[EM] Strategy free method proposal
rob brown
rob at karmatics.com
Thu Dec 15 13:33:22 PST 2005
I'm submitting the following method that I believe to be strategy-free.
There is no randomness involved (although like any method without
randomness, it is subject to true ties). I believe it is immune to
unresolvable cycles, with the exception of true ties. (I have not proven
this last point)
I should probably make clear that I submit this not so much as a possible
system to be used for real public elections, but as something to use a
hypothetical for comparison to and evaluation of other methods. I expect
that it is very likely to produce similar results to some Condorcet methods.
First, I want to restate Approval Strategy A (which Rob LeGrand first named,
I believe), as well define as "Range Strategy A" (which I am naming):
Approval Strategy A
If an Approval voter knows with 100% assurance who would be the winner
without his vote, and who will be the runner up, his strategy is very
straightforward: give a vote of yes to every candidate he prefers to the
leading candidate, and also give a yes to the leading candidate if that
candidate is preferred to the runner up. All other candidates should be
given a no.
Range Strategy A
If a voter is allowed to vote with intermediate "shades" between yes and no,
for instance being allowed any floating point value in between zero and one,
Approval strategy A still is the best strategy as long as the voter knows
100% who would be the winner and runner up without his vote. If he does not
know, he can give intermediate votes based on the percent chance of the
various outcomes. In the table below, I show a situation where there are 4
possible outcomes, each with associated probabilities. In the rightmost
column is the Approval ballot showing optimum strategy (Approval strategy A)
were this outcome known to be the case.
Voter preference:
B>C>A
Leader Runner-up Likelihood Optimum ballot
A B 73% A:0, B:1, C:1
B A 15% A:0, B:1, C:0
A C 10% A:0, B:1, C:1
C A 2% A:0, B:1, C:1
Given the above, we can create a composite (weighted average) ballot as
follows:
A: 0, B: 1, C: 0.85
C is given a .85 because there is an 85% chance that 1 (yes) is the best
vote, and 15% chance that 0 (no) is the best vote.
We will call this way of making a composite ballot to use in a Range
election "Range Strategy A".
-------
Given the above definitions, here is the system I propose.
Voters rank candidates in order of preference (as they would with Condorcet,
IRV or Borda elections).
In multiple rounds, each ranked ballot is converted into a Range ballot,
using Range strategy A. On the first round, each candidate is assumed to
have an equal probability of winning as every other candidate, and Range
ballots submitted accordingly. The winner and runner up are calculated
(simply summing all votes for each candidate), and another round is
conducted with it being assumed that those are 100% likely to be winner and
runner up. On subsequent rounds, it uses an average of all past rounds to
determine the percent chance of each candidate being winner or runner up,
and each ranked ballot converted into a Range ballot accordingly.
Eventually, an equilibrium should be reached, settling on a winner and a
runner up, and with each ballot now approaching a "pure" Approval ballot,
with all candidates given either a 0 or 1 on all ballots.
---------
A few notes:
This method uses the concept of a "software agent", which is a program or
part of a program that operates solely with the interests of a single
"client", in this case a single voter. The process of converting the ranked
ballot to a Range ballot must be done without regard for the "collective
good", otherwise, the voter will find that he can do better if he tries to
"trick" the agent into converting it in ways more aligned to his own
interests. This way, there is no reason for a voter to lie to his agent,
since his agent is always working towards his interests alone. Of course,
we hope the "collective good" is still achieved, but it is done in a way
that is fair and stable as it will not give people incentive to be
dishonest.
Note that Range Strategy A is not necessarily the "best strategy", if used
in a normal election. It would still probably be strategically advantageous
to vote all 0's and 1's in a final election, regardless of whether we know
100% the outcome or not. However, in the above method, Range ballots are
never used in a final round, but are only used in preliminary rounds (by the
time an equilibrium is reached, the ballots will always have only zeros and
ones, effectively being Approval ballots). In the case of preliminary
rounds, though, it makes strategic sense to vote with "shades", because it
increases the chance that an equilibrium will be found that is agreeable to
this voter.
To speed up processing, all identical ballots can be grouped together and
processed at once.
I have not covered what will happen in the case of ties during preliminary
rounds, nor covered how to handle equality within ballots. Both of these
are easy to handle in a consistant and logical way, but I avoided discussing
them so as not to distract from the basic description. I will be happy to
explain in detail if anyone cannot see how they should be handled.
It is possible that the second round could be done in a better way. Instead
of submitting ballots based on the assumption that the winner and runner up
from the first round are 100% likely, this could be "averaged in" with our
previous round's assumption, that all candidates are equally likely to win.
Again, I glossed over that to keep the description fairly simple, and
because I don't think it would ever affect the final result.
While I have not proven that this system will always find an equilibrium, I
believe that by eliminating "cusps" I have removed the main thing that
causes similar methods to get stuck in cycles.
I'm not sure if this should be considered a DSV method, because voters
simply rank the candidates. I chose not to have voters give each candidate
a Range-style "rating" on their ballots, for a few reasons:
1) I want it to be easier to directly compare to Condorcet methods
2) I prefer ranking rather than rating from a UI perspective. I feel that
rating will always feel like there is strategy involved (whether or not
there really is).
3) From a practical perspective, ranked ballots are easier to compress and
optimize. Because it is easier to "package up" the complete set of ballots
of an election (say, to allow them to be downloaded on the internet), it has
the ability to be more transparent.
4) If an equilibrium is found (where all voters are submitting Approval
Strategy A ballots), ratings will not make a difference, only rankings.
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