[EM] Compatibility

[Juho]

Here is some discussion on full and approximate compatibility with  
election method related criteria. My aim is to prove that sometimes it  
is better to fail to meet some criteria (that are good in the sense  
that one wants to meet them) than to meet them.

First a story from real life. A mother of three children buys three  
cakes for the kids on her way home. She puts the cakes on the dinner  
table for the kids to eat. But then their dog steals and eats one of  
the cakes. All the kids try to grab one of the cakes and make sure  
that at lest they will get their share. The mother interrupts the kids  
and decides to cut the cakes in parts so that each kid gets 2/3 of a  
cake. The wrong solution would have been to meet the requirements of  
two of the kids. The best solution was not to meet the requirements of  
any of the kids.

Let's say that there are three different criteria that an election  
method should meet. They are related to three different strategic  
voting related problems. As in the world of security also election  
methods are as vulnerable to strategic voting as their weakest link.  
In this situation there may be a solution that almost meets all the  
three criteria but does not meet any of them fully. That solution is  
often (but not always) the best solution.

So, there are patterns where a good method should meet only a strict  
subset of the requirements that some competing methods meet. One may  
thus improve a method by making it incompatible to some (good)  
criterion that it earlier met. The point is that compatibility with  
various criteria should often not be an on/off comparison but a richer  
analysis where also partial and almost complete compatibilities are  
counted.

Juho



P.S. Note that there are also criteria that one should meet in most  
situations but that in some special situations should not be met. A  
good method may sometimes fail these criteria, but not to give up  
something in order to give space to meeting better / more accurately  
some other criteria as discussed above but to directly improve the  
method.