
What happens when you ask people to pick the lowest unique prime number. - pavel_lishin
http://www.robinsloan.com/note/penumbra-primes/
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rmcclellan
This was done on a much larger scale (without primes) as a lottery game in
Sweden called "Limbo" or "LUPI" (lowest unique positive integer). Several game
theorists have analyzed the data with some interesting results:

<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.

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monk_the_dog
This reminds me of the game "clomp" (which I may or may not have invented+)
but I used to play with my brother on car rides. Here are the rules:

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".

~~~
swolchok
so...is there an incentive to deviate from "clomper picks 1, opponent picks 0,
first clomper wins?"

~~~
bornhuetter
The way I read the rules, this would be the strategy:

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.

~~~
monk_the_dog
Yes, this is how it's played. It really is a fun little game, especially if
you and your opponent are the type of people who over-think things. Try it
sometime!

~~~
bornhuetter
I think I will - it sounds fun!

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simonster
This is not quite asking people to pick the lowest unique prime number,
because it was clear in advance that the prize would go to the individuals who
picked the five lowest unique prime numbers, rather than the single individual
who picked the lowest. This means that there is an incentive to err on the
side of picking a prime number that's too high, rather than a prime number
that's too low, which probably skews the distribution.

~~~
Estragon
The moderators have struck again. The title used to be "What happens when you
ask people to pick the lowest prime number _no one else will pick._ "

~~~
3143
That's just as bad. The problem is in the word "lowest," because it is
understood to mean "the one lowest".

~~~
mistercow
It's not just as bad; it gives a closer to accurate picture, and it's pretty
obnoxious of the mods to change it to be less illuminating. The way it's
written now, I thought it was going to be about asking people a nonsensical
mathematical question (which still has a correct answer), and seeing how the
nonsensical curve-ball affects their answer.

~~~
HorizonXP
Agreed. When I read the title, I thought, "WTH, what's so hard about picking
the number 2? Is this article about how few people know this fact or
something?"

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zeteo
There's no reason to require prime numbers, the game works the same with
picking the lowest unique positive integer. We actually played this game at
our department's research symposium two years ago with about a hundred people.
Nobody picked 5 and the winning entry was 11.

~~~
TylerE
Actually, there is a reason. It forces people ( _most_ people, at any rate),
to actually look at a list of numbers, rather than simply _thinking_ of one.

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.

~~~
jonnathanson
_"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."_

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).

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itsadok
He mentioned that someone picked 333,667 and notes that it's a Sexy Prime, but
333,667 is also a Unique Prime (<http://en.wikipedia.org/wiki/Unique_prime>),
which makes me think someone was either being cute, or they suspected that it
was a trick question.

~~~
pvidler
Based on the end of the post, they might well have 'won', given the special
mention in the article.

"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…"

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jholman
Wow. TFA was a fine article, but I followed a link or two, and ended up
reading the posted fulltext of My. Penumbra's 24-Hour Bookstore, and that was
some Good Reading.

~~~
andrewcooke
doesn't it sound familiar? apart from the gooey ending and the google twist, i
could swear i had read that - a story, with a bookstore, with that meaning and
process - before.

[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....]

~~~
planetguy
It felt like Borges wrote the first half and Neal Stephenson wrote the second.

I liked the first half better -- the mystery of the situation was so fabulous
that any possible resolution would be a letdown.

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Symmetry
Unsurprisingly, there's a big spike at 17 since its well known to be the most
random number.
[http://blogs.discovermagazine.com/cosmicvariance/2007/01/30/...](http://blogs.discovermagazine.com/cosmicvariance/2007/01/30/the-
power-of-17/)

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sente
This reminds me of a question asked in a Game Theory class server years ago:

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:

<http://sente.cc/wsgi/number_poll/>

~~~
thaumasiotes
How could the strategy here be anything other than "pick 0"?

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).

~~~
sente
I had 10 minutes to create the poll before the library closed...I definitely
could have done a better job describing the problem, you're right.

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

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dewarrn1
This is an example of a Keynesian beauty contest [0].

0: <http://en.wikipedia.org/wiki/Keynesian_beauty_contest>

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simonbrown
What about negative primes? According to Wolfram Alpha:

<http://www.wolframalpha.com/input/?i=-2+prime>

~~~
tikhonj
I've always heard "prime number" defined to be greater than 1. WolframAlpha is
probably just being crazy.

Wikipedia [1], and, more amusingly, Wolfram MathWorld [2] both agree with this
definition.

[1]: <http://en.wikipedia.org/wiki/Prime_number>

[2]: <http://mathworld.wolfram.com/PrimeNumber.html>

~~~
drostie
Strictly speaking, the moment you ask "Is -2 prime?" you are asking a
fundamentally different question from "is 2 prime?" because -2 is not in N =
{0?, 1, 2, 3, ...}, the set over which the notion "prime" is defined.

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.

~~~
dbaupp
Another definition[1] of prime in an arbitrary ring is

    
    
      p is prime if and only if p|ab implies p|a or p|b
    

which is equivalent to the prime ideal concept. Interestingly, these
definitions mean that 0 is actually a prime in Z!

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>

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gkoberger
There are quite a few of those scammy online auction sites that do "Unique bid
auctions" -- same concept, just not restricted to prime numbers.

Since there is (possibly) a bit of skill involved, it makes it interesting
from a legal perspective.

<http://en.wikipedia.org/wiki/Unique_bid_auction>

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vhf
The challenge to win a book was really interesting, as was the results. And
nice and smart way to advertise his book.

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 !

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tzs
Why primes?

~~~
planetguy
To make the entrants feel smarter. The whole thing is, of course, completely
isomorphic to a guess-an-integer competition, but the rather minimal effort
you need to go to to figure out the Nth prime makes it all worthwhile.

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.

~~~
robinsloan
Author here: I am also fascinated by this sense of certain numbers sounding
prime (or more prime than others). For instance -- maybe this is just me -- I
think 109 (the first winner) doesn't sound prime at all.

But 17! Yes! It can't shut up about how prime it is!

~~~
parfe
Did you receive any emails choosing composite numbers? To me, 51 sounds prime.

~~~
baddox
A number is divisible by 3 if the sum of its digits is divisible by 3.

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gwern
Hm, I wonder what Douglas Hofstadter's superrational
(<http://www.gwern.net/docs/1985-hofstadter>) strategy would be.

~~~
praptak
From what I understand the "super-" only makes a difference in games where
cooperation/defection make sense. The superrational player chooses cooperation
because they know that all the other players are also superrational and will
also do so. This is not the case in the game described, so I think that the
superrational player would play like a plain old rational one.

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KaoruAoiShiho
2 people on 61 is thinking, damn it, almost!

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planetguy
I would assume that the right strategy here varies depending on how many other
people are going to enter the competition.

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".

