About 2⁄3 of people prefer Kasich to any other candidate on offer [...]
I think this is misleadingly phrased. It’s true that Kasich wins by 2⁄3 to 1⁄3 no matter which other candicate you pit him against, but it’s not true that 2⁄3 of people prefer him to any other candidate on offer. Only 14% + 17% = 1⁄3 of people have him as their first choice.
Your thesis stands, though, and I’ve updated on it.
The statement is ambiguous, and depends on whether you’re binding “2/3 of people” or “any other candidate” to a value first. It can mean either the intended:
for each other candidate on offer:
about 2/3 of people prefer Kasich to that candidate
Or:
there exists a group of about 2/3 of people that:
for each other candidate on offer:
everyone in the group prefers Kasich to that candidate
I think the ambiguity is clear in context, however, since the latter is so clearly false.
A 2⁄3 majority prefer Kasich to any other candidate. The point is that Independence of Irrelevant Alternatives dictates that if a group prefers X to Y, and prefers X to Z, that they logically must prefer X if all three are options. It’s like if I ask you to choose between chocolate and vanilla and you pick chocolate; if I tell you strawberry is also an option, that shouldn’t make you switch to vanilla.
There’s no group that prefers Kasich to Trump and also prefers Kasich to Clinton. It’s 2⁄3 in each case, but those two groups of 2⁄3 only have an overlap of 1⁄3.
I’m not familiar with voting theory, so I might be missing the point, but the sentence “there exists a 2⁄3 majority of the voting population all of whom prefer Kasich to any other candidate” is false. (The problem might be the ambiguity of the English language: it is true that “for any candidate besides Kasich, there exists a 2⁄3 majority who prefers Kasich to that candidate”.)
I think this is misleadingly phrased. It’s true that Kasich wins by 2⁄3 to 1⁄3 no matter which other candicate you pit him against, but it’s not true that 2⁄3 of people prefer him to any other candidate on offer. Only 14% + 17% = 1⁄3 of people have him as their first choice.
Your thesis stands, though, and I’ve updated on it.
The statement is ambiguous, and depends on whether you’re binding “2/3 of people” or “any other candidate” to a value first. It can mean either the intended:
Or:
I think the ambiguity is clear in context, however, since the latter is so clearly false.
A 2⁄3 majority prefer Kasich to any other candidate. The point is that Independence of Irrelevant Alternatives dictates that if a group prefers X to Y, and prefers X to Z, that they logically must prefer X if all three are options. It’s like if I ask you to choose between chocolate and vanilla and you pick chocolate; if I tell you strawberry is also an option, that shouldn’t make you switch to vanilla.
There’s no group that prefers Kasich to Trump and also prefers Kasich to Clinton. It’s 2⁄3 in each case, but those two groups of 2⁄3 only have an overlap of 1⁄3.
I’m not familiar with voting theory, so I might be missing the point, but the sentence “there exists a 2⁄3 majority of the voting population all of whom prefer Kasich to any other candidate” is false. (The problem might be the ambiguity of the English language: it is true that “for any candidate besides Kasich, there exists a 2⁄3 majority who prefers Kasich to that candidate”.)
That is irrelevant.