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What does this space witchcraft have to do with the Condorcet Paradox?
This was asked to me in a microeconomics problem set, and I am going to share the answer with you. Follow the thread
.
This was asked to me in a microeconomics problem set, and I am going to share the answer with you. Follow the thread
.
The Shepard tone illusion ( https://en.wikipedia.org/wiki/Shepard_tone) can be viewed as an aggregation of individually transitive relations, which fails transitivity when put together, creating the illusion of a tone that continually ascends or descends in pitch, yet which ultimately seems
to get no higher or lower. For the sake of simplicity, assume that there are only 3 tones, (A, B, C), which are played simultaneously. Each instrument plays a scale elevating the tone until the end of the scale and then starts again.
Illustratively, instrument one plays A first, then B, then C, elevating the pitch, and then A again, coming back to the first pitch. We can say that A L B L C, where L is a relation indicating that A is a lower tone than B.
Simultaneously, instrument two also plays the same sequence, but in this case, A is one octave above, so B L C L A. Also, instrument three also plays A, then B, then C , but in this case, A and B is one octave above, so C L A L B.
Now, we say that the "band" (instrument 1, 2, and 3) elevates the tone if at least two sources are playing a higher tone than the previous one. Note that, in the way we constructed our "song", there will be at least two instruments transiting to a higher tone,
so the "band" will be always elevating the tone, giving these endless loops sensations.
This happens because the aggregation of acyclic relations is not necessarily acyclic. But what does this have to do with the Condorcet Paradox?
This happens because the aggregation of acyclic relations is not necessarily acyclic. But what does this have to do with the Condorcet Paradox?
The Condorcet Paradox ( https://en.wikipedia.org/wiki/Condorcet_paradox) poses that collective preferences can be cyclic, even if the preferences of individual voters are not cyclic. To see this at play, just go through Shepard's tone illustrative example again +
swapping "tones" for "candidates", "band" for "collective", "instrument" for "voter", and "is a lower tone than" for "is preferred than". In this framing, candidate A is preferred to B, which is preferred to C, which is preferred to A. BAAAAM... QED
outro dia, acho que foi o @PedrooCava que tava brincando de música e matemática por aqui, talvez vc vá gostar.
