> for the reasons of symmetry given above. The more plausible  
> situation is if there's a potential to strategically create a cycle,  
> but no real cycle. For instance, say the election is a one- 
> dimensional, partisan ABC battle, with B in the middle; you are a C  
> voter; you know that C is not the likely winner; you have reason to  
> believe that there are more BCA than BAC votes; and you're willing  
> to dishonestly vote CAB and risk A beating B (that is, you don't  
> really care about the difference between A and B). This is still  
> just a bit implausible - if C voters think that B is closer to A  
> than C, why do B voters think that C is closer to them than A?  
> Moreover, most Condorcet tiebreakers would tend to give this  
> election to B or A, even in the face of strategy, unless the fake- 
> CAB strategists were surprisingly unified. If the strategy does  
> "succeed" in picking C, that means that C would have won a plurality  
> election. So this is the "best case" real-life strategy, and it's  
> pretty weak - unlikely to succeed and likely to backfire - and the  
> result is no more "pathological" than what we're already used to. I  
> can imagine it happening rarely, I can imagine being very frustrated  
> by it if I were on the wrong side - but honestly, I can't say that  
> the C voters wouldn't deserve the victory. That is, if this strategy  
> is rational, it is likely that C is the true Range winner, though  
> not the honest Condorcet winner.

Yes. You already identified a number of problems that the C supporters  
will face, so I don't need to try to find more.

In order to prove that Condorcet is not safe enough in typical  
elections it would be enough to present one concrete and rational  
example (= working election specific strategy) that the campaign  
offices could recommend and voters could then implement, or that the  
voters could independently apply. Any of the discussed categories  
would do. But it seems that such examples are not very easy to find.  
Anyone, any good examples in your mind?

Juho



>
>
>
> Juho
>
>
> On Jan 29, 2010, at 6:17 PM, Jameson Quinn wrote:
>
>>
>>
>> 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. 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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