8 ms·

Can I get a simple wikipedia explanation? That was the most challenging wikipedia article I've ever read.

by c-slice 11y ago

vezzy-fnord 11y ago
Try the Stanford Encyclopedia of Philosophy's take on it: http://plato.stanford.edu/entries/arrows-theorem/ http://plato.stanford.edu/entries/arrows-theorem/ It's actually not that "simple", but it's overall less technical.
- theVirginian 11y ago
  It's technical which is good but much better written than Wikipedia.
bradleyjg 11y ago
It is a theorem about the properties of all possible voting systems with preference rankings. The basic idea is there are a few desirable properties for any such voting system, but that it is mathematically impossible for any voting system to satisfy all of them.
- dragonwriter 11y ago
  > It is a theorem about the properties of all possible voting systems with preference rankings. More accurately, it is a theorem about the properties of all possible voting systems where the input is voters preference rankings and the output is also a preference ranking.
JadeNB 11y ago
It's not Wikipedia, but http://tech.mit.edu/V123/N8/8voting.8n.html http://tech.mit.edu/V123/N8/8voting.8n.html may be more intelligible (especially regarding the particular desireable conditions that Arrow's theorem says are mutually incompatible). There's also http://dev.whydomath.org/node/voting/Arrow's_Impossibility_Theorem.html http://dev.whydomath.org/node/voting/Arrow's_Impossibility_T..., which includes some intuition-building exercises.
schmit 11y ago
In short: For a voting system (ranking of some candidates based on preferences of voters), it would be nice if: - A single voter cannot determine the ranking (as a dictator) - For every possible set of voter preferences, there is an outcome (not random) - If everyone likes candidate A over candidate B, then in the final ranking candidate A should be ranked higher than candidate B - If one prefers A over B when comparing just A and B, then one should also prefer A over B when an additional option C is offered Sounds like some reasonable properties for a voting system, right? Well, the theorem states that if there are more than 2 candidates, then there is no voting system that has all 4 properties above.
- jonahx 11y ago
  Are the counterexamples offered by the theorem pathological, in the sense that they are unlikely to occur in practice but are theoretically possible? Or would they arise in practice frequently using standard rank voting systems?
  - dragonwriter 11y ago
    > Are the counterexamples offered by the theorem pathological, in the sense that they are unlikely to occur in practice but are theoretically possible? Or would they arise in practice frequently using standard rank voting systems? Which particular problems occur, and the frequency with which they occur, depend on the particular voting system. Plurality and majority/runoff (which are ranked preference voting systems with a vary narrow constraint on the preferences that are expressed on the input ballots, which is pretty much the same constraint as on the preferences reflected in the output of any single-winner voting system) hits problems fairly frequently in practice, but most of the common voting systems that most people think of as ranked-preference systems (IRV, etc.) still hit them in practice as well, though not generally as frequently and in ways which create as clear incentives to tactical voting.
  - stephencanon 11y ago
    http://en.wikipedia.org/wiki/Burlington_mayoral_election,_2009 http://en.wikipedia.org/wiki/Burlington_mayoral_election,_20...
kelvintran 11y ago
This is how I was taught it (or understood I was taught it) - at law school, so it might have been dumbed down. The impossibility is the impossibility of ensuring rational (transitive) outcomes amongst ranked preferences and adhering to a set of fair and democratic norms. A rational transitive outcomes is one in which votes result in option A being preferred over option B and option B being preferred over option C, such that A is preferred over C (eg, A > B > C). Option A is known as the Condorcet winner. But there may be cases where the vote yields no Condorcet winner (eg, A > B > C > A). This is illustrated by the following table: Preferences Voter 1 Choc Vanil Strwb Voter 2 Vanil Strwb Choc Voter 3 Strwb Choc Vanil Two voters prefer C over V and two prefer V over S, but two also prefer S over C. To ensure transitivity, we can introduce voting rules, but it is impossible to introduce rules that do not violate the fair and democratic norms (referred to as the pre-specified criteria in the Wikipedia article: unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives).
- SilasX 11y ago
  Yeah, but that takes the punch out of the theorem. It's saying, "hey, sometimes you have really screwy preferences, too bad." Realistically, that kind of situation doesn't break a voting system. We can say "we don't care about that case -- just pick a random winner then", but it's no longer deterministic. Is there a stronger version of the theorem that says there's no sane procedure even ignoring those cases?
  - gweinberg 11y ago
    No, of course not. It's really easy to come up with a system that always comes up with "good" results if you rule out "screwy" voter preferences, with a sufficiently restrictive value of "screwy".
    - SilasX 11y ago
      "Screwy" in this context means the sort of intransitive situations referred to in the parent of my last comment.
  - 11y ago
lambda 11y ago
There is a simple explanation in the introduction: In short, the theorem states that no rank-order voting system can be designed that always satisfies these three "fairness" criteria: 1. If every voter prefers alternative X over alternative Y, then the group prefers X over Y. 2. If every voter's preference between X and Y remains unchanged, then the group's preference between X and Y will also remain unchanged (even if voters' preferences between other pairs like X and Z, Y and Z, or Z and W change). 3. There is no "dictator": no single voter possesses the power to always determine the group's preference. A "rank-order voting system" is one in which all voters order the possible choices, from most preferred to least preferred. The idea is that based on everyone's votes, and some set of rules on how to evaluate those votes, you can come up with a decision for the entire group that respects those preferences, in some way that is considered fair. All of these requirements listed above seem like fairly simple requirements for a voting system to have; if there's unanimous agreement about the ordering of two of the choices, then the voting system should order them that way. If a voter changes preferences about one choice C (maybe changing between them being ranked above or below D, or above or below one of A or B), without changing their ordering between choices A and B, that shouldn't affect the outcome of the A/B ordering that the voting system gives you. And finally, there is not dictator; no single voter is able to always determine the group ordering unilaterally. The first requirement seems pretty obvious; if everyone is unanimous that we should pick one option over another, then the group should pick that option. The second is a little trickier; but it basically states that there shouldn't be able to be "spoiler"; someone whose ordering in people's preferences affects the ordering of to other choices. Think about, say, the 2000 election, with Bush and Gore and a couple of third party candidates; now, that used just single choices per voter rather than a ranking, but imagine it used a ranking. So let's say Bush and Gore are fairly close, but 51% of voters prefer Gore over Bush. Their preferences for other candidates should not affect the fact that Gore wins over Bush. If I rank candidates Nader > Gore > Bush, that should not change the Gore/Bush result compared to if I ranked the candidates Gore > Bush > Nader. And the last is pretty trivially desirable; generally a democracy wants to democratically make a choice, in which everyone's vote has the same weight in affecting the final outcome. The fact that these conditions are impossible to achieve together is fairly profound, and means that almost any voting system will have serious flaws under certain circumstances.
logicchop 11y ago
Here's the version of the story with respect to voting: Democracy requires voting. Voting involves an objective measurement of group preference with respect to choices. Arrow's theorem and related theorems (see Gibbard-Satterthwaite theorem) show that there is no way to objectively measure group preferences.

Can I get a simple wikipedia explanation? That was the most challenging wikipedia article I've ever read.

by c-slice 11y ago

Can I get a simple wikipedia explanation? That was the most challenging wikipedia article I've ever read.

Try the Stanford Encyclopedia of Philosophy's take on it: http://plato.stanford.edu/entries/arrows-theorem/ http://plato.stanford.edu/entries/arrows-theorem/ It's actually not that "simple", but it's overall less technical.

theVirginian 11y ago
It's technical which is good but much better written than Wikipedia.

bradleyjg 11y ago

It is a theorem about the properties of all possible voting systems with preference rankings. The basic idea is there are a few desirable properties for any such voting system, but that it is mathematically impossible for any voting system to satisfy all of them.

dragonwriter 11y ago
> It is a theorem about the properties of all possible voting systems with preference rankings. More accurately, it is a theorem about the properties of all possible voting systems where the input is voters preference rankings and the output is also a preference ranking.

JadeNB 11y ago

It's not Wikipedia, but http://tech.mit.edu/V123/N8/8voting.8n.html http://tech.mit.edu/V123/N8/8voting.8n.html may be more intelligible (especially regarding the particular desireable conditions that Arrow's theorem says are mutually incompatible). There's also http://dev.whydomath.org/node/voting/Arrow's_Impossibility_Theorem.html http://dev.whydomath.org/node/voting/Arrow's_Impossibility_T..., which includes some intuition-building exercises.

schmit 11y ago

In short: For a voting system (ranking of some candidates based on preferences of voters), it would be nice if: - A single voter cannot determine the ranking (as a dictator) - For every possible set of voter preferences, there is an outcome (not random) - If everyone likes candidate A over candidate B, then in the final ranking candidate A should be ranked higher than candidate B - If one prefers A over B when comparing just A and B, then one should also prefer A over B when an additional option C is offered Sounds like some reasonable properties for a voting system, right? Well, the theorem states that if there are more than 2 candidates, then there is no voting system that has all 4 properties above.

jonahx 11y ago
Are the counterexamples offered by the theorem pathological, in the sense that they are unlikely to occur in practice but are theoretically possible? Or would they arise in practice frequently using standard rank voting systems?
- dragonwriter 11y ago
  > Are the counterexamples offered by the theorem pathological, in the sense that they are unlikely to occur in practice but are theoretically possible? Or would they arise in practice frequently using standard rank voting systems? Which particular problems occur, and the frequency with which they occur, depend on the particular voting system. Plurality and majority/runoff (which are ranked preference voting systems with a vary narrow constraint on the preferences that are expressed on the input ballots, which is pretty much the same constraint as on the preferences reflected in the output of any single-winner voting system) hits problems fairly frequently in practice, but most of the common voting systems that most people think of as ranked-preference systems (IRV, etc.) still hit them in practice as well, though not generally as frequently and in ways which create as clear incentives to tactical voting.
- stephencanon 11y ago
  http://en.wikipedia.org/wiki/Burlington_mayoral_election,_2009 http://en.wikipedia.org/wiki/Burlington_mayoral_election,_20...

kelvintran 11y ago

This is how I was taught it (or understood I was taught it) - at law school, so it might have been dumbed down. The impossibility is the impossibility of ensuring rational (transitive) outcomes amongst ranked preferences and adhering to a set of fair and democratic norms. A rational transitive outcomes is one in which votes result in option A being preferred over option B and option B being preferred over option C, such that A is preferred over C (eg, A > B > C). Option A is known as the Condorcet winner. But there may be cases where the vote yields no Condorcet winner (eg, A > B > C > A). This is illustrated by the following table: Preferences Voter 1 Choc Vanil Strwb Voter 2 Vanil Strwb Choc Voter 3 Strwb Choc Vanil Two voters prefer C over V and two prefer V over S, but two also prefer S over C. To ensure transitivity, we can introduce voting rules, but it is impossible to introduce rules that do not violate the fair and democratic norms (referred to as the pre-specified criteria in the Wikipedia article: unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives).

SilasX 11y ago
Yeah, but that takes the punch out of the theorem. It's saying, "hey, sometimes you have really screwy preferences, too bad." Realistically, that kind of situation doesn't break a voting system. We can say "we don't care about that case -- just pick a random winner then", but it's no longer deterministic. Is there a stronger version of the theorem that says there's no sane procedure even ignoring those cases?
- gweinberg 11y ago
  No, of course not. It's really easy to come up with a system that always comes up with "good" results if you rule out "screwy" voter preferences, with a sufficiently restrictive value of "screwy".
  - SilasX 11y ago
    "Screwy" in this context means the sort of intransitive situations referred to in the parent of my last comment.
- 11y ago

lambda 11y ago

There is a simple explanation in the introduction: In short, the theorem states that no rank-order voting system can be designed that always satisfies these three "fairness" criteria: 1. If every voter prefers alternative X over alternative Y, then the group prefers X over Y. 2. If every voter's preference between X and Y remains unchanged, then the group's preference between X and Y will also remain unchanged (even if voters' preferences between other pairs like X and Z, Y and Z, or Z and W change). 3. There is no "dictator": no single voter possesses the power to always determine the group's preference. A "rank-order voting system" is one in which all voters order the possible choices, from most preferred to least preferred. The idea is that based on everyone's votes, and some set of rules on how to evaluate those votes, you can come up with a decision for the entire group that respects those preferences, in some way that is considered fair. All of these requirements listed above seem like fairly simple requirements for a voting system to have; if there's unanimous agreement about the ordering of two of the choices, then the voting system should order them that way. If a voter changes preferences about one choice C (maybe changing between them being ranked above or below D, or above or below one of A or B), without changing their ordering between choices A and B, that shouldn't affect the outcome of the A/B ordering that the voting system gives you. And finally, there is not dictator; no single voter is able to always determine the group ordering unilaterally. The first requirement seems pretty obvious; if everyone is unanimous that we should pick one option over another, then the group should pick that option. The second is a little trickier; but it basically states that there shouldn't be able to be "spoiler"; someone whose ordering in people's preferences affects the ordering of to other choices. Think about, say, the 2000 election, with Bush and Gore and a couple of third party candidates; now, that used just single choices per voter rather than a ranking, but imagine it used a ranking. So let's say Bush and Gore are fairly close, but 51% of voters prefer Gore over Bush. Their preferences for other candidates should not affect the fact that Gore wins over Bush. If I rank candidates Nader > Gore > Bush, that should not change the Gore/Bush result compared to if I ranked the candidates Gore > Bush > Nader. And the last is pretty trivially desirable; generally a democracy wants to democratically make a choice, in which everyone's vote has the same weight in affecting the final outcome. The fact that these conditions are impossible to achieve together is fairly profound, and means that almost any voting system will have serious flaws under certain circumstances.

logicchop 11y ago

Here's the version of the story with respect to voting: Democracy requires voting. Voting involves an objective measurement of group preference with respect to choices. Arrow's theorem and related theorems (see Gibbard-Satterthwaite theorem) show that there is no way to objectively measure group preferences.