[EM] Re: Approval Strategy A- Question for Rob LeGrand

[Rob LeGrand]

David wrote:
> Thanks for the information. So am I right in thinking that strategy A
> gets to the Condorcet winner by a process of iteration. In response to a
> series of Approval polls the voters alter their choices and end up voting
> in such a way that they elect the Condorcet winner. Or is it  more
> complex than this in theory (I know it's more complex in reality)?

That's correct.  Following Lorrie Cranor's Declared Strategy Voting
(http://lorrie.cranor.org/dsv.html), I distinguish between ballot-by-ballot
mode and batch mode.  You've been using batch mode (all voters react to the
last results at the same time), which in this example indeed leads to
cycling between winners A and B.  Strategy A doesn't always lead to an
equilibrium in batch mode even when a Condorcet winner exists (see
http://groups.yahoo.com/group/election-methods-list/message/9713), but it's
extremely likely to when voters have fully-ranked preferences.  When many
voters don't, as in your example, equilibria are less common and don't
always elect the Condorcet winner.  My simulations sometimes generated tied
preferences but not often enough to produce this kind of situation.

Ballot-by-ballot mode (voters take turns reacting to the latest results)
would eventually find an equilibrium, but it won't necessarily elect the
Condorcet winner either:

A   380   approve A
A>B  28   approve A
A>C   9   approve AC
B    80   approve B
B>A   2   approve B
B>C 133   approve B
C     4   approve C
C>A  13   approve CA
C>B 351   approve CB

This equilibrium elects B (A 430, B 566, C 377).  Strategy A is still
optimal here in the sense that none of the nine blocs can change its vote
and improve the result from its perspective.  In fact, as far as I can see,
there's no coalition of blocs that can band together and change the result
to the coalition's advantage, which makes it a strong Nash equilibrium. 
Steven Brams has proved that every Approval strong Nash equilibrium elects
a Condorcet winner . . . when all voters have fully-ranked preferences. 
When they don't, obviously strange things can happen.  Consider a simpler
example:

45:Reagan>Anderson=Carter
20:Anderson>Carter>Reagan
35:Carter>Anderson>Reagan

Now there are two equilibria, one that elects Anderson and one that elects
Carter.  When the three blocs are considered players, this election reduces
to the game of chicken.  The Reagan voters are effectively sitting the
election out; this strangeness goes away when the Reagan voters discover a
preference between Carter and Anderson and become kingmakers.

As far as I can tell, strategy A does as well for a voter as any other
Approval strategy that considers only current "poll" results, own
preferences and last own vote.  If anyone has a better one, or even an
interesting new one, please let me know.

=====
Rob LeGrand, psephologist
rob at approvalvoting.org
Citizens for Approval Voting
http://www.approvalvoting.org/

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