[EM] Equilibrium analyses

[Kristofer Munsterhjelm]

On 2024-02-24 20:39, Closed Limelike Curves wrote:
> Does anyone know any papers that look at Myerson-Weber equilibria for 
> different systems? I know about Burt Monroe's "nonelection of turkeys" 
> paper, but not much else. I'd be very interested in any work on 
> calculating the regret for each method.

There are some papers that analyze Nash and M-W voting equilibria for 
voting methods, but they're usually very specific, as calculating Nash 
over a multiplayer game with simultaneous moves is in general very hard. 
(By multiplayer I mean more than two distinct players or types.) 
Monroe's paper uses only two types of players, and I suspect that's the 
reason.

Example of Nash equilibria in voting: 
https://webdoc.sub.gwdg.de/ebook/serien/e/CORE/dp9931.pdf

https://www.nber.org/system/files/working_papers/w23898/revisions/w23898.rev0.pdf 
(Chicken dilemma situation with more constrained behavior on the voters)

https://link.springer.com/article/10.1007/BF02425262 (sci-hub: 
https://sci-hub.se/https://link.springer.com/article/10.1007/BF02425262)

https://link.springer.com/chapter/10.1007/978-3-642-02839-7_9 (sci-hub 
analogous)

As for Monroe, I would further suspect that for some methods, modifying 
a method X to be Condorcet//X instead will preserve NIA, but not for all 
such methods. For instance, I'd imagine Condorcet//IRV passes and 
Condorcet//Plurality fails. But again, I haven't proven this.

E.g. for Condorcet//IRV: Honest voting gives a tie between 1 and 2. If 
any voter raises 3 to second to try to set up a cycle, the method falls 
through to IRV, where 3 is still eliminated. This is basically DH3, and 
C//IRV resists DH3 so it passes in this case.

Proving a single scenario doesn't prove the whole space, of course. It 
would be interesting to determine if DMTBR (or electing from the 
resistant set) implies Monroe's NIA, but I think doing so formally is a 
bit out of my grasp. The converse, NIA->DMTBR, is false (see 
Electowiki). Other interesting things would include proving e.g. DMTBR 
implying resistant-efficiency.

In any case, it should be possible to analyze two-voter situations by 
using linear programming. But I don't know of any papers where that has 
been done more comprehensively.

-km