[EM] Viability (finding the odds after you've voted)

[Kevin Venzke]

I have a decent idea for a refinement, which might permit CR to boil down
to Approval ballots filled out per the Better-Than-Expectation strategy.
You could also do the Maximum Power strategy.

Suppose there are two factions that vote roughly as follows:
50%: A 10, others 0
50%: B 10, C 10, others 0

If you use the normal viability definition (Borda score, or CR sum), it looks
like the candidates have an equal chance of winning.  But intuitively this is
not so.  A has a 50% chance of winning, and B and C each have about a 25%
chance.  The math to produce these odds is pretty simple:

Instead of giving a candidate the score it was given from a certain ballot, give 
it the score AS A PERCENTAGE of the total points awarded from that ballot.  (That
means total points awarded from a ballot equal 1.0, regardless of how many slots
were on that ballot.)

Now that the score is measured with more sophistication, we can actually guess
at each candidates' odds of winning.  No need to worry about front-runners.

Let me attempt an example.

50: A 10, E 3
50: B 10, C 10, D 10, E 6

A gets (10/13)*50 or 38.46% odds
B C and D each get (10/36)*50 or 13.88% odds
E gets (3/13)*50 plus (6/36)*50 or 19.87% odds

Odds total 100% (or should).

First faction's expectation:
10*38.46% + 3*19.87% = 3.846 + .5961 = 4.4421
Second faction's expectation:
3(10*13.88%) + 6*19.87% = 4.164 + 1.1922 = 6.086

Resulting approval ballots:
50: A
50: BCD

I do find it slightly bizarre that E gets "more odds" from the first faction
than the second.


For the Maximum Power strategy:
Instead of calculating expectation, you try to approve candidates with a total
of 50% odds.

First faction:
A alone is 38.46 (11.54 from 50), A and E is 58.33 (8.33 away).
Second faction:
BCD sum to 41.64 (8.36 off); adding in E gives 61.51 (11.51 off).

So the result is the same, except that the first faction also approves E.

Any thoughts?


Kevin Venzke
stepjak at yahoo.fr


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