[EM] Re: simulating an Approval campaign/election

Rob LeGrand honky1998 at yahoo.com
Tue Feb 1 16:06:44 PST 2005

Russ wrote:
> Interesting. Do you mind if I ask why you are interested in
> Declared-Strategy Voting as opposed to Undeclared-Strategy
> Voting?

DSV is the invention of Lorrie Cranor and the subject of her
dissertation (http://lorrie.cranor.org/dsv.html).  "Declared" just
means that a voter declares a strategy for the DSV system to use to
vote for him in the simulated election(s).

> Does another strategy converge even if no Condorcet winner
> exists?

In the example

     A   B   C
9: 100   0  90  (9 voters have utility 100 for A, etc.)
8:  90 100   0
6:   0  10 100

the only equilibrium when all voters use strategy T is


The same is true for strategies B (Poll Assumption (Approval) from
7.4 of Brams and Fishburn's Approval Voting) and I (change the
approvals from your last ballot just enough so that you approve of
one of the two frontrunners and not the other).  Note that the
equilibrium disappears if the 9 A-favorite voters realize that they
can improve the outcome from their perspective by approving C in
addition to A, as strategy A would recommend.

> I assume you mean that plurality can be manipulated by throwing
> in spoilers (e.g., Nader or Perot).

No, I mean that when everyone votes strategically (giving "spoiler"
candidates no votes), those that are voting insincerely are
manipulating the election, usually into equilibria that allow
little chance for candidates that might turn out to be Condorcet
winners.  Not that there's anything wrong with this manipulation--
it's the smart way to vote if you want to affect the outcome.

> And as for multiple equilibria, it seems to me that all but one
> of those equilibria is practically inaccessable if it requires a
> third party to switch places with one of the two dominant
> parties.

I agree that plurality leads to a static two-party structure.
Other systems that lead to equilibria more often than Approval also
lead to party systems that limit entrance of new candidates and
parties.  Approval is the most stable voting system I know that
still allows a dynamic (and party-less?) political system; at the
extremes, plurality gives you stability at the expense of openness
and Borda gives you dynamism (too much!) at the expense of ultra-
instability.  Also, plurality encourages each ideology to run at
most one candidate (and often none), Borda encourages each ideology
to run many candidates, and Approval encourages running those
candidates that have a chance to win.

> You seem to have confirmed my hypothesis that, in the idealized
> case (DSV batch mode), Approval voting almost always converges on
> the Cordorcet winner if one exists, but rarely (never?) converges
> if one does not exist.

Yes, that's true, if all voters use strategy A or something very
much like it, which according to my investigations is in their best

> If that is true, then it seems to me that Approval may be roughly
> equivalent to Condorcet with random selection of the winner from
> the Smith set. Do you agree with that?

Only very roughly.  The selection from the Smith set is random
(assuming that the number of rounds is large and at least
pseudorandom, e.g. based on the number of voters in a large
election in a way that makes it very difficult to predict), but
some members of the Smith set may have no chance of winning.

> If so, has anyone shown that the Condorcet winner based on a
> "good" Condorcet resolution method would at least be favored in
> the random selection process?

That depends on the random selection process.  If you use regular
batch DSV and stop after a predetermined number of rounds, you'll
get different winning probabilities than if you use cumulative
batch mode or ballot-by-ballot mode.  According to my simulations,
the above example election run in ballot-by-ballot mode for a large
random number of rounds would give A, B and C win probabilities of
approximately 45.8%, 32.0% and 22.2%.  Most of us would agree that
A has the best claim to victory, followed by B.  Batch mode gives A
50%, B 25% and C 25% (tight loop); cumulative batch mode gives
roughly A 30.4%, B 47.8% and C 21.7%.  (The win probabilities for
cumulative batch mode are relatively difficult to measure because
the intervals between leader changes become longer and longer as
the number of rounds increases.)  To me, these results confirm my
intuition that ballot-by-ballot (with a random voter order in each
round) is a fairer way to find a winner than the batch modes given
a large number of rounds.

Rob LeGrand, psephologist
rob at approvalvoting.org
Citizens for Approval Voting

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