ACC, more strategy-free?

[Adam Tarr]

Mike wrote:

>That hadn't occurred to me--that there's another non-reversed
>defensive strategy that always thwarts offensive order-reversal
>in the wv methods: equal-top-ranking of the CW.
>
>Of course the Nader voters might insist that the Gore voters
>announce publicly that they're going to defensively truncate,
>because that's an easier or more mild strategy. Given that announcement,
>the Bush voters wouldn't order-reverse.

For reference, Mike is referring to my example, where the sincere 
preferences are:

49: Bush>Gore>Nader
12: Gore>Bush>Nader
12: Gore>Nader>Bush
27: Nader>Gore>Bush

...making Gore the Condorcet winner.  But the Bush voters are threatening 
to reverse:

49: Bush>Nader>Gore
12: Gore>Bush>Nader
12: Gore>Nader>Bush
27: Nader>Gore>Bush

...which gives the election to Bush (N>G 76, B>N 61, G>B 51).  I suggested 
that the Nader voters could equal-rank Gore:

49: Bush>Nader>Gore
12: Gore>Bush>Nader
12: Gore>Nader>Bush
27: Nader=Gore>Bush

(B>N 61, G>B 51, N>G 49), and now Gore wins in a winning-votes 
method.  Mike suggests that another way for the Gore camp to achieve their 
desired results is if the Gore>Bush>Nader voters announce their intention 
to truncate.  If the Bush voters still order reverse, you get

49: Bush>Nader>Gore
12: Gore>Bush=Nader
12: Gore>Nader>Bush
27: Nader>Gore>Bush

(N>G 76, G>B 51, B>N 49) and now Nader wins.  Since this is the 
least-desired result of the Bush camp, they will fear such an outcome and 
lose their incentive to order-reverse, and will in stead vote honestly:

49: Bush>Gore>Nader
12: Gore>Bush=Nader
12: Gore>Nader>Bush
27: Nader>Gore>Bush

And now Gore is once again the Condorcet winner.

So, which one of our two equilibriums is more meaningful?  Both are Nash 
equilibria, even if we define the players to be entire sets of voters.  (I 
believe Mike coined the phrase "many-voter equilibrium" for this.)

I'd say that both equilibria are illustrative, for different 
reasons.  Mike's is nice because it only involves one simple truncation at 
the bottom of the rankings, by one camp, and everyone else votes 
honestly.  Nobody has any need to insincerely rank another candidate equal 
to their favorite.  It is realistic that you could convince these Gore 
voters to drop Bush from their ballots, since Nader is unlikely to win anyway.

The reason I also like the other equilibrium, however, is that it is more 
free of concern about what the other camp does.  This makes it more 
stable.  If the Gore>Bush voters are told to bullet vote Gore, but the Bush 
voters announce that they plan to order-reverse anyway, then you get into a 
game of strategic brinksmanship.  Both camps would prefer each other to 
Nader, but if they protect against Nader (by not truncating in the Gore 
camp, or by not reversing in the Bush camp) they run the risk that the 
other side will use the more aggressive strategy and win the election.

Looking at this as a two-player game sheds some light on the 
situation.  The players are the 49% Bush>Gore>Nader voters, and the 12% 
Gore>Bush>Nader voters.  Each has two strategies.  The Bush camp's 
strategies are Bush>Nader>Gore and Bush>Gore>Nader ("reverse" and "no 
reverse"), while the Gore camp's strategies are Gore>Bush=Nader and 
Gore>Bush>Nader ("truncate" and "no truncate").  The outcomes of the game are:

  Truncate | No Truncate
|---------+-------------|
|         |             |
|  Nader  |   Bush      |  Reverse
|         |             |
|---------+-------------|------------
|         |             |
|  Gore   |   Gore      |  No Reverse
|         |             |
-------------------------

(I fully realize that the above table looks horrible in most 
readers.  Hopefully you get the idea.  If it's too garbled to make out, cut 
and paste it into a simple text viewer and it will look fine.)

There are two Nash Equilibria here: (no-reverse, truncate) and (reverse, no 
truncate).  One gives the election to Gore, and the other to Bush.  Hence, 
the game of brinkmanship to see which camp will be able to bully the other 
camp into letting them win to stay away from Nader.

That's not all the information we can garner from this table, 
however.  Game theorists refer to a strategy as a "dominated strategy" if, 
for every response the other player gives, one strategy is as good or 
better than another.  In this case, truncation is a dominated strategy for 
the Gore camp.  If the Bush camp reverses, they do better by not 
truncating, and if the Bush camp does not reverse, then they do just as 
well by not truncating as by truncating.  This suggests that the Bush camp 
has an advantage over the Gore camp in the game of brinkmanship.

I fully recognize that this does not mean the Bush camp will succeed in 
convincing the Gore camp to give in and rank Bush second.  The leaders of 
the Bush camp seem to have a much more difficult job, trying to convince 
their supporters to uprank Nader, even though this runs the risk of handing 
Nader the election.  The leaders of the Gore camp just have to convince 
their supporters to bullet vote, which seems to me a much easier job.

Furthermore, it seems that the Bush voters would be even more averse to 
Nader than the Gore voters, which puts them at a disadvantage in the 
brinkmanship/bullying contest.  This won't necessarily be true in a more 
general case, though.

At any rate, back to my original point.  This is why I like the other 
equilibrium, where the Nader voters announce that they plan to equal-rank 
Gore and Nader.  This effectively nullifies all offensive strategy on the 
part of the Bush camp.  Of course, it requires insincere equal 
first-ranking, so it has problems of its own.  But it has none of the 
stability issues of the other equilibrium.

-Adam

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