[EM] Better cardinal methods?
Andy Jennings
elections at jenningsstory.com
Sun Sep 26 23:43:30 PDT 2021
Kristofer,
I, too, find myself going back to the risk-neutral lottery-based definition
of utility. I feel like it goes so naturally with "random ballot".
Suppose V1 has a ranking of A > B > C and V2 has a ranking of C > B > A. In
random ballot, V1's vote becomes a lottery ticket that causes A to win and
V2's vote becomes a lottery ticket that causes C to win. Let us ask when
would V1 and V2 both agree to trade their "one chance of A winning and one
chance of C winning" for "two chances of B winning".
We know we can say little about the "absolute" or "interpersonal" utilities
of how U_V1(A), U_V1(B), and U_V1(C) compare to U_V2(A), U_V2(B), and
U_V2(C).
But asking each voter to quantify exactly where B lies on the spectrum
between A and C, as a number between 0 and 1, is completely meaningful (in
the risk-neutral lottery paradigm).
Let b_1 = (U_V1(B) - U_V1(C)) / (U_V1(A) - U_V1(C))
Let b_2 = (U_V2(B) - U_V2(A)) / (U_V2(C) - U_V2(A))
(In other words, rescale each voter's utility so their favorite candidate
is at 1.0 and their least favorite is at 0.0 and examine their utility
estimations of B.)
If b_1 = 0.5 and b_2 = 0.5, we propose they trade "one chance of A winning
and one chance of C winning" for "two chances of B winning". The voters are
actually completely neutral toward this trade, though as an outsider I much
prefer the lowered entropy. b_1 and b_2 would have to be strictly greater
than 0.5 for both voters to be excited about the transaction.
It works for other fractions, too. If b_1 = 0.6 and b_2 = 0.4, the
utility-neutral trade is "one chance of A winning and one chance of C
winning" for "1.666 chances of B winning and 0.333 chance of C winning".
(If b_1 > 0.6 and b_2 > 0.4, then the trade is positive-sum.)
Can we actually set up this market (declared-strategy style), let all
voters submit their three-candidate ranking and a utility (between 0 and 1)
for their middle candidate, then we simulate all the trades and come up
with a final, optimal lottery?
A > B > C voters and C > B > A voters would trade with each other. A > C >
B voters and B > C > A voters would trade with each other. B > A > C voters
and C > A > B voters would trade with each other.
It seems obvious to me that in an election where 50% of the voters want A >
B > C and 50% want C > B > A, if you can get a number from each voter on
where B is on their scale from 0 to 1, that information is useful AND
meaningful. I mean, if all the voters say 0.9 then clearly we should just
elect B as the compromise candidate. And if all the voters say 0.1, then
giving them a 50/50 lottery between A and C is probably the best we can do.
Why should we decline to collect and use this "utility of the middle
candidate" information?
How can we simulate those trades? Line up all the A > B > C voters in order
of decreasing "B" utility and line up all the C > B > A voters in order of
increasing "B" utility and match up the two lines somehow? What about the
mismatch in length?
One problem I see is that whenever a transaction is perfectly fair, it is
utility-neutral, and the two parties are indifferent to whether the trade
actually happens. A trade that is positive-sum, on the other hand, has some
surplus utility and we could be unfair about which voter captures it.
If b_1 = b_2 = 0.6, then trading "one chance of A and one chance of C" for
any of the following would be utility-neutral or -positive for both voters:
- 1.666 chances of B winning and 0.333 chance of C winning
- 2 chances of B winning
- 1.666 chances of B winning and 0.333 chance of A winning
Obviously, as neutral election administrators, we should choose the middle
option. But I think this illustrates the opportunity for strategic voting
in this system. If you, as a voter, have perfect information about the
other voters, maybe your utility for B is 0.6 and you see that you can
decrease your declared utility for B to 0.400001 and still get a trade. It
will be a trade the other person barely agrees to, and you'll maximize your
utility, capturing all the surplus from the transaction.
Is there something else we could do as election administrators to make
dishonesty less profitable? Does it depend on the way we line up and match
up the opposing voters? If we always try to make sure that we match up
voters with a "sum of compromise utility" that is greater than one but as
small as possible, does that help somehow?
Perhaps in a large election, it will be difficult to know enough
information about the other voters and the benefits will be small enough
that voters will just be honest?
Or maybe we just discard the concept of matching up individual voters, look
at all the data, and come up with a "market-clearing price" for turning A
and C chances into B chances? Does that fix anything, or just leave a lot
of positive-sum transactions unfulfilled?
Can it be generalized to more than three candidates?
~ Andy
On Sun, Aug 15, 2021 at 3:02 PM Kristofer Munsterhjelm <km_elmet at t-online.de>
wrote:
> Suppose we take the risk neutral lottery-based definition of utility as
> a basis for honesty. (That is, if you're indifferent between a 100%
> chance of choice X and a 70% chance of Y, 30% chance of Z, then your
> utility for X is equal to 0.7 * u(Y) + 0.3 * u(Z).)
>
> What kind of cardinal system could incentivize voters to report this
> kind of information? It seems very hard to do it with any of the broad
> IIA class (score for X is just a function of ratings for X, highest
> score wins, and increasing your rating for X never decreases X's chance
> of winning), because those methods encourage minmax strategy.
>
> Perhaps some kind of cumulative voting? There's Hay, but it sucks.
>
> The lottery definition above can't determine both a natural zero and
> unit value, because if you scale all utilities by some constant, the
> lottery equations remain the same. So any method that takes this kind of
> input should pass a kind of "irrelevance of constant scaling" property,
> which says that if every voter v scales his ballot by some (private)
> constant factor v_F, then the outcome remains the same.
>
> In a Range-type method, that means the system should scale every ballot
> so that one candidate is max rated and another is min rated (I think).
> This would probably lead to IIA because in a two-candidate election,
> you'd get majority rule (whichever candidate voter v prefers gets max
> rating, and the other one gets min rating).
>
> Maybe it's possible to preserve IIA, but I doubt it.
>
> Any thoughts on how a method with better "honesty" incentives could be
> designed for lottery-type cardinal ballots?
>
> -km
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>
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