[EM] Delete this if you are offended by non-deterministic methods

Simmons, Forest simmonfo at up.edu
Fri Nov 18 15:29:43 PST 2005

I wrote ...

Disclaimer:  Doubtless some situations require deterministic election methods.  This message has nothing to do with any such situations.

Simple Lottery Method:

 If there is an alternative that pairwise defeats the approval winner A, then toss a coin to decide between A and the most approved alternative that pairwise defeats A.  Otherwise just elect A.

... and then I gave some examples.  Now (under the same disclaimer) I continue ...
The following modification works identically on those examples, but is an improvement on an example that I supply below:
If there is an alternative that is not majority defeated by the approval winner A, then toss a coin to decide between A and the most approved alternative (besides A itself) that is not majority defeated by A.  Otherwise, just elect A.
Among other properties, this method satisfies the Favorite Betrayal Criterion; there is never any advantage to ranking your favorite alternative below some other alternative.
In this version a single alternative gets 100% of the probability only if that alternative is the approval winner and majority defeats every other alternative, a position that indeed should be totally immune and unassailable.  
And while a marginally acceptable Condorcet Winner may or may not end up sharing the probability with an alternative that has a broader base of support, it is certain that under this method an Approval winner with less than 50% approval will have to share the coin toss with another alternative, as in the following example:
49 B
24 A
27 C

Alternative B is both the Condorcet and Approval winner, but does not majority defeat either of the other two candidates, so C gets to share the probability with B.
This would tend to discourage the truncations, since, for example, the A supporters, acting unilaterally could get A into the winning circle with either B or C by ranking the preferred of these (unless foiled by stronger moves to the contrary by the other two camps).
Now it seems to me that under this method ...
1.  if an alternative cannot muster the support needed to gain 100% probability, then that alternative does not deserve 100% of the probability, and furthermore,
2. whatever probability an alternative gets, it is likely to deserve more than any alternative that gets less.
I would be interested in possible counterexamples to these two statements.
Also, I would be interested if somebody came up with a simpler method (deterministic or not) that satisfied these two statements to a higher degree.
Simple random ballot would satisfy (2)  but not (1), for example.
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