[EM] The election methods trade-off paradox/impossibility theorems paradox.

Kristofer Munsterhjelm km_elmet at t-online.de
Tue Jul 4 16:47:02 PDT 2017

On 06/22/2017 08:30 AM, Richard Lung wrote:
> The election methods trade-off paradox/impossibility theorems paradox.
> For the sake of argument, suppose a trade-off theory of elections that
> there is no consistently democratic electoral system: the impossibility
> supposition.
> That supposition implies some conception (albeit non-existent) of a
> consistently derived right election result.
> If there is no such measure, then there is no standard even to judge
> that there is a trade-off between electoral systems.
> Suppose there is a consistent theory of choice, setting a standard by
> which electoral systems can be judged for their democratic consistency.
> It follows that the election result will only be as consistent as the
> electoral system, and there is no pre-conceivably right election result,
> because that presupposes a perfection not given to science as a
> progressive pursuit.

I don't think that's quite right. Impossibility proofs like Arrow's 
generally say something along this vein:

- Here are some properties that it seems reasonable that all voting 
methods should have (IIA etc for Arrow; strategy resistance and 
determinism for Gibbard).

- But as long as the method is of a certain form (ordinal for Arrow, 
pretty much every method for Gibbard), it's impossible to have all of these.

- Thus we're in a bind, because we'd like to have all of them.

It is then possible to construct subsets that you can have: e.g. 
dictatorship gives you all the properties in Arrow's theorem except for 
non-dictatorship; random pair gives you all the properties except 
determinism, and so on).

You could measure any given election against the measures, even though 
it's impossible to attain all of them. For instance, to measure how 
strategy resistant a given method is for a given election, you could 
determine how many ways there are for voters in favor of some party X to 
alter their votes so that X wins instead of whoever won. The fact that 
it's impossible to make a method completely resistant to strategy 
doesn't make it impossible to measure how far the system is from 
attaining (impossible) perfection.

In a similar vein for something like Bayesian Regret, no method would 
have a BR of zero, and so no method can attain perfection. But the 
actual BR might still be of interest.

The trick, if there is one, is in that the impossibility proofs don't 
specify in detail what a hypothetical perfect method would look like, 
only some properties it should reasonably satisfy; and then as long as 
it's possible to show that the properties can't all be satisfied, we 
know that there can't be a perfect system.

There is one way that "there is no objective trade-off" is correct, 
however. Suppose we have two methods:

A is very vulnerable to voter strategy (compromising, burial) but not to 
candidate strategy (cloning);
B is the other way around.

Which one is the better method? If we can't have both, then that depends 
on the situation. There's a boundary beyond which you can't improve some 
quality without giving up some other quality, but which point to pick 
along that boundary depends on the context, or what the use the method 
is being put to.

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