I’m having some trouble understanding your post, and I think it could be useful to:
State the exact problem setting you are addressing,
State Gibbard’s theorem, and
Show how exactly machine learning has solutions for that problem.
As far as I understand Gibbard’s theorem, it does not only apply to ranked lists (as opposed to Arrow’s theorem). Rather, it applies for any situation where each participant has to choose from some set of personal actions, and there’s some mechanism that translates every possible combination of actions to a social choice (of one result from a set). In this context either the mechanism limits the choice to only two results; or there’s a dictatorship, or there’s an option for strategic voting.
This isn’t my area so I’m taking a risk here that I might say something stupid, but: The existence of strategic voting is a problem, since it disincentivises truthfulness; But intuitively that doesn’t mean it’s the end of the world—all voting mechanisms we’ve had so far were very obviously open to strategic voting, and we still have functioning governments. Also choices that are limited to two options are often important and interesting, e.g. whether we improve the world or harm it, whether we expand into space or not, or whether the world should act to limit population growth.
1] “State the exact problem setting you are addressing,”
- “There is an unexplored domain, for which we have not made any definitive conclusions.” I then hypothesize how we might gain some certainty regarding the enlarged class of voting algorithms, though I am likely wrong! [at the top, in epistemic status]
2] “State Gibbard’s theorem, and”
- Gibbard’s Theorem proved that no voting method can avoid strategic voting (unless it is dictatorial or two-choices only) [In the TL;DR at the top]
- He restricted his analysis to all “strategies which can be expressed as a preference n-tuple”… a ranked list. [Second sentence of the first paragraph under “Gibbard’s Theorem” header, at the beginning of the body of the post]
3] “Show how exactly machine learning has solutions for that problem.”
- This is proof by existence that “An algorithm CAN function properly when fed latent-space vectors, DESPITE that algorithm failing when given ONLY a ranked-list.” So, the claim of Voting Theorists that “if ranked-lists fail, then everything must fail” is categorically false. [The third paragraph of the “Gibbard’s Theorem” section, first sentence]
4] “Rather, it applies for any situation where each participant has to choose from some set of personal actions, and there’s some mechanism that translates every possible combination of actions to a social choice (of one result from a set).”
No, specifically, Gibbard frames all those choices as a particular data-type: “the domain of the function gconsisting of all n-tuples...”, (p.589) and he presumed that such a data-type would be expressive enough to find any voting algorithm that would be non-strategic, if any such an algorithm could exist. By restricting himself to that data-type, he missed the proof by existence I mentioned above.
5] “that doesn’t mean it’s the end of the world”
At no point did I claim this was an existential risk—neither is shrimp welfare. I’m not sure what point you’re trying to make with this comment. At the bottom of my post, the section titled “Purpose!” I outline the value statement: “considering that governments’ budgets consume a large fraction of the global economy, there are likely trillions of dollars which could be better-allocated. That’s the value-statement, justifying an earnest effort to find such a voting algorithm OR prove that none can exist in this case, as well.”
I’m not sure why I was able to answer all your questions with only quotes from my post. Did I clump the thoughts into paragraphs in a way that all of them were missed?
I think Guy Raveh is right, at least about what Gibbard’s theorem states. The second paragraph of Gibbard’s theorem (Gibbard, Allan. “Manipulation of Voting Schemes: A General Result.” Econometrica, vol. 41, no. 4, The Econometric Society, July 1973, pp. 587–601. JSTOR, https://doi.org/10.2307/1914083.) begins: “The result on voting schemes follows from a theorem I shall prove which covers schemes of a more general kind. Let a game form be any scheme which makes an outcome depend on individual actions of some specified sort, which I shall call strategies. A voting scheme, then, is a game form in which a strategy is a profession of preferences, but many game forms are not voting schemes. Call a strategy dominant for someone if, whatever anyone else does, it achieves his goals at least as well as would any alternative strategy.”
I’m trying to understand his proof, since otherwise I’m a bit skeptical that it’s actually valid. One thing that tripped me up is that (1b) does not state transitivity of strict preference, but merely that everything has to be either greater than the minimum of a pair or less than their maximum. However, its contrapositive states transitivity of non-strict preference: ~xPy | ~yPz → ~xPz, i.e. yRx | zRy → zRx. One key thing it implies is that a preference cannot be formed from a chain of solely indifference.
But by also using (1a), we can deduce transitivity from (1b): Assume xPy and yPz. Since xPy, by (1b), xPz | zPy. Meanwhile, by (1a), ~(yPz & zPy), i.e. ~yPz | ~zPy. Since yPz, ~yPz is impossible, so ~zPy. Bringing them back together, since xPz | zPy but ~zPy, xPz. QED.
Thank you for diving into the details! And, to be clear, I am not taking issue with any of Gibbard’s proof itself—if you found an error in his arguments, that’s your own victory, please claim it! Instead, what I point to is Gibbard’s method of DATA-COLLECTION.
Gibbard pre-supposes that the ONLY data to be collected from voters is a SINGULAR election’s List of Preferences. And, I agree with Gibbard in his conclusion, regarding such a data-set: “IF you ONLY collect a single election’s ranked preferences, then YES, there is no way to avoid strategic voting, unless you have only one or two candidates.”
However, that Data-Set Gibbard chose is NOT the only option. In a Bank, they detect Fraudulent Transactions by placing each customer’s ‘lifetime profile’ into a Cluster (cluster analysis). When that customer’s behavior jumps OUTSIDE of their cluster, you raise a red flag of fraud. This is empirically capable of detecting what is mathematically equivalent to ‘strategic voting’.
So, IF each voter’s ‘lifetime profile’ was fed into a Variational Auto-Encoder, to be placed within some Latent Space, within a Cluster of similarly-minded folks, THEN we can see if they are being strategic in any particular election: if their list of preferences jumps outside of their cluster, they are lying about their preferences. Ignore those votes, safely protecting your ballot from manipulation.
Do you see how this does not depend upon Gibbard being right or wrong in his proof? As well as the fact that I do NOT disagree with his conclusion that “strategy-proof voting with more than two candidates is not possible IF you ONLY collect a SINGLE preference-list as your one-time ballot”?
I’m having some trouble understanding your post, and I think it could be useful to:
State the exact problem setting you are addressing,
State Gibbard’s theorem, and
Show how exactly machine learning has solutions for that problem.
As far as I understand Gibbard’s theorem, it does not only apply to ranked lists (as opposed to Arrow’s theorem). Rather, it applies for any situation where each participant has to choose from some set of personal actions, and there’s some mechanism that translates every possible combination of actions to a social choice (of one result from a set). In this context either the mechanism limits the choice to only two results; or there’s a dictatorship, or there’s an option for strategic voting.
This isn’t my area so I’m taking a risk here that I might say something stupid, but: The existence of strategic voting is a problem, since it disincentivises truthfulness; But intuitively that doesn’t mean it’s the end of the world—all voting mechanisms we’ve had so far were very obviously open to strategic voting, and we still have functioning governments. Also choices that are limited to two options are often important and interesting, e.g. whether we improve the world or harm it, whether we expand into space or not, or whether the world should act to limit population growth.
I can point you to where I did those things...
1] “State the exact problem setting you are addressing,”
- “There is an unexplored domain, for which we have not made any definitive conclusions.” I then hypothesize how we might gain some certainty regarding the enlarged class of voting algorithms, though I am likely wrong! [at the top, in epistemic status]
2] “State Gibbard’s theorem, and”
- Gibbard’s Theorem proved that no voting method can avoid strategic voting (unless it is dictatorial or two-choices only) [In the TL;DR at the top]
- He restricted his analysis to all “strategies which can be expressed as a preference n-tuple”… a ranked list. [Second sentence of the first paragraph under “Gibbard’s Theorem” header, at the beginning of the body of the post]
3] “Show how exactly machine learning has solutions for that problem.”
- This is proof by existence that “An algorithm CAN function properly when fed latent-space vectors, DESPITE that algorithm failing when given ONLY a ranked-list.” So, the claim of Voting Theorists that “if ranked-lists fail, then everything must fail” is categorically false. [The third paragraph of the “Gibbard’s Theorem” section, first sentence]
4] “Rather, it applies for any situation where each participant has to choose from some set of personal actions, and there’s some mechanism that translates every possible combination of actions to a social choice (of one result from a set).”
No, specifically, Gibbard frames all those choices as a particular data-type: “the domain of the function g consisting of all n-tuples...”, (p.589) and he presumed that such a data-type would be expressive enough to find any voting algorithm that would be non-strategic, if any such an algorithm could exist. By restricting himself to that data-type, he missed the proof by existence I mentioned above.
5] “that doesn’t mean it’s the end of the world”
At no point did I claim this was an existential risk—neither is shrimp welfare. I’m not sure what point you’re trying to make with this comment. At the bottom of my post, the section titled “Purpose!” I outline the value statement: “considering that governments’ budgets consume a large fraction of the global economy, there are likely trillions of dollars which could be better-allocated. That’s the value-statement, justifying an earnest effort to find such a voting algorithm OR prove that none can exist in this case, as well.”
I’m not sure why I was able to answer all your questions with only quotes from my post. Did I clump the thoughts into paragraphs in a way that all of them were missed?
I think Guy Raveh is right, at least about what Gibbard’s theorem states. The second paragraph of Gibbard’s theorem (Gibbard, Allan. “Manipulation of Voting Schemes: A General Result.” Econometrica, vol. 41, no. 4, The Econometric Society, July 1973, pp. 587–601. JSTOR, https://doi.org/10.2307/1914083.) begins: “The result on voting schemes follows from a theorem I shall prove which covers schemes of a more general kind. Let a game form be any scheme which makes an outcome depend on individual actions of some specified sort, which I shall call strategies. A voting scheme, then, is a game form in which a strategy is a profession of preferences, but many game forms are not voting schemes. Call a strategy dominant for someone if, whatever anyone else does, it achieves his goals at least as well as would any alternative strategy.”
I’m trying to understand his proof, since otherwise I’m a bit skeptical that it’s actually valid. One thing that tripped me up is that (1b) does not state transitivity of strict preference, but merely that everything has to be either greater than the minimum of a pair or less than their maximum. However, its contrapositive states transitivity of non-strict preference: ~xPy | ~yPz → ~xPz, i.e. yRx | zRy → zRx. One key thing it implies is that a preference cannot be formed from a chain of solely indifference.
But by also using (1a), we can deduce transitivity from (1b): Assume xPy and yPz. Since xPy, by (1b), xPz | zPy. Meanwhile, by (1a), ~(yPz & zPy), i.e. ~yPz | ~zPy. Since yPz, ~yPz is impossible, so ~zPy. Bringing them back together, since xPz | zPy but ~zPy, xPz. QED.
Thank you for diving into the details! And, to be clear, I am not taking issue with any of Gibbard’s proof itself—if you found an error in his arguments, that’s your own victory, please claim it! Instead, what I point to is Gibbard’s method of DATA-COLLECTION.
Gibbard pre-supposes that the ONLY data to be collected from voters is a SINGULAR election’s List of Preferences. And, I agree with Gibbard in his conclusion, regarding such a data-set: “IF you ONLY collect a single election’s ranked preferences, then YES, there is no way to avoid strategic voting, unless you have only one or two candidates.”
However, that Data-Set Gibbard chose is NOT the only option. In a Bank, they detect Fraudulent Transactions by placing each customer’s ‘lifetime profile’ into a Cluster (cluster analysis). When that customer’s behavior jumps OUTSIDE of their cluster, you raise a red flag of fraud. This is empirically capable of detecting what is mathematically equivalent to ‘strategic voting’.
So, IF each voter’s ‘lifetime profile’ was fed into a Variational Auto-Encoder, to be placed within some Latent Space, within a Cluster of similarly-minded folks, THEN we can see if they are being strategic in any particular election: if their list of preferences jumps outside of their cluster, they are lying about their preferences. Ignore those votes, safely protecting your ballot from manipulation.
Do you see how this does not depend upon Gibbard being right or wrong in his proof? As well as the fact that I do NOT disagree with his conclusion that “strategy-proof voting with more than two candidates is not possible IF you ONLY collect a SINGLE preference-list as your one-time ballot”?