[EM] Hylland's theorem

[Kristofer Munsterhjelm]

On 2023-12-25 05:32, Rob Lanphier wrote:
> Hi folks,
> 
> I've seen many references to "Hylland's theorem" in recent papers and 
> other places, and yet, this theorem seems to be a partial mystery to the 
> Internet, and no one has bothered to write a Wikipedia article about it:
> https://en.wikipedia.org/wiki/Hylland%27s_theorem 
> <https://en.wikipedia.org/wiki/Hylland%27s_theorem>
> 
> Way back in 2005, I found a discussion of "Hylland's theorem" (and 
> "May's theorem") on the EM-list:
> http://lists.electorama.com/pipermail/election-methods-electorama.com//2005-January/thread.html#79759 <http://lists.electorama.com/pipermail/election-methods-electorama.com//2005-January/thread.html#79759>
> 
> That led me to this paper:
> "Strategy Proofness of Voting Procedures with Lotteries as Outcomes and 
> Infinite Sets of Strategies" -- Aanund Hyllund -- January 1980
> 
> ...which seems to be archived here (among other places, I hope):
> https://www.sv.uio.no/econ/personer/vit/aanundh/upubliserte-artikler-og-notater/Strategy%20Proofness%5B1%5D.pdf <https://www.sv.uio.no/econ/personer/vit/aanundh/upubliserte-artikler-og-notater/Strategy%20Proofness%5B1%5D.pdf>
> 
> Am I following the breadcrumbs properly?  Is there a different "Hylland" 
> that deserves to have a theorem named after them?

My impression is that "Hylland's theorem" refers to one of the results 
in that paper, although which result it is may vary.

For instance, Dutta et al. (http://dx.doi.org/10.1007/s00355-006-0152-9, 
https://ageconsearch.umn.edu/record/269616/files/twerp722.pdf) refer to 
the paper you linked to, and then say "Despite this difficulty, Hylland, 
in an important and regrettably unpublished paper,showed that the random 
dictatorship result holds even if the decision scheme is allowed to use 
cardinal information. In this paper, we have two main objectives. First, 
we providean alternative and considerably simpler proof of Hylland’s 
theorem. [...]"

Schulze says that Hylland proved that "when there are only two 
candidates and the used single-winner election method is strategyproof 
then the result depends only on whether the individual voter strictly 
prefers candidate A to candidate B, strictly prefers candidate B to 
candidate A or is indifferent between candidate A and candidate B". This 
is implied by the actual results in the paper, as far as I understand, 
but not actually given as a main result.

The paper deals with methods that are immune to every kind of strategy. 
Hylland free riding is a different concept: it's a particular kind of 
strategy, and methods may be immune to it without being fully 
strategyproof. So the different Hylland concepts aren't really related, 
apart from both dealing with how methods respond to strategy.

-km