BR & the CW

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BR & the CW

Most Score advocates emphasize the minimization of Bayesian Regret. I assume that Bayesian Regret (BR) is the sum of everyone's disutility, as measured by distance in issue-space., between the person's position and that of the winner.

Though Score & Approval are excellent methods, quite aside from the matter of BR, I'd like to comment on BR minimization.

The CW is always the BR-minimizer.

No one would deny that the CW is always the BR-minimizer when candidates and voters are on a 1-dimensional political spectrum, or when distances issue-space are measured by the city-block metric.

Well, to a large extent, voters and candidates _are_ on a 1-dimensional political-spectrum. That's why rank-balloting political polls always have a CW.   ...never have a top-cycle for 1st place.

And city-block distance (as opposed to Euclidian distance) is the measure of a voter's disutility.

That's because the separate disutilities resulting from the voter-candidate distance in the various dimensions are additive.

The issue-distances (between the voter & the winning candidate)  that affect the voter affect hir additively.

What about issues that don't affect the voter hirself, but which s/he is altruistically concerned about...such as disutility to groups of people In other countries, or to animals?

Because BR is a simple sum of everyone's disutilities, then a simple sum (rather than a sum of squares) of those disutilities to particular other groups (measured by certain issue-distances for that voter) is what's relevant for BR.

So it's city-block distances. A sum of city-block distances is the measure of BR.

So, because the CW minimizes the sum of city-block voter-to-winner distances, then the sincere CW is the BR minimizer.

Of course any method that meets the Condorcet Criterion (CC) will find the voted CW.

Of course strategy problems could prevent a Condorcet method from findings the _sincere_ CW.

So, what is desired is a CC-complying method that is strategy-proof.

Two such methods are Benham's method and Woodall's method.

Because Benham is more briefly defined, it would be much easier to propose & explain:

 

Benham:

Do IRV till there's an un-eliminated candidate who pairwise-beats each of the other un-eliminated candidates. Elect hir.

[end of Benham definition]

Obviously, if there's a voted CW, then Benham's "until" condition is already met, and therefore no IRV is done, and the voted CW is elected.

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So, with an antagonistic electorate, in which similar parties are even more antagonist toward eachother, strategy is a valid concern, including chicken-dilemma.

Benham has no chicken dilemma, and is, ins general, strategy-proof.

In the CES polling project, the fact that Benham reliably finds the sincere CW makes it valuable for determining whether Score & Approval have minimized BR.  ...by electing the sincere CW.

Michael Ossipoff

 

 

 

 

 

 

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