There are land-division algorithms that have some relationship to Qix.
Oh, here’ a diagram from Stromquist’s paper:
(The big sword is wielded by a referee.)
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well this should generally not happen since the size of each piece is dictated by the most conservative guest. Since it is a ridiculous premise to frame an interesting problem, I don’t feel like it labors the puzzle too much to presume that the basic piece size estimating skills of the guests would vary somewhat but generally center somewhere near correctness. Some will overestimate how much 1/n is, some will underestimate; the underestimators will get their pieces first, avoiding your scenario. For it to be a problem, the guests would have to be mostly composed of overestimators, which may well be the expected reality in a random sample of humans. I wouldn’t know or care to guess as to that, but I would say that any party where this algorithm is resorted to probably is attended by the sort of person who is actually pretty good at this sort of thing.
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What are these?
What is this?
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That looks fun though I don’t remember playing.
There are many such axioms, depending on the situation. For example, Arrow’s Impossibility Theorem considers a situation where everyone in a community ranks n alternatives (candidates, building plans, ice cream flavors, whatever) and asks whether there is a deterministic algorithm which takes the rankings/ballots as input, and outputs a ranking satisfying the following:
- No dictator: the algorithm doesn’t simply consist of taking one prespecified person’s rankings (eg, the mayor) and discarding the rest.
- Irrelevant alternatives are irrelevant: if some people change some of their rankings, but the relative positions of choice A and choice B do not change on anyone’s ballot, then the final algorithm does not change the relative positions of A and B
- Unanimity: If everyone prefers A to B, so must the final ranking.
Arrow’s Theorem is that these three are not mutually consistent. There are lots of other kinds of axioms, and it is generally the case that once you get a critical mass of them you can’t meet them.
For voting systems (where the output is supposed to be a single candidate, or maybe a list of top m candidates) the fairness criteria tend to be a little different, eg the majority condition is that if X is the top choice of a majority of the voters then X should win. These have been developed over a period of 200+ years - there used to be fistfights in the French Academy over voting systems - though the idea that you can’t have any good system is 20th century.
The Wikipedia article on voting systems used to be pretty good if you want to see some of the other criteria used, and some voting system alternatives to plurality/majority. In particular, “single transferrable vote” is pretty popular and worth a look.
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That’s interesting to see the problems represented as logic puzzles.
The term fairness is also important for legal and political philosophy like Rawls, Dworkin, Okin and critical theorists like Nancy Fraser. Weighted voting systems that correct for underrepresentation of particular racial groups or consensus (negotiation) decision-making models are examples.
The cake cutting question is a fun way to tilt some of those questions. How would you cut the cake to compensate for prior cake inequality? Or how would you cut the cake to accommodate exacting, idiosyncratic and conflicting conditions?
Rawls gives fairness principles and other guidance, but his theory is also criticized for inadequate consideration of some problems, like intolerant groups and gendered institutions.
The idea that rational decision-making or policy might require some mathematical analysis is not at all new (see the Talmud reference above), and while some of the most important ideas required mathematics that didn’t exist until the 20th century, analysis of voting systems is pretty mature field at this point. It only requires policymakers to decide to incorporate it into their decisionmaking. It doesn’t help when dingbats like Andrew Hacker have decided to run around advocating that the only math people need to be good citizens is the most basic kind of counting and graphing. It is fine to drop Calculus if you must, but this other stuff is freaking useful, and is best understood with algebra in one’s background.
(Ironically, at the undergraduate level it is far more commonly taught to non-STEM students than it is to math or science majors. I never saw any of it until I was a fairly advanced graduate student.)
By the way, for some other examples of mathematical puzzles applied in the real world, I recommend Bad Acts and Guilty Minds: Conundrums of the Criminal Law by Leo Katz. (I’m not just recommending it because he quotes an article of mine in its entirety in the book.)
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Give morbidly obese people smaller slices, starving people larger ones.
If nobody’s fat or skinny, the inequality is not worth the tyranny required to address it; simply have whoever baked the cake give people whatever s/he thinks they deserve, and just slap any crybabies who won’t stop whinging and grizzling, on the premise that they really needed something to cry about.
Everything is easy if you apply the Rule of My Mother!
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