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Why Democracy Is Mathematically Impossible [video]

24 points by redbell 2y ago

DoctorOetker 2y ago
I have read too many texts on Arrows impossibiity theorem. The examples are always artificially constrained, like 3 candidate supermarket locations around a lake, with inhabitants distributed around it. consider 3 groups objectively each preffering: 1) A over B over C 2) B over C over A 3) C over A over B Then 2 thirds of the population agrees A is better than B, but also that B is better than C yet also that C is better than A! In fact the world and human incentives are much more high dimensional than the problem statement. (people could propose a rent reduction when further away from the supermarket, if not democratically then economically, so instead of a democratic issue, its housing owners bickering over the coming supermarket location, which just reveals housing is not very democratically distributed, in a democracy the prices could be agreed to depend on supermarket distance). In fact there is only good news for democracy: by detecting preference cycles, we KNOW the kind of thing that is going on. And we know that A,B,C is a false trilemma, and formal statements depicting rules or laws can be mathematically much much richer in structure than a simple limited list of "choices". I value the contribution of Arrow, but at the same time one shouldn't try to understand the concept of democracy by reading the Arrow impossibility literature. The moment democracy has been reduced to simply a predetermined multiple choice form, is the moment democracy is no longer understood.
deleted 2y ago
[deleted]
corimaith 2y ago
I actually did some research on this topic on the gulf between voters' actual preferences and their ordinal votes. Tldr; While the mathematics do prove the impracticality of guaranteeing the socially optimal candidate in ALL situations, realistically in the majority of cases plurality can achieve a candidate with a high social utility even if they aren't the best. In fact in most average situations it can outperform Harmonic or Borda voting procceses because of the lower spread on total scores leading to more deterministic outcomes for close candidates.
passion__desire 2y ago
I think there is a stoic quote which says that "If every person had access to same reasoning process and facts as me, they will reach the same conclusion as mine". As we know, people don't have access to same set of facts and even if they do, they don't have time to reason over them. I believe if each person could stop the world, spend time on going over all the knowledge accumulated by humanity and then make a decision, democracy is possible in such a scenario. But that is not true, we are made to play many timed chess matches where we have only few microseconds to make a decision before some other issue (another chess match) calls for our decision.
- karmakaze 2y ago
  This assumes that people use facts and reasoning to vote. The truth is closer to voting for "who you like better" which is anything but.
- breuleux 2y ago
  We could, in theory, pick a random sample of people and give them the time and money to actually think about the issues. Statistically, their interests should match those of the general population, but they will be better tooled to actually attend to them. I do think it would ultimately be much better to pick representatives at random than elect them.
  - rexpop 2y ago
    > give them the time and money Sounds like a republic, which most of us live under. The issue is that "the time and money" is exactly what "the general population" doesn't have (per your example), and so their interests diverge.
dirtdobber 2y ago
The part of the impossibility proof that deals with dictators is subtle and neat. At first, I was convinced that the dictator was actually a "swing voter" whose power was entirely conditioned on the other voter's preferences, but that's not actually the case. Still, my guess is that these types of edge cases could be handled by using some randomness and low probability arguments. For example, claiming that such dictators exist with negligible probability given a small-ish fraction of voters that are considered "random." While you would never want to inject random votes into an election, it's probably reasonable to assume that some fraction of voters behave naturally random (e.g., people get sick and don't vote, or are just politically fickle). This probably conflicts with the property of "unrestricted domain" but it seems like a reasonable concession in practice.
- Wowfunhappy 2y ago
  > The part of the impossibility proof that deals with dictators is subtle and neat. At first, I was convinced that the dictator was actually a "swing voter" whose power was entirely conditioned on the other voter's preferences, but that's not actually the case. Based on the video, I am still convinced of this. Why is it not the case?
  - dirtdobber 2y ago
    The "no dictator" property states that a single voter should not override the preferences of the majority. For example, if every voter (excluding the dictator) prefers candidate A over candidate C, then the dictator should not be able to control the outcome so that candidate C is ranked higher than candidate A. If you follow the proof closely (it's very subtle) you can see that despite every candidate ranking A over candidate C, the actual preferential outcome will have candidate C ranked higher than candidate A (and this is entirely dependent on the dictator's choice). In other words, there's a counterexample where every single voter decides to rank A > C, and yet the dictator alters the outcome so that C > A. This is different from a swing voter that pushes one candidate over the top in a very close race.
karmakaze 2y ago
Related: Why it’s mathematically impossible to share fair[0] (Stand-up Maths aka Matt Parker). [0] https://www.youtube.com/watch?v=GVhFBujPlVo https://www.youtube.com/watch?v=GVhFBujPlVo
mannyv 2y ago
When there's a consensus democracy is easy. When there's no consensus democracy is a mess. When it's messy people complain that it's failing. But the messiness is really a reflection of society and/or the media coverage.
amai 2y ago
This assumes that elections are necessary for democracy. However this is wrong and the ancient greek already knew that: "It is democratic for state offices to be filled by lot, and oligarchic for them to be filled by election" (Aristotle, Politics) This idea was the majority view until the French revolution: "The decision by lot corresponds to the nature of democracy, the decision by election to that of aristocracy." (Montesquieu, The Spirit of Law) So, just because voting systems cannot be perfect doesn't mean democracy is mathematically impossible. In fact mathematically and statistically democracy is only possible without any elections, simply by choosing candidates by lot, see https://en.wikipedia.org/wiki/Sortition https://en.wikipedia.org/wiki/Sortition .

24 points by redbell 2y ago

I have read too many texts on Arrows impossibiity theorem. The examples are always artificially constrained, like 3 candidate supermarket locations around a lake, with inhabitants distributed around it. consider 3 groups objectively each preffering: 1) A over B over C 2) B over C over A 3) C over A over B Then 2 thirds of the population agrees A is better than B, but also that B is better than C yet also that C is better than A! In fact the world and human incentives are much more high dimensional than the problem statement. (people could propose a rent reduction when further away from the supermarket, if not democratically then economically, so instead of a democratic issue, its housing owners bickering over the coming supermarket location, which just reveals housing is not very democratically distributed, in a democracy the prices could be agreed to depend on supermarket distance). In fact there is only good news for democracy: by detecting preference cycles, we KNOW the kind of thing that is going on. And we know that A,B,C is a false trilemma, and formal statements depicting rules or laws can be mathematically much much richer in structure than a simple limited list of "choices". I value the contribution of Arrow, but at the same time one shouldn't try to understand the concept of democracy by reading the Arrow impossibility literature. The moment democracy has been reduced to simply a predetermined multiple choice form, is the moment democracy is no longer understood.

deleted 2y ago

[deleted]

corimaith 2y ago

I actually did some research on this topic on the gulf between voters' actual preferences and their ordinal votes. Tldr; While the mathematics do prove the impracticality of guaranteeing the socially optimal candidate in ALL situations, realistically in the majority of cases plurality can achieve a candidate with a high social utility even if they aren't the best. In fact in most average situations it can outperform Harmonic or Borda voting procceses because of the lower spread on total scores leading to more deterministic outcomes for close candidates.

passion__desire 2y ago

I think there is a stoic quote which says that "If every person had access to same reasoning process and facts as me, they will reach the same conclusion as mine". As we know, people don't have access to same set of facts and even if they do, they don't have time to reason over them. I believe if each person could stop the world, spend time on going over all the knowledge accumulated by humanity and then make a decision, democracy is possible in such a scenario. But that is not true, we are made to play many timed chess matches where we have only few microseconds to make a decision before some other issue (another chess match) calls for our decision.

karmakaze 2y ago
This assumes that people use facts and reasoning to vote. The truth is closer to voting for "who you like better" which is anything but.
breuleux 2y ago
We could, in theory, pick a random sample of people and give them the time and money to actually think about the issues. Statistically, their interests should match those of the general population, but they will be better tooled to actually attend to them. I do think it would ultimately be much better to pick representatives at random than elect them.
- rexpop 2y ago
  > give them the time and money Sounds like a republic, which most of us live under. The issue is that "the time and money" is exactly what "the general population" doesn't have (per your example), and so their interests diverge.

dirtdobber 2y ago

The part of the impossibility proof that deals with dictators is subtle and neat. At first, I was convinced that the dictator was actually a "swing voter" whose power was entirely conditioned on the other voter's preferences, but that's not actually the case. Still, my guess is that these types of edge cases could be handled by using some randomness and low probability arguments. For example, claiming that such dictators exist with negligible probability given a small-ish fraction of voters that are considered "random." While you would never want to inject random votes into an election, it's probably reasonable to assume that some fraction of voters behave naturally random (e.g., people get sick and don't vote, or are just politically fickle). This probably conflicts with the property of "unrestricted domain" but it seems like a reasonable concession in practice.

Wowfunhappy 2y ago
> The part of the impossibility proof that deals with dictators is subtle and neat. At first, I was convinced that the dictator was actually a "swing voter" whose power was entirely conditioned on the other voter's preferences, but that's not actually the case. Based on the video, I am still convinced of this. Why is it not the case?
- dirtdobber 2y ago
  The "no dictator" property states that a single voter should not override the preferences of the majority. For example, if every voter (excluding the dictator) prefers candidate A over candidate C, then the dictator should not be able to control the outcome so that candidate C is ranked higher than candidate A. If you follow the proof closely (it's very subtle) you can see that despite every candidate ranking A over candidate C, the actual preferential outcome will have candidate C ranked higher than candidate A (and this is entirely dependent on the dictator's choice). In other words, there's a counterexample where every single voter decides to rank A > C, and yet the dictator alters the outcome so that C > A. This is different from a swing voter that pushes one candidate over the top in a very close race.

karmakaze 2y ago

Related: Why it’s mathematically impossible to share fair[0] (Stand-up Maths aka Matt Parker). [0] https://www.youtube.com/watch?v=GVhFBujPlVo https://www.youtube.com/watch?v=GVhFBujPlVo

mannyv 2y ago

When there's a consensus democracy is easy. When there's no consensus democracy is a mess. When it's messy people complain that it's failing. But the messiness is really a reflection of society and/or the media coverage.

amai 2y ago

This assumes that elections are necessary for democracy. However this is wrong and the ancient greek already knew that: "It is democratic for state offices to be filled by lot, and oligarchic for them to be filled by election" (Aristotle, Politics) This idea was the majority view until the French revolution: "The decision by lot corresponds to the nature of democracy, the decision by election to that of aristocracy." (Montesquieu, The Spirit of Law) So, just because voting systems cannot be perfect doesn't mean democracy is mathematically impossible. In fact mathematically and statistically democracy is only possible without any elections, simply by choosing candidates by lot, see https://en.wikipedia.org/wiki/Sortition https://en.wikipedia.org/wiki/Sortition .