[EM] Fwd: SODA, negotiation, and weak CWs

[fsimmons at pcc.edu]

Jameson asked for "thoughts."

My first thought is that this kind of analysis is exactly what we need.

My second thought is that so far SODA has held up well under all the probes for weakness that anybody has 
come up with.  SODA seems to be a very robust method.

My third thought is that I have never seen this kind of soul searching or probing directed at IRV by IRV 
enthusiasts.


> Date: Sun, 25 Dec 2011 11:28:24 -0600
> From: Jameson Quinn 
> To: EM 
> Subject: [EM] Fwd: SODA, negotiation, and weak CWs
> Message-ID:
> 
> Content-Type: text/plain; charset="iso-8859-1"
> 
> 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 
> Date: 2011/12/25
> Subject: SODA, negotiation, and weak CWs
> To: EM 
> 
> 
> 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 
> scenarioshave 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 
> propensitydepended on the utility payoff of a strategy; but the 
> strategicpossibilities 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 
> successfullyfind 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 
> switchedfrom 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 
> assessingthe relative payoffs between those candidates than 
> group a, distracted by
> the A/C rivalry, had been.
> 
> Thoughts?
> 
> Jameson