What happens when you ask people to pick the lowest unique prime number.
robinsloan.com
robinsloan.com
http://swopec.hhs.se/hastef/abs/hastef0671.htm
Calculating the equillibrium strategy for rational actors is difficult because each player doesn't know how many other players there are. In the paper above, game theorists calculate it and show that the distributions seen in the lottery match up fairly well to a rational strategy.
1) Pick any integer. Your partner picks an integer without knowing yours. The first person to think of a number says "clomp". They are locked in. The other person can say their number any time after "clomp". Honor system. No cheating.
2) If your number is less then your opponent, but not one less, then add your number to your score.
3) If your number is one more than your opponent, then add your number and your opponents number together and add it to your score.
4) Otherwise no score.
5) First to 21 or more wins.
+I may have stolen this from Martin Gardner, I honestly don't remember if I came up with it or I read it. My brother came up with the name "clomp".
It reminds me of a drinking game played in China (more or less the same as Liar's Dice from Red Dead Redemption) involving dice and cups. There's a street of bars ("Bar Street" I think it might be called) in Lijiang where almost every table is full of people buying beers half-a-dozen at a time and playing this game loudly.
If you pick 1, and opponent picks 1, then no score.
But if opponent picks 1, and you pick 2, then you will win 3 points.
If your opponent thinks you are going to pick 2, he can pick 3 for 5 points. But if you go 1 instead, then you win 1.
You sit in a group, elbows on the table, fists in the air.
You take in in turns to go, clockwise.
When it is your go, you say a number that is a multiple of 5, from 0 to number of people x 10
As you say the number, everyone can either keep one or both of their fists closed, or spread their five fingers out (i.e each hand either counts as zero or five).
If the number you say is equal to the number of fingers shown, next time you use one hand not two. And then the next go, if you get it right again, you are out. Play continues until there is one person left, who then drinks the forfeit.
Your strategy isn't to pick the lowest prime. It's to pick a prime that less than 5 other people would pick lower than - so you will tend to pick a higher number. The larger the group of pickers the higher you should pick.
Like, imagine you asked a group of people to pick a number between 1-100. Some numbers are going to occur out of proportion to a random distribution. If it's a geeky crowd, you'll probably get at least 10% picking 42.
Whereas if you give them a list of the first 100 primes, and ask them to pick one, that sort of cultural bias should be mostly gone.
I realize that's not exactly the scenario involved here, but the concept still applies imo.
This is so true that, for many years, magicians would use it as a "mind reading" parlor trick. The number 7 comes up with outsized frequency when people are asked to choose a number between 1 and 10 (the further out you expand the set, the lower the probability of their picking 7 -- but not by as much as you might expect). So much so, that you could fairly reliably ask someone to pick a number, guess that they'd picked 7, and be correct.
This is because most people don't actively think about the answer. They just select the number that comes most readily to mind. In Western culture, 7 is ingrained fairly heavily as a lucky/special number, and we encounter it with enough frequency that it'll be a likely choice for the brain's equivalent of the auto-fill feature.
You're much more likely to get a truly random sampling if your question requires thought or calculation, or if the bounds are unusual enough to warrant active thought. For instance, "Pick a number between 2 and 99" isn't drastically different from "Pick a number between 1 and 100." But it's different enough that it cues the brain to stop for a millisecond and actually think about the question. As a result, it's slightly more likely to generate a unique/uncommon number. (An even stranger set, such as "Pick a number between 6 and 73," will force the person to think even longer).
"P.S. There might, in fact, be more than five primes out there. There might, after all, have been special shipments to rogue recipients. You never know…"
[maybe this sounds catty, but if you had to write a pastiche of borges you'd probably come up with this, so perhaps two people simply hit the same idea....]
I liked the first half better -- the mystery of the situation was so fabulous that any possible resolution would be a letdown.
I guess they must be related--I'm certainly interested in reading the longer version when it is released.
(Rated G. May contain creepy imagery and disturbing data visualizations. Free, CC-licensed. Escape Pod is awesome.)
Pick a number Whoever's number is closest to one half the average of all submitted numbers wins!
I just whipped together a tiny wsgi app for this:
Even the other degenerate strategy of "pick something which is like negative infinity" is a guaranteed loser, and it's only interesting if you assume everyone else will pick positive numbers (which your web app specifies, though your problem statement doesn't).
Each person was told to pick a number between 0 and 100
The number I would pick would be dependent on who else is taking the poll, how much credit do you give them, etc
I'm in my phone now with a low battery, otherwise I'd type
But one of the problems with selecting a strategy is assuming that everyone will act perfectly rationally. This is hardly ever the case.
(the other link doesn't restrict each IP to only 1 vote)
Wikipedia [1], and, more amusingly, Wolfram MathWorld [2] both agree with this definition.
There is a more general notion of primes which can be applied to any ring (a set with "addition-like" and "multiplication-like" operations). That is the notion of "prime ideals": https://en.wikipedia.org/wiki/Prime_ideal
On this account, there are "primes" for Z, and -2 is the "same prime" as 2.
p is prime if and only if p|ab implies p|a or p|b
As explanation: the only number 0 divides is 0, and Z is an integral domain[2], i.e. ab = 0 implies a = 0 or b = 0, thus 0 divides at least one of a and b if 0 divides ab.
[1]: https://en.wikipedia.org/wiki/Prime_element [2]: https://en.wikipedia.org/wiki/Integral_domain
Since there is (possibly) a bit of skill involved, it makes it interesting from a legal perspective.
I really wonder if he will be able to keep tracks of these 5 books, since some could consider them "collector" of "deserved" and would tend to keep them instead of passing them over. I'll certainly stay tuned !
Perhaps it also indicates something about which primes sound primest, though. 29 is an outlying underachiever, suggesting that "29" doesn't spring to mind when you ask someone to name a prime... certainly I had to think for a moment to make sure it really was a prime. 23, 17, 37 are overachievers -- these are really prime-sounding primes, no doubt about it.
But 17! Yes! It can't shut up about how prime it is!
Theorem: Anything less than 91 which looks prime is prime.
Proof: To do this, we will calculate the smallest number that looks prime but isn't. The numbers 2, 3, 4, 5, and 6 all have easy and quick divisibility rules, so our number cannot be divisible by any of them. 7, however, is hard to check for - so clearly the first number that looks prime but isn't will be divisible by 7. Sadly, everyone realizes 49 isn't prime, because it's a square, so we need to find the next difficult prime factor. 8, 9, 10, 11, and 12 are all also subject to quick tests and divisibility rules, so discount those. 13, however, is ugly - thus, the first number that looks prime but isn't is 7 times 13, which equals 91. Q.E.D. (My professor insisted this stood for "Quite Easily Done".)
Actually, a "guess how many people will enter this competition" competition would be fun.
What would be interesting now is to repeat the competition after publishing these results; in fact, make sure these results are given on the entry page for the new competition. What would happen this time? Well, a bunch of people would start picking numbers around 109 this time. Except everybody expects everybody to do that, so maybe they'll start picking lower. Meanwhile smartasses will still be picking "2", and smartasses who think they can outsmart those smartasses will be picking "3".