An unfair coin (potentially EXTREMELY unfair) will be flipped fifty times. Your goal is to correctly predict as many of these flips as possible, by deducing the nature of the unfairness as quickly as possible.
[Predict Heads] [Predict Tails]
You can play this (in-browser, very short) game here; the rule governing the unfairness is automatically revealed after flip 50. Followups with different governing rules are here, here, here, here and here.
Note that this is NOT an “unfair” or even “EXTREMELY unfair” coin. It’s not a fixed bias or even a wear pattern or other understandably-moving probability. There is no possible way for a real coin to have that distribution.
It would be a more interesting prediction challenge if the range of posssible patterns were known. “I’ve coded something that doesn’t make sense; go!” is not that helpful. Or even just say “predict the output of this program, which is absolutely not a coin”.
There is no possible way for a real coin to have that distribution.
Unless the person throwing read Jaynes:
a person familiar with the laws of mechanics can toss a biased coin so that it will produce predominantly either heads or tails, at will. [...] From the fact that we have seen a strong preponderance of heads, we cannot conclude legitimately that the coin is biased; it may be biased, or it may have been tossed in a way that systematically favors heads. Likewise, from the fact that we have seen equal numbers of heads and tails, we cannot conclude legitimately that the coin is ‘honest’. It may be honest, or it may have been tossed in a way that nullifies the effect of its bias.
More on how:
An important feature of this tumbling motion is conservation of angular momentum;
during its flight the angular momentum of the coin maintains a fixed direction in space (but
the angular velocity does not; and so the tumbling may appear chaotic to the eye). Let us
denote this fixed direction by the unit vector n; it can be any direction you choose, and it
is determined by the particular kind of twist you give the coin at the instant of launching.
Whether the coin is biased or not, it will show the same face throughout the motion if viewed
from this direction (unless, of course, n is exactly perpendicular to the axis of the coin, in
which case it shows no face at all).
Therefore, in order to know which face will be uppermost in your hand, you have only
to carry out the following procedure. Denote by k a unit vector passing through the coin
along its axis, with its point on the ‘heads’ side. Now toss the coin with a twist so that k and
n make an acute angle, then catch it with your palm held flat, in a plane normal to n. On
successive tosses, you can let the direction of n, the magnitude of the angular momentum,
and the angle between n and k, vary widely; the tumbling motion will then appear entirely
different to the eye on different tosses, and it would require almost superhuman powers of
observation to discover your strategy.
Thus, anyone familiar with the law of conservation of angular momentum can, after some
practice, cheat at the usual coin-toss game and call his shots with 100% accuracy. You can
obtain any frequency of heads you want – and the bias of the coin has no influence at all
on the results!
Just like life, then!
I had an expectation that it could be a very weird type of bias given that the text says to predict the nature of the unfairness, not just a direction or something like that. I agree that calling it a “coin” is quite misleading though.
This would be more to my taste (I can’t speak for anyone else’s) if we were told more about the space of possible unfairnesses. In particular, it wasn’t clear to me whether
the coin flips are allowed to depend on our predictions (and, if so, which predictions)
nor whether
we were actually looking for a genuinely probabilistic rule (in which case it would have to be a very simple one for there to be any chance of guessing it) or for a deterministic one, possibly depending on the predictions (in which case it might be more complicated).
Again, I don’t claim that my taste is anyone else’s; but my reaction to the extreme open-endedness is along the lines of “this could be practically anything, and some varieties of thing-it-could-be are obviously not deducible with any confidence from 50 bits of information, and this has probably been designed so that it’s solvable if you correctly guess what space of possibilities the creator had in mind but I don’t feel like trying to read his mind”.
The spoiler-blocks above aren’t very spoilery since they are just asking questions. But for the benefit of anyone who feels the same way as I do, here are what I now believe to be the answers to those questions:
The coin does not know anything about your predictions; you are trying to model a thing that autonomously emits coin flips, not something that actively responds to your probing. Unsurprisingly-given-that, the coin is (at least potentially) probabilistic.
You link to index C twice, rather than linking to index D. (And index D was such an interesting one too.)
Anyways, this is very fun. I made a couple (fairly easy) coins of my own, should be runnable by pasting into the console in your browser’s dev tools (while you’re on the fifty flips page, of course):
eval(unescape(escape`𩡬𪑰𫡯🐰𫱲𬡥𨱴𫡯🐰𬡥𩁩𨱴𩑤𬰽𦱝𨱴𭑡𫁳👛𧐻𩡵𫡣𭁩𫱮𘁦𫁩𬀨𬁲𩑤𪑣𭁥𩀩𮱩𩠨𩡬𪑰𫡯🀵𮱦𫁩𬁮𫰫🐱𩠨𣑡𭁨𨑮𩁯𫐨𩡬𪑰𫡯𛰵𮱡𨱴𭑡𫀽𘡔𨑩𫁳𘡽𩑬𬱥𮱡𨱴𭑡𫀽𘡈𩑡𩁳𘡽𬁲𩑤𪑣𭁥𩁳𭑳𪀨𬁲𩑤𪑣𭁥𩀩𨱴𭑡𫁳𭑳𪀨𨑣𭁵𨑬𪑦𬡥𩁩𨱴𩑤🐽𨑣𭁵𨑬𨱯𬡲𩑣𭁮𫰫🐱𫱣𭑭𩑮𭀮𩱥𭁅𫁥𫑥𫡴𠡹𢑤𬡥𬱵𫁴𬰢𪑮𫡥𬡈𥁍𣀽𣱮𘁦𫁩𬀠𘠫𩡬𪑰𫡯𛀠𮑯𭐠𬁲𩑤𪑣𭁥𩀠𘠫𬁲𩑤𪑣𭁥𩀫𘠻𘁴𪁥𘁣𫱩𫠠𨱡𫑥𘁤𫱷𫠠𘠫𨑣𭁵𨑬𩁯𨱵𫑥𫡴𩑴𡑬𩑭𩑮𭁂𮑉𩀨𘡳𨱯𬡥𘠩𫡮𩑲𢁔𣑌🐨𘡙𫱵𘁧𭑥𬱳𩑤𘀢𫱲𬡥𨱴𫡯𛰢𫁩𬁮𫰫𘠠𨱯𬡲𩑣𭁬𮐮𘠩𫱣𭑭𩑮𭀮𩱥𭁅𫁥𫑥𫡴𠡹𢑤𬡥𨱯𬡤𘠩𫡮𩑲𢁔𣑌🁢𬠾𘠫𩡬𪑰𫡯𛀢𬡥𩁩𨱴𩑤𛀢𨱴𭑡𫀩𩠨𩡬𪑰𫡯🐽𝐰𩁯𨱵𫑥𫡴𩑴𡑬𩑭𩑮𭁂𮑉𩀨𘡴𪁥𤡵𫁥𘠩𫡮𩑲𢁔𣑌🐧𥁨𩐠𬡵𫁥𘁷𨑳𞠠🁢🠢𥁨𩐠𬁲𫱢𨑢𪑬𪑴𮐠𫱦𘁴𨑩𫁳𘁩𬰠𩡬𪑰𘰠𛰠𝐰𨠾`.replace(/u../g,'')))
eval(unescape(escape`𩡬𪑰𫡯🐰𫱲𬡥𨱴𫡯🐰𬡥𩁩𨱴𩑤𬰽𦱝𨱴𭑡𫁳👛𧐻𩡵𫡣𭁩𫱮𘁦𫁩𬀨𬁲𩑤𪑣𭁥𩀩𮱩𩠨𩡬𪑰𫡯🀵𮱦𫁩𬁮𫰫🐱𩠨𩡬𪑰𫡯🐽𝐰𨑣𭁵𨑬🐢𣡯𬁥𘡽𩑬𬱥𮱡𨱴𭑡𫀽𬁲𩑤𪑣𭁥𩁽𬁲𩑤𪑣𭁥𩁳𭑳𪀨𬁲𩑤𪑣𭁥𩀩𨱴𭑡𫁳𭑳𪀨𨑣𭁵𨑬𪑦𬡥𩁩𨱴𩑤🐽𨑣𭁵𨑬𨱯𬡲𩑣𭁮𫰫🐱𫱣𭑭𩑮𭀮𩱥𭁅𫁥𫑥𫡴𠡹𢑤𬡥𬱵𫁴𬰢𪑮𫡥𬡈𥁍𣀽𣱮𘁦𫁩𬀠𘠫𩡬𪑰𫡯𛀠𮑯𭐠𬁲𩑤𪑣𭁥𩀠𘠫𬁲𩑤𪑣𭁥𩀫𘠻𘁴𪁥𘁣𫱩𫠠𨱡𫑥𘁤𫱷𫠠𘠫𨑣𭁵𨑬𩁯𨱵𫑥𫡴𩑴𡑬𩑭𩑮𭁂𮑉𩀨𘡳𨱯𬡥𘠩𫡮𩑲𢁔𣑌🐨𘡙𫱵𘁧𭑥𬱳𩑤𘀢𫱲𬡥𨱴𫡯𛰢𫁩𬁮𫰫𘠠𨱯𬡲𩑣𭁬𮐮𘠩𫱣𭑭𩑮𭀮𩱥𭁅𫁥𫑥𫡴𠡹𢑤𬡥𨱯𬡤𘠩𫡮𩑲𢁔𣑌🁢𬠾𘠫𩡬𪑰𫡯𛀢𬡥𩁩𨱴𩑤𛀢𨱴𭑡𫀩𩠨𩡬𪑰𫡯🐽𝐰𩁯𨱵𫑥𫡴𩑴𡑬𩑭𩑮𭁂𮑉𩀨𘡴𪁥𤡵𫁥𘠩𫡮𩑲𢁔𣑌🐧𥁨𩐠𬡵𫁥𘁷𨑳𞠠🁢🠢𥁨𪑳𘁣𫱩𫠠𪑳𘁡𘁨𩑬𬁦𭑬𘁣𪁥𨑴𩑲𛀠𨑮𩀠𨑬𭱡𮑳𘁬𨑮𩁳𘁴𪁥𘁳𨑭𩐠𭱡𮐠𮑯𭐠𬁲𩑤𪑣𭁥𩀮𘠼𛱢🠧`.replace(/u../g,'')))
>You link to index C twice, rather than linking to index D.
Whoops! Fixed now, thank you.
I assumed that if you are flipping a coin, trials would be independent events and each flip would have a fixed rule (which is what happens when you flip a single coin). Instead, the coin had a different rule for odd and even numbered flips. I think that the language of the website should be amended to reflect this.
Do you mean the coin has a fixed probability of heads for each of the 50 flips (perhaps drawn from some distribution at the beginning of the game)?
Or can the probability change for each flip based on some more complicated rule?
The coin does not have a fixed probability on each flip.
Boy howdy was I having trouble with spoiler text on markdown.
I liked this! The game was plenty interesting and reasonably introduced. It’s a fun twist on induction games with the addition of reasoning over uncertainty rather than exactly guessing a rule, though it does have the downside the relatively small number of samples can make the payoff dominated by randomness.
To offer one small piece of constructive advice on the execution, I did wish the flip history autoscrolled to the newest entry.
I got alternating THTHTHTHTH… for the first 28 flips, which I would have thought would be very unlikely on priors for the 80% rule. Are you sure that’s an accurate description of the rule? It doesn’t change halfway through?
It’s pretty unlikely, but not extraordinarily unlikely. I wouldn’t be surprised to learn that the site has been run a few hundred times, which would make it about 50-50 for someone to see that.
This was a fun little exercise. We get many “theory of rationality” posts on this site, so it’s very good to also have some chances to practice figuring out confusing things also mixed in. The various coins each teach good lessons about ways the world can surprise you.
Anyway, I think this was an underrated post, and we need more posts in this general category.