[EM] Sprucing up vs. Condorcet Lottery vs. immunity: The "twisted prism" example

[Jobst Heitzig]

Dear Forest!

Your sprucing up technique is a very nice idea since it can simplify the 
tallying of those methods which fulfil beat-clone-proofness and 
uncoveredness. However, some of which you wrote has confused me 
completely: Did I understand you right in that you claim that the 
technique should reduce everything to the 3-candidate case? I wonder how 
that could possibly be when, in general, there is neither a proper beat 
clone set nor a covered element! Second, I don't understand how the 
method I posted yesterday under the name "Condorcet Lottery" could be 
the same as anything based on Random Ballot???

In order to study these questions I provide the following example which 
is also interesting with respect to immunity:


The "twisted prism" example:
----------------------------
Six candidates A1,A2,A3,B1,B2,B3, 15 voters. Sincere preferences:
4 A1>B2>A2>B1>B3>A3
4 A2>B3>A3>B2>B1>A1
4 A3>B1>A1>B3>B2>A2
1 B1>B3>B2>A1>A2>A3
1 B2>B1>B3>A2>A3>A1
1 B3>B2>B1>A3>A1>A2
Defeats:
A1>A2>A3>A1, strength 10 (the "upper clockwise" 3-cycle)
B1<B2<B3<B1, strength 10 (the "lower counter-clockwise" 3-cycle)
Bi>Ai, strength 9 (the 3 "straight upward" beats)
Ai>Bj (i!=j), strength 8 (the 6 "diagonal downward" beats)

The following diagram shows the two cycles and the upward beats:

A1
10/| \ clockwise cycle
/ | \
A3------A2
| | | upward beats
|9 | | (diagonal downward beats omitted)
| B1 |
| / \ | counter-clockwise cycle
| /10 \|
B3------B2

What do the various methods do with this?

1. From the symmetry it is obvious that all neutral and anonymous 
methods (that is, all methods we usually discuss) give each Ai the same 
probability and each Bi the same probability.

2. Since there are no downward beats of strength 9, the immune elements 
are B1,B2,B3, hence all immune methods (RP, Beatpath, River, ...) elect 
Bi with probability 1/3.

3. Smith Set = Uncovered Set = Iterated Uncovered Set = all 6.

4. No proper beat clone set, hence no possibility to spruce up anything! 
In particular, spruced up Random Ballot equals Random Ballot here and 
elects Ai with prob. 4/15 and Bi with prob. 1/15.

5. Banks Set = Tournament Equilibrium Set = Dutta's Minimal Covering Set 
= Bipartisan Set = {A1,A2,A3}. Hence all methods which guarantee the 
winner to be in one of these sets elect Ai with prob. 1/3, including 
ROACC and Condorcet Lottery.

6. Also Plurality, IRV, Approval, Copeland and probably Kemeny elect Ai 
with prob. 1/3.

So in this example the immune methods are alone in electing the Bs while 
all other methods favour the As.


Now the crucial point is perhaps what we can say about strategizing. I 
provide a start for the discussion:

7. Assume a method gives the Bs a non-zero probability. Then the 12 
A-voters have a strong incentive to assure that the Ai win by 
strategically voting
4 A1>A2>A3>B2>B1>B3
4 A2>A3>A1>B3>B2>B1
4 A3>A1>A2>B1>B3>B2,
that is, each of them reverses only 4 of his 15 pairwise preferences.
The result will be preferable to each of the 12 since the probability 
that the individual voter will prefer the strategical winner to the 
sincere winner is larger than 1/2. When we assume additionally that the 
sincere "utilities" (if you believe that there is such a thing at all) 
of the candidates are 5,4,3,2,1, and 0 for the 12 voters, then also 
their expected utility is increased by voting like this.

That is: 80% of the electorate have incentive to vote insincerely when 
an immune method is used here!

So bad, so good. But what strategic incentives exist under the other 
methods??

8. ...to be continued by you!


Sincerely, Jobst
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