[EM] SODA, negotiation, and weak CWs

[Jameson Quinn]

In order to have optimum Bayesian Regret, a voting system should be able to
not elect a Weak Condorcet Winner (WCW), that is, a CW whose utility is
lower than the other candidates. Consider the following payout matrices:


Group

size
can did ate

Scenario 1 (zero sum)

A

B

C

a

4

4

1

0

b

2

0

3

2

c

3

0

2

4

Total payout

16

16

16

Scenario 2 (positive sum)

A

B

C

a

4

3

1

0

b

2

0

3

1.5

c

3

0

2

3

Total payout

12

16

12

Scenario 3 (negative sum)

A

B

C

a

4

4

0.5

0

b

2

0

3

2

c

3

0

1

4

Total payout

16

11

16

All three scenarios consist of 3 groups of voters: groups a, b, and c, with
4, 2, and 3 voters respectively, for a total of 9 voters. All scenarios
have 3 candidates: A, B, and C, who favor their respective groups. And in
all three scenarios, candidate B is the CW, because the preference matrix
is always

4: A>B
2: B>C
3: C>B

But in scenario 1, the utilities of the three candidates are balanced; in
scenario 2, B has the highest utility; and in scenario 3, A and C have the
highest utilities.

Obviously, any purely preferential system will tend to give the same result
in all three scenarios. This might not be 100% true if strategy propensity
depended on the utility payoff of a strategy; but the strategic
possibilities would have to be just right for a method to "get it right"
for this reason.

It's easy to see how Range could "get it right" in scenarios 2 and 3. With
just a bit of strategy, it's also easy to see how it could successfully
find the CW in scenario 1.

You can also construct plausible stories of how Approval or MJ could "get
it right" in all 3 scenarios, although it probably involves adding some
random noise to voting patterns rather than assuming pure "honest" votes.

But what about SODA? As a primarily preferential system, it seems that it
should give the same result in all three scenarios. If candidates all
rationally pursue the interests of their primary constituency, then A will
approve B to prevent B from having to approve C, leaving a win for B.

But if candidate A decides to make an ultimatum, things could go
differently. A says to B: "Make some promise that transfers 0.5 point of
utility to each member of group a, or I will not approve you." Assume that
B can make a promise to transfer utility from one group to another at 80%
efficiency; and that such promises are not strictly enforceable. Thus, if A
gets too greedy, B can simply promise the moon and not keep the promise;
but if A asks for something reasonable, B will see honesty as worth it.

B could promise to transfer 0.5 point of utility from groups b and c to
group a. Since utility transfers are assumed to be only 80% efficient, that
transfer of 2.5 utility points would result in a net loss of 0.5. So the
payoffs would be:


Group

size
can did ate

Scenario 1 (zero sum)

A

B

C

a

4

4

1.5

0

b

2

0

2.5

2

c

3

0

1.5

4

Total payout

16

15.5

16

Scenario 2 (positive sum)

A

B

C

a

4

3

1.5

0

b

2

0

2.5

1.5

c

3

0

1.5

3

Total payout

12

15.5

12

Scenario 3 (negative sum)

A

B

C

a

4

4

1

0

b

2

0

2.5

2

c

3

0

0.5

4

Total payout

16

10.5

16

Note that in scenarios 1 and 2, this utility transfer has left B giving the
same utility to groups a and c, while in scenario 3, B has switched from
favoring group c over group a, to favoring group a over group c. Also, note
that in scenario 2, group b still gets a full point of advantage with
candidate B versus what they would get with candidate C, whereas in the
other two scenarios, group b only gets half a point of advantage there. If
group b demands a full point of advantage, then B could only meet the
ultimatum in scenario 1 by taking all the utility from group c, which would
leave c with less utility than a.

I believe that these factors tend to make it more likely that B would meet
the ultimatum in scenario 2 than in the other scenarios. Of course, C could
realize this, and simply not attempt to make the ultimatum in scenarios 1
and 3; and then B would still win. But C's utility payouts show that they
honestly have no preference between groups b and c, so I think that it is
not unreasonable to imagine that they'd make the ultimatum in all three
cases.

The upshot is, there is a plausible (though perhaps not too likely)
mechanism for SODA to avoid electing a CW specifically in cases where that
CW is intrinsically weak. And that's with perfect information; I'd argue
that the mechanism would work even better in cases where B's strength were
illusory; that is, where groups a and c were overestimating their payoff
from candidate B because of the A/C rivalry. Candidate A, realizing that
they were choosing between B and C, would be more careful about assessing
the relative payoffs between those candidates than group a, distracted by
the A/C rivalry, had been.

Thoughts?

Jameson
All three scena
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