[EM] Efforts to improve on CR's strategy

[Ken Johnson]

>Date: Wed, 19 May 2004 23:24:29 -0700
>From: Bart Ingles <bartman at netgate.net>
>
>...  In your 10 candidate, 1 issue trial, are you able to account
>for why sincere CR, exaggerated CR, Condorcet, Borda, IRV, and Plurality
>all yield exactly the same average across 100,000 elections?  It looks
>like top-two Runoff is within 0.1% of the same score.
>
I think it's simply the case that with 1 issue, all voters' CR profiles 
are precisely correlated (i.e., any two profiles differ only by a 
multiplicative scale factor), so all these methods become equivalent.

>
>Stranger still, exaggerated CR should be equivalent to Approval, but the
>scores here are wildly dissimilar.
>
What I call "ExaggerateCR" is not actually the optimal zero-info CR 
strategy, which would be equivalent to Approval.

>
>I'm afraid I don't trust the simulation.  There are too many cases where
>widely different methods return exactly the same results.  I would
>expect this with two candidates (if the optimal strategies are done
>correctly), but not with three or more.
>
>Also, I think the Approval strategy used is not what is generally
>recognized as optimal zero-info strategy.  You have voters approving all
>candidates where CR is >= 0.  The usual "sincere strategy" is to approve
>all candidates where CR is greater than the mean CR of all candidates. 
>In the two candidate cases, this should give Approval the same score as
>the other methods, as one would expect.
>
"Sincere strategy"? From my perspective all strategies are insincere.
Based on the optimum zero-info strategy, should the approval cutoff be 
at the mean CR of ALL candidates, or of just the highest- and 
lowest-rated candidates? (I assumed the latter for "ExaggerateAV".)

Here's a conceptual example that I think better illustrates the problem 
that I observed. Suppose you vote in an election in which there are 6 
candidates and you have no idea how anyone else votes. Your sincere CR 
profile for candidates A ... F is
    SincereCR: A(0.7), B(0.5), C(0.3), D(0.1), E(-0.1), F(-0.3)
(This assumes signed CR's, with an approval cutoff of zero.) What I call 
"ExaggerateCR" simply applies a linear transformation so that the max 
and min CR's are +1 and -1:
    ExaggerateCR: A(1.0), B(0.6), C(0.2), D(-0.2), E(-0.6), F(-1.0)
Sincere Approval is based on the Sincere CR profile:
    SincereAV: A(1), B(1), C(1), D(1), E(0), F(0)
Strategic Approval is based on the ExaggerateCR profile:
    ExaggerateAV: A(1), B(1), C(1), D(0), E(0), F(0)
All the rank methods sort the candidates by CR:
    A > B > C > D > E > F

Now suppose the ballots get counted and it turns out that the total 
ballot count is 1. No one else bothered to vote, so your ballot 
determines the election. With the exception of Approval, all methods 
give A as the winner, whose CR is 0.7. SincereAV gives a 4-way tie 
between A, B, C and D, and ExaggerateAV gives a 3-way tie between A, B, 
and C. In computing the winner's CR, I assumed that a tie is broken by 
random choice, and I report the probability-weighted CR as the result. 
For example, with SincereAV A, B, C and D each has a 25% chance of 
winning, so I report the winner's CR as the average of A, B, C and D - 
which is 0.4. Similarly, for ExaggerateAV the winner's CR is reported as 
0.5.

The 1-ballot assumption is obviously unrealistic, but I think this 
example conveys the essence of what was happening with my simulations.

Ken Johnson