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

[Forest Simmons]

On Fri, 21 Nov 2003, Rob LeGrand wrote:

> 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.

CRAB stands for Cumulative Repeated Approval Balloting.

The ballot-by-ballot mode is automatically "Cumulative."

The "batch mode" of Lorrie Cranor is not cumulative.

Lorrie found that ballot-by-ballot was better than batch, but that
students in her survey didn't like that it required randomness (in the
order of counting the ballots).

For that reason I proposed CRAB, counting all ballots together but
accumulating the results over many iterations as in the batch mode.

I haven't tried CRAB on David's example, but note that in David's A,B,A,B
...  cycle (in his non-cumulative version of repeated approval voting) the
Condorcet winner is always in second place.

It would be impossible to have an A,B,A,B,... cycle in CRAB if C were
always in second place (in cumulative votes) since eventually when the
winner changes from A to B, A's cumulative votes will be enough to keep A
in second place for awhile.

In other words, after sufficiently many votes have accumulated, the
deposed winner will enjoy second place status for awhile, so in CRAB there
can be no two candidate cycle based on a third candidate always being in
second place.

I believe that CRAB will eventually find an equilibrium with C in first
place in David's example.

I'll go through the computation later this weekend.


> 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.

What is the function of "last own vote?"

Forest