I know Bayesian Regret is a metric some people like to use to judge the effectiveness of a voting system. I have concerns with it because it uses a utility score baseline for a voter's true position. With that as a source of truth, it only makes sense that score voting systems would rank high.Has there been any Bayesian Regret analysis done of voting systems using a different baseline measurement of truth? For example, I would be interested in the results with a Condercet/score ballot where a voter indicates their strength of preference for every combination of two candidates on the ballot.
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For one thing, VSE is not as biased towards score voting as you might think; it's definitely possible to make non-absurd assumptions about utilities, knowledge, and strategies that leave other systems doing better than score. Because of normalization, it's even possible to do this with honest voting.
I am working on a paper improving on Warren's 2000 results, and yes, I do manage to vary the assumptions by enough to see that score voting no longer dominates. And yet, there are still clear take-aways; plurality is clearly horrible, IRV is not much better, and Borda does badly with strategy.
Imagine that honest utilities are as follows:
40: A60 B50 C50
60: A50 B60 C10
"Honest" score ballots would be normalized to a 0-100 scale:
40: A100 B0 C0
60: A80 B100 C0
Totals: A8800, B6000, C0
A wins in this "honest" score election despite being a decisive loser in both utility and Condorcet.
That's what I mean by normalization.
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If A⊃B and B⊃C then B ∼ p?A:C for at least one p with 0<p<1.That line assumes human preferences are liner and transitive. With that assumption, utility functions are easily derived.
I don't think it assumes linearity. Linearity comes from:
- Lottery utilities are expected utilities (Lin):
- u(p?a:b) = p·u(a) + (1-p)·u(b).
On Friday, October 21, 2016 at 8:36:41 PM UTC-7, Brian Kelly wrote:If A⊃B and B⊃C then B ∼ p?A:C for at least one p with 0<p<1.
That line assumes human preferences are liner and transitive. With that assumption, utility functions are easily derived.
I don't think it assumes linearity.
- Transitivity is necessary for any meaningful definition of welfare. Otherwise you could make yourself happier by constantly cycling through X/Y/Z/X..
I don't think it assumes linearity. Linearity comes from:Lottery utilities are expected utilities (Lin):u(p?a:b) = p·u(a) + (1-p)·u(b).
Sorry, that's just a statement of the theorem. I can't remember how the stated axioms above imply that.
I seem to remember something about how non-linear utilities would mean you could add a constant to the utilities in two compared lotteries and change which one was preferred.
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It's certainly possible to have coherent accounts of human preferences that are not linear utilities. But any such coherent account involves making additional assumptions about when and how linearity is broken; assumptions which invite argument from both empirical and ethical standpoints.
Utilitarianism may not be the truth, but I think it's the most we can hope to agree on, and even that agreement will be only for the sake of argument. Still, that makes it very useful, and the right standard for measuring VSE.
2016-10-22 18:23 GMT-04:00 Brian Kelly <bkell...@gmail.com>:
On Saturday,October 22, 2016 at 9:14:39 AM UTC-7, Clay Shentrup wrote:On Friday, October 21, 2016 at 8:36:41 PM UTC-7, Brian Kelly wrote:If A⊃B and B⊃C then B ∼ p?A:C for at least one p with 0<p<1.That line assumes human preferences are liner and transitive. With that assumption, utility functions are easily derived.Quick note: I see a problem with my statement (beyond spelling). The second sentence should say "With that assumption, linear utility functions are easily derived."I don't think it assumes linearity.It requires it. You can define a line with points A and C. This axiom says point B must be between A and C. Any point between A and C is on the AC line.
- Transitivity is necessary for any meaningful definition of welfare. Otherwise you could make yourself happier by constantly cycling through X/Y/Z/X..
Not at all. I think your perspective stems from a belief that welfare and utility must be on a one-dimensional spectrum. If you accept the notion that Humans have three or more independent desires then it is easily provable that such concepts are more complicated than that.
On Saturday, October 22, 2016 at 10:26:41 AM UTC-6, Clay Shentrup wrote:I don't think it assumes linearity. Linearity comes from:Lottery utilities are expected utilities (Lin):u(p?a:b) = p·u(a) + (1-p)·u(b).Sorry, that's just a statement of the theorem. I can't remember how the stated axioms above imply that.It is pretty clear to me this proof assumes that Human preference is linear. I only object at the continuity axiom because it is the first place where more than two points are considered.I seem to remember something about how non-linear utilities would mean you could add a constant to the utilities in two compared lotteries and change which one was preferred.Which still makes perfect sense to me. Remember I wrote to you about the Ultimatum Game (https://en.wikipedia.org/wiki/Ultimatum_game)? People are less likely to reject proposals when the amount of money offered increases. That is an example of how adding a constant can change people's preferences.
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It requires it. You can define a line with points A and C. This axiom says point B must be between A and C. Any point between A and C is on the AC line.
- Transitivity is necessary for any meaningful definition of welfare. Otherwise you could make yourself happier by constantly cycling through X/Y/Z/X..
Not at all. I think your perspective stems from a belief that welfare and utility must be on a one-dimensional spectrum. If you accept the notion that Humans have three or more independent desires then it is easily provable that such concepts are more complicated than that.
On Saturday, October 22, 2016 at 3:23:30 PM UTC-7, Brian Kelly wrote:It requires it. You can define a line with points A and C. This axiom says point B must be between A and C. Any point between A and C is on the AC line.Consider the social welfare function is sum of utility^2. That's not linear.
- Transitivity is necessary for any meaningful definition of welfare. Otherwise you could make yourself happier by constantly cycling through X/Y/Z/X..
Not at all. I think your perspective stems from a belief that welfare and utility must be on a one-dimensional spectrum. If you accept the notion that Humans have three or more independent desires then it is easily provable that such concepts are more complicated than that.It doesn't matter how many independent desires you have, when you make decisions you have to sum them all up into a single overall value.
I agree that to compare desires you need to flatten the data to a single value.Do you agree that mapping an individual's preference onto a one-dimensional utility scale is a lossy operation?
Um... yes? Anything in voting theory is a lossy operation.I agree that to compare desires you need to flatten the data to a single value.Do you agree that mapping an individual's preference onto a one-dimensional utility scale is a lossy operation?
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