[EM] Fwd: SODA, negotiation, and weak CWs
Jameson Quinn
jameson.quinn at gmail.com
Sun Dec 25 09:28:24 PST 2011
I'm resubmitting this in a text-friendly format, at Forest's request. I'll
also take the opportunity to add one paragraph about how rated methods can
fail to find the highest-utility candidates in scenarios like this. Added
text is marked ADDED.
---------- Forwarded message ----------
From: Jameson Quinn <jameson.quinn at gmail.com>
Date: 2011/12/25
Subject: SODA, negotiation, and weak CWs
To: EM <election-methods at lists.electorama.com>
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 Candidate Utilities
Scenario 1 (zero sum) A B C
a 4 4 1 0
b 2 0 3 2
c 3 0 2 4
Total utility 16 16 16
Scenario 2 (pos. sum) A B C
a 4 3 1 0
b 2 0 3 1.5
c 3 0 2 3
Total utility 12 16 12
Scenario 3 (neg. sum) A B C
a 4 4 0.5 0
b 2 0 3 2
c 3 0 1 4
Total utility 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.
ADDED: Of course, Range, Approval, and MJ can all get these scenarios
"wrong" too. Because the scenarios present a classic chicken dilemma
between B and C, these rated systems could all end up electing A,
regardless of utility.
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 Candidate Utilities
Scenario 1a(zero sum) A B C
a 4 4 1.5 0
b 2 0 2.5 2
c 3 0 1.5 4
Total utility 16 15.5 16
Group Size Candidate Utilities
Scenario 1b(zero sum) A B C
a 4 4 1.5 0
b 2 0 3 2
c 3 0 1.1 4
Total utility 16 15.3 16
Scenario 2a(pos. sum) A B C
a 4 3 1.5 0
b 2 0 2.5 1.5
c 3 0 1.5 3
Total utility 12 15.5 12
Scenario 3a(neg. sum) A B C
a 4 4 1 0
b 2 0 2.5 2
c 3 0 0.5 4
Total utility 16 10.5 16
Scenario 3b(neg. sum) A B C
a 4 4 1 0
b 2 0 3 2
c 3 0 0.1 4
Total utility 16 10.3 16
Note that in scenarios 1a and 2a, this utility transfer has left B giving
the same utility to groups a and c, while in scenario 3a, B has switched
from favoring group c over group a, to favoring group a over group c. Also,
note that in scenario 2a, 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 Xa 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, as in
scenarios 1b and 3b. Again, this 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 (because they'd be
reluctant to anger group c by "unfairly" favoring group a). Of course, A
could realize this, and simply not attempt to make the ultimatum in
scenarios 1 and 3; and then B would still win. But A'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
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