Better than Condorcet?
A few weeks ago I argued that Bayesian decision theory was the right
approach to voting (assuming sincere voters). I then went away and
implemented a semi-Bayesian voting method to test my claim, comparing it
against Condorcet methods under a carefully selected spatial model. The
semi-Bayesian method gives 95.7% accuracy where Condorcet methods must
fall in the range from 90.5% to 91%.
My model uses an infinite population of voters whose distribution is a
mixture of 3 components, each of which is uniform over a unit circular
disc. There are 4 candidates drawn randomly from the voter distribution
(which was probably a bad decision).
The semi-Bayesian method is not very Bayesian at all; it simply solves
the equations relating the model parameters to the ballot frequencies,
and elects the best candidate under the implied model. It would be
possible to improve on it by using prior information to choose between
alternative solutions. A truly Bayesian method would integrate over the
solution space.
I can's say that I think this proves very much. It may be of some
interest to have an example in which Condorcet methods are beaten by a
non-Condorcet method under a spatial model, but not many people would
regard this as impossible. I wouldn't dream of suggesting that any
Bayesian method was worth considering in practice, and I doubt than a
reasonable approximation can be derived.
Anyone interested in the gory details will find them on a web page:
http://www.masterlyinactivity.com/condorcet/semibayes.html
CJC
On 01.02.2022 11:13, Colin Champion wrote:
A few weeks ago I argued that Bayesian decision theory was the right
approach to voting (assuming sincere voters). I then went away and
implemented a semi-Bayesian voting method to test my claim, comparing it
against Condorcet methods under a carefully selected spatial model. The
semi-Bayesian method gives 95.7% accuracy where Condorcet methods must
fall in the range from 90.5% to 91%.
My model uses an infinite population of voters whose distribution is a
mixture of 3 components, each of which is uniform over a unit circular
disc. There are 4 candidates drawn randomly from the voter distribution
(which was probably a bad decision).
The semi-Bayesian method is not very Bayesian at all; it simply solves
the equations relating the model parameters to the ballot frequencies,
and elects the best candidate under the implied model. It would be
possible to improve on it by using prior information to choose between
alternative solutions. A truly Bayesian method would integrate over the
solution space.
I can's say that I think this proves very much. It may be of some
interest to have an example in which Condorcet methods are beaten by a
non-Condorcet method under a spatial model, but not many people would
regard this as impossible. I wouldn't dream of suggesting that any
Bayesian method was worth considering in practice, and I doubt than a
reasonable approximation can be derived.
Anyone interested in the gory details will find them on a web page:
http://www.masterlyinactivity.com/condorcet/semibayes.html
There's a result that if the method X is of the form where a majority
can force an outcome, then Condorcet,X is no more strategically
susceptible than X. But no such result has (to my knowledge) been found
for VSE, and I would be surprised if there is any at all.
The best VSE-wise method I know of is Sinkhorn, which is not Condorcet
either. Perhaps you could compare your semi-Bayes method performance to
its performance?
The Sinkhorn method is implemented in Warren Smith's IEVS:
https://rangevoting.org/IEVS/IEVS.c
-km
Kristofer wrote: "The best VSE-wise method I know of is Sinkhorn, which
is not Condorcet either..." I added it to my big evaluation and found it
to be somewhat weaker than Borda.
The reason for expecting better from it is Warren D. Smith's own
evaluation in which the Borda count itself outperforms Condorcet
methods.[1] It appears that Smith's voting model is not spatial in the
conventional sense, but instead an unexplained multidimensional model in
which a dot product takes the place of distance; this model is outlined
in his "Range voting" paper of 2000. So the poor showing of the Sinkhorn
method in my own evaluation is not inconsistent with its good showing in
Smith's.
CJC
[1]. https://rangevoting.org/StratHonMix.html, brought to my notice by
Kristofer.
On 05.02.2022 19:02, Colin Champion wrote:
Kristofer wrote: "The best VSE-wise method I know of is Sinkhorn, which
is not Condorcet either..." I added it to my big evaluation and found it
to be somewhat weaker than Borda.
The reason for expecting better from it is Warren D. Smith's own
evaluation in which the Borda count itself outperforms Condorcet
methods.[1] It appears that Smith's voting model is not spatial in the
conventional sense, but instead an unexplained multidimensional model in
which a dot product takes the place of distance; this model is outlined
in his "Range voting" paper of 2000. So the poor showing of the Sinkhorn
method in my own evaluation is not inconsistent with its good showing in
Smith's.
Fair enough. This is why reproduction is important :-)
-km