[EM] Introduction (cont.)

[Richard Moore]

Roy wrote:
 > I disagree. Strategy would indicate that they would 
"normalize" their
 > votes to fill the range: last choice gets lowest rating, 
first choice
 > gets top rating, but in-between choices would be 
distributed. This
 > seems fair and desirable to me. Every voter has an equal 
voice, and is
 > allowed to indicate his level of support for each candidate.

To see how strategic voting would work, consider a very
simple example with three candidates. Let's say your sincere
ratings for the three are A=100, B=50, and C=0. Now you
would obviously vote A full-scale positive and C full-scale
negative. What about B?

Voting a cardinal rating of X for B would increase B's
chances of winning by some probability increment, X*deltaPb.
(This linear relationship is accurate as long as the 
population is not very small). It will also decrease A's and 
C's chances by X*deltaPa and X*deltaPc, where deltaPa + 
deltaPb + deltaPc = 0. The expectation of utility, which is 
what we try to maximize when we seek a mathematically 
optimum strategy, will change by an amount given by

	deltaEU = X*( 100*deltaPa + 50*deltaPb )

(remember that deltaPa is negative). Consider that if
100*deltaPa + 50*deltaPb is positive, you will get a
positive strategic benefit from your vote for B. That 
benefit is maximized if X is set to the full-scale positive 
rating. On the other hand, if the value in parentheses is 
negative, you get a negative benefit from your B vote. In 
this case you would want to minimize the negative impact by 
voting B at the lowest end of the scale.

 > If you believe they would turn it into Approval voting, 
you must
 > believe the same thing about allowing ties in any ranking 
system: all
 > voters would tie their approved choices for #1 and their 
disapproved
 > for #2.

No, there is a difference between rating and ranking. Ranked 
systems such as Borda and Condorcet generally penalize this 
sort of collapsing. There are cases in Condorcet in which 
there is an incentive for strategic collapsing or reversal 
but that depends on the completion method and the particular 
circumstances. In CR the incentive is always there.

Actually, it's possible to use a CR method based on L2 norms
(square root of the sum of the squares). In this method, if 
you increase the rating of your second favorite, the rating 
of your favorite decreases to keep the L2 norm of your votes 
constant. All allowed votes then lie on a unit sphere, and 
the optimum vote -- the one that gives the best strategic 
value -- doesn't always coincide with the sphere's 
intersection with one of the axes of the coordinate system. 
You would want to vote in a way that is proportional to the 
strategic value of each candidate to you (which is a 
function of the candidate's utility and the probability 
deltas associated with each vote/candidate pair). Note that 
this still isn't guaranteed to be a sincere vote except in 
zero-information elections.

Richard