Arrow's impossibility theorem • Hacker News Board

Re: Arrow's impossibility theorem

#91

Post by kijin » Fri, Jun 05, 2015, 4:36 AM UTC

logicchop wrote: post #82

kijin wrote: post #55

Earlier quoted context omitted.

If we read early works (from the late 80s) in what is now called "deliberative democracy", we can see that it began as a response to certain models of democracy that emphasize preferences and procedures -- up to and including the participatory models of the 70s and early 80s. Although deliberative democracy is not a direct response to Arrow's impossibility theorem in particular, it was intended to sidestep its troubl…

I think you are a bit confused. What would Arrow's theorem explain? There is no empirical phenomena that we are puzzled about that Arrow's theorem solves (unless you are wondering: Why is it so hard to come up with a voting system that doesn't have the potential for goof-ball results? - Answer, because it is impossible..) I don't know what you mean by the "trinity" in which "satisfaction of all pre-existing preferenc…

If we're not interested in a system that satisfies preferences, then Arrow's impossibility theorem is irrelevant.

The conditions, of course, sound obvious and intuitive. Of course we don't want a dictatorship, and of course we want everyone to count as equally as possible. But it takes a very specific interpretation of your third condition to bootstrap the rest of Arrow's theorem. You have to interpret it in a way that emphasizes translating fixed individual preferences into group preferences as straightforwardly as possible. If you care about that, then yes, you should be worried about Arrow's theorem. Otherwise, Arrow's theorem is just a cool thought experiment that helps explain why said interpretation is wrong.

Deliberative democracy currently happens to be the most popular model of democracy among political theorists, and they don't care about Arrow's theorem because whatever preferences people have before they enter the democratic "procedure" isn't worth jack shit to them. As a result, Arrow's theorem is much less relevant to the possibility and fate of democracy as currently understood than it was 60 years ago.

Re: Arrow's impossibility theorem

#92

Post by ClayShentrup » Sat, Jun 06, 2015, 3:31 PM UTC

logicchop wrote: post #88

ClayShentrup wrote: post #85

Earlier quoted context omitted.

No. Utility is not about ordering. E.g. suppose you prefer X over Y over Z, and can have a guarantee of Y, or a 50/50 probability of X or Z. If your preference for Y is greater than the average of X and Z, then you would want to take Y. If Y is less than that average, then you want the lottery. You can arbitrarily change the 50/50 probability to e.g. 40/60 or what have you, to make it equally preferable to any guaran…

Yes, we know that preference involves more than ordering and that "strengths" are relevant to decision making. We know this because it explains an obvious empirical matter, that there are scenarios like the one you cite where we prefer X over Y, but end up choosing Y over X. Not a puzzle; not a paradox; how it is. And of course the way we tease out the "strength" of your preference involves, e.g., looking at how your…

> I don't know how "10 to A, 20 to B" could indicate anything beyond the simple fact that the voter prefers B to A.

Scores are actual utilities with added loss, from:

  - ignorance    
  - normalization    
  - rounding error    
  - tactical voting

The way you measure the combined impact of that loss is with Bayesian regret. http://ScoreVoting.net/BayRegsFig.html

So those scores HAVE meaning. That meaning is muddied by the aforementioned loss, but it is lunacy to say it only "could indicate anything beyond the simple fact that the voter prefers B to A".

Re: Arrow's impossibility theorem

#93

Post by ClayShentrup » Sat, Jun 06, 2015, 3:38 PM UTC

_dps wrote: post #89

ClayShentrup wrote: post #85

Earlier quoted context omitted.

No. Utility is not about ordering. E.g. suppose you prefer X over Y over Z, and can have a guarantee of Y, or a 50/50 probability of X or Z. If your preference for Y is greater than the average of X and Z, then you would want to take Y. If Y is less than that average, then you want the lottery. You can arbitrarily change the 50/50 probability to e.g. 40/60 or what have you, to make it equally preferable to any guaran…

You are implicitly presuming an expected utility framework (more generally, that people rank outcome distributions based solely on their long run central tendencies). This isn't the case (or even uniquely formalizable) in base (non-e.u.) utility theory which requires choice-order preservation under arbitrary monotonic transformations of the scoring function. Expectation ordering is not necessarily preserved under suc…

> You are implicitly presuming an expected utility framework

This isn't an assumption. Organisms have specifically evolved to maximize the expected number of copies of their genes they make. Or rather, _genes_ have specifically survived in proportion to their expected impact on the number of copies of themselves they make, and therefore we are to a very close approximation utility maximizers.

> ..linear utility of money

Utility is approximately log(money).

A caveat here is that certain decisions themselves are costly (particularly in terms of the most precious resource: time) and hence we can sometimes make clearly "irrational" choices because the expected utility of spending more time choosing was lower than the expected utility of making the more optimal choice. This leads to apparent paradoxes like the Allais Paradox.

Here's some background on utility theory. http://scorevoting.net/UtilFoundns.html