Rigging elections with integer linear programming
drmaciver.com
drmaciver.com
In particular, the median voter theorem states: "The median voter theorem states that 'a majority rule voting system will select the outcome most preferred by the median voter'" - IF (big if) "voters can place all election alternatives along a one-dimensional political spectrum."
Median voter theorem also, to be relevant other than in the trivial sense (in which a majority-rule election by definition, choses the formal option which was voted for by a majority of voters, which, assuming unimodal preferences but not necessarily unidimensionality, includes the median voter), requires the substantive policy outcomes to be both determined by the majority-rule election and transparent to voters at the time of the election. As substantive policy is usually indirectly set by aggregate results of of multiple elections that would be, in the best case, majority-of-majorities (but usually is less majorities than that at both levels), and because substantive policy outcomes are often not transparent to voters at the time of elections, the theorem is mostly an empty intellectual exercise.
On top of all that, median voter theorem tends to be even more irrelevant in the US because the US doesn't actually use majority-rule systems as much as people seem to think (it uses plurality rule more often for single-winner elections.)
Median voter theorem is most applicable in the real world in systems with strongly proportional representation, no separation of powers (e.g., parliamentary supremacy), high party discipline, and transparent party platforms that make clear both party positions and relative priorities.
"Two doctors give you an independent diagnosis: One doctor says there's a 99% chance it's disease A, but there's a 1% chance it's disease B. The other doctor says there's 99% chance it's disease C, but there's a 1% chance it's disease B. Question: What's the most likely outcome? Answer: It's most likely disease B, the point where both doctors/experts agree (even though they both only give it a 1% chance of being true)."
Does anyone recall where that example comes from? I wanted to cite it the other day.
We know that there are only P(C | X) = 0, meaning that you cannot have disease C because the first doctor is positive (and therefore has incontrovertible evidence) that it is not C. Similarly, P(A | Y) = 0, meaning that you cannot have disease A. As a result, the only possibility with a non-zero probability is that you have disease B.
The complete/original example goes as follows:
"Suppose that one has two equi-reliable doctors and one doctor believes a patient has either a brain tumor, with a probability (i.e. a basic belief assignment—bba's, or mass of belief) of 0.99; or meningitis, with a probability of only 0.01. A second doctor believes the patient has a concussion, with a probability of 0.99, and believes the patient suffers from meningitis, with a probability of only 0.01. Applying Dempster’s rule to combine these two sets of masses of belief, one gets finally m(meningitis)=1 (the meningitis is diagnosed with 100 percent of confidence)."
https://en.wikipedia.org/wiki/Dempster–Shafer_theory#Example...
For example, suppose we have a prior probability of: Tumor - .49, Concussion - .49, meningitis - .02.
The first doctor performs an MRI that conclusively proves that there is no Tumor. He now conludes that there is a 96% chance of concussion, and 4% chance of meningitis.
The second doctor performs a cognitive test that conclusively proves there is no concussion, but a 96% chance of tumor, and 4% chance of menigitis.
If we combine these two tests, we have disproven the two post likely causes, so the less likely cause becomes reasonable.
This doesn't work out quite so nicely in reality because of false negatives and (as you point out) the non 0% chance of an unspecified disease. Also, ideally you would tell the second doctor about the first test.
Say, there are 3 candidates and only 1 agenda point: 1 supports the idea and 2 are against it. The later 2 get 33% of the votes each and the first gets 34%. The unpopular idea wins while 66% was against it.
It sure seems like one can get a high degree of control over the results by simply adding candidates until the unpopular idea is sufficiently under represented.
It's like, if you have 100 Bernie Sanders sharing the votes you may never get socialism.
Anything good and free? I understand that commercial solvers are much better, but the price is too high for me.