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 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”?