On Tue, May 27, 2003 at 10:18:18AM -0400, Andrew Pimlott wrote:
> sincere strategy by BAD
>
> 9 ABD 9 ABD
> 6 BAD 6 BDA
> 3 DAB 3 DAB
> 2 DBA 2 DBA
> Condorcet+SSD A A
> approval+ A B
A beats B 12:8 12:8
B beats D 15:5 15:5
A versus D 15:5 9:11
Now, the 6 BAD voters (teehee) don't really know how the any of the 9 ABD
voters are actually going to vote. Let's give them 0.5 odds of voting ADB.
In that case you have:
1 9x ABD B wins (B:D = 15:5)
9 8x ABD 1x ADB B wins (B:D = 14:6)
36 7x ABD 2x ADB B wins (B:D = 13:7)
84 6x ABD 3x ADB B wins (B:D = 12:8)
126 5x ABD 4x ADB B wins (B:D = 11:9)
126 4x ABD 5x ADB D wins (B:D = 10:10)
84 3x ABD 6x ADB D wins (B:D = 9:11)
36 2x ABD 7x ADB D wins (B:D = 8:12)
9 1x ABD 8x ADB D wins (B:D = 7:13)
1 9x ADB D wins (B:D = 6:14)
So voting as a bloc against A, without knowing how the bloc of A voters will
vote gives them equal odds of B winning, or D winning. Which means that if
they value A, B and D as:
D: 0
A: x
B: x+y
then their satisfaction shifts from "x", to 0.5(x+y). If the only question
in their mind is whether the bloc of nine A voters will or won't try the
same strategy, it's worth them voting that way if they value B more than
twice as much as they value A; if they don't, it's not in their interests.
That's ignoring a lot of other possibilities: they run the risk of some
of their bloc not voting that way, that some of the 5 other voters might
vote for A, that more people might vote, etc.
But the point is, that voting as a bloc might increase your chance of getting
your favourite option up (from 0 to 0.5 in this example above), but they
also increase the odds of the default option getting up (again, from 0 to
0.5). That's a gamble, not a strategy.
(Compare it against vote splitting in a first-past-the-post election: if
you can get your opponents to divide their vote amongst two candidates,
and can stop your allies from doing the same, you do nothing but increase
your odds of outright success; also, compare it to a pure CpSSD vote of
40 A B C
30 B C A
20 C B A
where A beats B, 60:30, B beats C 70:20, and C beats A 50:40; CvA gets
dropped and A wins. If the 30 BCA voters vote as a bloc, they can make C
the Condorcet winner, by voting 30 CBA, and get a more acceptable result;
but it's a gamble, because it relies on everyone else voting the same
way -- they're trading off the chance of B winning, for a chance of C
winning, which is a losing proposition if you prefer C)
Cheers,
aj
--
Anthony Towns <aj@humbug.org.au> <http://azure.humbug.org.au/~aj/>
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