How to vote in Approval

[Joe Weinstein]

Let C1, C2, ... be the N candidates ranked in descending order of your 
utility measures U1, U2, ... for their respective victories.  (These 
measures presumably are reflected by your ratings.)

A rational strategy will have you voting for your first so-many favorites, 
down to a cutoff utility value U0.   That is, on candidate Ci, vote YES if 
Ui>U0, NO if Ui<U0, and vote to whim if Ui=U0.
The question is, how to choose U0?

Rational Strategy.  Totally rationally, one should choose U0 = expected 
value UE of the Ui. (Here, for definiteness,  ‘expected’ is under the 
baseline condition that you yourself don’t vote: a quibble unless the 
electorate is quite small)

As to ‘expected’, Rob’s question implies that your one source of information 
is the snapshot given by the poll data.   Let P1, P2, ... be the respective 
probabilities - imputed by you from the poll data - of the candidate’s 
victory (again, assuming you don’t vote).    Then the Pi, and your Ui, 
together establish UE.  In fact, even if you do not have exact values for 
the Pi, you may still be able to infer partial information sufficient for 
useful conclusions about UE.

How do Rob’s three proposals stack up against the Rational Strategy?  They 
may all do very poorly when the two leading candidates in fact do not have 
overwhelming probability of success yet are among the worst (or maybe the 
best) alternatives.

For instance, imagine a close race among three good guys - C1 (U1=100, 
P1=20%, C2 (U2=95, P2=19%), C3 (U3=90, P3=18%) - and two bad guys C4 (U4=10, 
P4=22%) and C5 (U5=0, P5=21%).

The leaders are C4 and C5.   Rational has you voting for just the three good 
guys.  All three proposed strategies make you vote for the bad leader C4 as 
well.

Now, suppose the implications of poll numbers shift slightly toward the good 
guys:  C4 and C5 each lose 2% and C1 and C2 each gain 2%.  Rational still 
has you voting for the three good guys.  All three proposed strategies have 
you voting just for C1.


High-Utility Victory-Median Strategy.  Alternatives proposed to the Rational 
Strategy have appeal mainly because in some situations you cannot readily 
compute UE to good approximation.  However, compared with the three 
proposals on Rob’s list, it seems that you would often do a lot better by 
simply finding the victory-probability median (midpoint) of your ranking, 
and taking as U0 the utility of the candidate in that position.

That is, as above, rank all candidates in descending order of preference, 
and estimate (from the poll) their respective probabilities Pi of victory.  
Proceding from top-rank down, find the first (i.e. highest ranked) candidate 
for which the sum of probabilities, from the top down to and including the 
candidate, is at least 50%.  Vote at whim for him and YES for candidates 
above him.

In the above example, with both the original and the shifted poll numbers, 
this strategy would have you vote YES for C1 and C2, and at whim for C3.

Kudos and apologies in case, as seems likely, someone else (Mike or ...?) 
has already noted and named this high-utility victory-median strategy.

Joe Weinstein
Long Beach CA USA




----Original Message Follows----
From: Rob LeGrand <honky1998 at yahoo.com>
Reply-To: honky98 at aggies.org
To: election-methods-list at eskimo.com
Subject: Re: How to vote in Approval
Date: Mon, 1 Apr 2002 22:57:54 -0800 (PST)

Okay, so far I've seen at least three proposals for Approval strategy:

1. Approve every candidate you prefer to the current first-placer; approve 
the
    current first-placer if you prefer him to the current second-placer.

2. Approve every candidate you like at least as well as your favorite of the
    current first-placer and current second-placer.

3. Approve every candidate you prefer to a 50-50 lottery between the current
    first-placer and current second-placer.

I agree with Mike that the second one isn't too good.  I think it would be
likely to give a strong advantage to current frontrunners, just like 
plurality
does, but I haven't simulated that strategy yet.  The third might work.  I 
know
using the first will always lead to and lock on a sincere Condorcet winner 
when
there is one.

Let's hear some more ideas!  My ground rules:  The voter rates each of the
candidates with a real number between 0 and 100 and has a full ranking of 
the
candidates from the latest poll.  How should he determine his Approval vote?

--
Rob LeGrand
honky98 at aggies.org
http://www.onr.com/user/honky98/rbvote/calc.html



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