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.
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.