7 ms·

Mathematicians roll dice and get rock-paper-scissors

90 points by nsoonhui 4y ago

julianeon 4y ago
Quanta magazine is doing amazing work; every time I see a link to it I know it will be good.
DieBruderBauer 4y ago
I wonder what the implications are for “intransitive” dice and Casino-level gambling potential. Wonderful article.
thaumasiotes 4y ago
> The three dice below are said to be "intransitive" -- no one of them is stronger than the others. Except that this is a definition of "stronger" quite different from the one in common use: > [Die] A beats [die] B on 58% of rolls. > B beats C on 58% of rolls. > C beats A on 69% of rolls. By any conventional metric, one of these dice is the strongest, and it's C. If you released something like this into the wild, you'd expect to come back later and find that C had done the best and A had done the worst. import random dice = [ [3,3,3,3,3,6], [2,2,2,5,5,5], [1,4,4,4,4,4] ] wins = [0,0,0] for t in range(10000): die1 = random.randrange(3) face1 = random.randrange(6) die2 = random.randrange(3) face2 = random.randrange(6) result1 = dice[die1][face1] result2 = dice[die2][face2] if result1 > result2: wins[die1] += 1 elif result2 > result1: wins[die2] += 1 print(wins) This actually doesn't show much of a pattern of C doing better than B. It does show A being much weaker than either of them. edit: The reason C and B look equally good in the code above is that B gets many fewer draws than the other dice do (because it's split 3/3 whereas they're both split 5/1). Those draws that don't happen are a win for B and a loss for [the other] B, but if you only count the win, it looks like B is benefiting. B wins about the same amount as C does, but it loses a lot more. A loses about the same amount as B does, but it wins a lot less.
- mendelab 4y ago
  B is actually the best if you play all three at the same time. Intransitive dice are still surprising and interesting to me, I Wonder if there are 3 such dice which are equally strong when played at the same time ? Another fun dice concept are go first dice: https://en.m.wikipedia.org/wiki/Go_First_Dice https://en.m.wikipedia.org/wiki/Go_First_Dice
- brookst 4y ago
  I think “strongest” refers to the Gates/Buffet bet at the beginning of the article. There is no die you can pick that will give you an advantage over someone who picks second. Contrast with a scenario where two dice are our normal 1-6 flavor and one die is all 6’s. In that case the all-6 die is stronger because it doesn’t matter which other dies it goes up against.
  - thaumasiotes 4y ago
    > I think “strongest” refers to the Gates/Buffet bet at the beginning of the article. There is no die you can pick that will give you an advantage over someone who picks second. But there is a die you can pick that is uniformly better than any alternative. That die is C. If your opponent is playing correctly, C will give you the optimal outcome. And if your opponent isn't playing correctly, C will also give you the optimal outcome. This is not true of the other two dice.
    - kadoban 4y ago
      There is an optimal choice of die, yes. But you still lose more often than you win if you pick first. The counterintuitive part is that the first player cannot win more often than they lose.
      - brookst 4y ago
        Perhaps better framed as “there is no optimal die until after the first player picks”.
    - 4y ago
bbstats 4y ago
It appears on the surface that this phenomenon is related to (if not exactly equal to) Simpson's paradox.
- pxx 4y ago
  what? no, it has nothing to do with simpson's paradox, which is a property that comes from conditional probabilities (of which there are none in this problem; each die roll is independently distributed).
zoklet-enjoyer 4y ago
I have a set of these https://mathsgear.co.uk/products/go-first-dice https://mathsgear.co.uk/products/go-first-dice
- 082349872349872 4y ago
  https://mathsgear.co.uk/collections/games/products/jigsaw-jigsaw https://mathsgear.co.uk/collections/games/products/jigsaw-ji... is brilliant
ramraj07 4y ago
I’m trying to understand what the implication of the Gates-Buffet story is: is it supposed to be an exaggerated story like a Mathematician joke? Or is it being implied here that Gates instantly deduced this intransitive property just merely looking at the dice?
- throw_pm23 4y ago
  Or maybe he just suspected there was something funny about the game, and that the second player had a winning play. Playing second he would also buy more time to figure out what's going on exactly.
- 082349872349872 4y ago
  It might just imply that Gates reads Martin Gardner (who had a column in a popular periodical) or the equivalent, but I don't see why he wouldn't have been able to figure expectation values in his head?
- mensetmanusman 4y ago
  It’s to offer substantial proof that Bill Gates is a God among men and Windows 11 should in fact have advertisements in the start menu.
- kadoban 4y ago
  It's really not that hard to work out that it's true when given 3 dice with this property, especially if you already know that it's possible or if your suspicion level is high due to the framing. All you need to do is look at each pair of dice and compare. Especially for some sets of dice the comparison is trivial. Like look at the ones in the article with five 3s and five 4s, you don't even need to think to figure out which one wins more.
- bntyhntr 4y ago
  People like apocryphal stories about successful people. Celebrities, billionaires, professors. imo the story serves as a way to connect the reader to the article. The HN crowd tends to care less about these kinds of things, which is fine, but the audience is much broader. See for example: http://www.borrett.id.au/computing/petals-bg.htm http://www.borrett.id.au/computing/petals-bg.htm or the umpteen versions of http://math.bme.hu/~petz/vnsumming.html http://math.bme.hu/~petz/vnsumming.html
mabbo 4y ago
I love my little set of transitive dice. I cannot wait until my child is old enough to be fooled by them.
pxx 4y ago
Wow that figure is bad. The table compares only the situation where both dice roll the same labeled side, which isn't meaningful (you would be able to conclude that a d6 + 1 is a worse die than a d6 this way...). But at least the numbers on the bottom are right.

90 points by nsoonhui 4y ago

julianeon 4y ago

Quanta magazine is doing amazing work; every time I see a link to it I know it will be good.

DieBruderBauer 4y ago

I wonder what the implications are for “intransitive” dice and Casino-level gambling potential. Wonderful article.

thaumasiotes 4y ago

> The three dice below are said to be "intransitive" -- no one of them is stronger than the others. Except that this is a definition of "stronger" quite different from the one in common use: > [Die] A beats [die] B on 58% of rolls. > B beats C on 58% of rolls. > C beats A on 69% of rolls. By any conventional metric, one of these dice is the strongest, and it's C. If you released something like this into the wild, you'd expect to come back later and find that C had done the best and A had done the worst. import random dice = [ [3,3,3,3,3,6], [2,2,2,5,5,5], [1,4,4,4,4,4] ] wins = [0,0,0] for t in range(10000): die1 = random.randrange(3) face1 = random.randrange(6) die2 = random.randrange(3) face2 = random.randrange(6) result1 = dice[die1][face1] result2 = dice[die2][face2] if result1 > result2: wins[die1] += 1 elif result2 > result1: wins[die2] += 1 print(wins) This actually doesn't show much of a pattern of C doing better than B. It does show A being much weaker than either of them. edit: The reason C and B look equally good in the code above is that B gets many fewer draws than the other dice do (because it's split 3/3 whereas they're both split 5/1). Those draws that don't happen are a win for B and a loss for [the other] B, but if you only count the win, it looks like B is benefiting. B wins about the same amount as C does, but it loses a lot more. A loses about the same amount as B does, but it wins a lot less.

mendelab 4y ago
B is actually the best if you play all three at the same time. Intransitive dice are still surprising and interesting to me, I Wonder if there are 3 such dice which are equally strong when played at the same time ? Another fun dice concept are go first dice: https://en.m.wikipedia.org/wiki/Go_First_Dice https://en.m.wikipedia.org/wiki/Go_First_Dice
brookst 4y ago
I think “strongest” refers to the Gates/Buffet bet at the beginning of the article. There is no die you can pick that will give you an advantage over someone who picks second. Contrast with a scenario where two dice are our normal 1-6 flavor and one die is all 6’s. In that case the all-6 die is stronger because it doesn’t matter which other dies it goes up against.
- thaumasiotes 4y ago
  > I think “strongest” refers to the Gates/Buffet bet at the beginning of the article. There is no die you can pick that will give you an advantage over someone who picks second. But there is a die you can pick that is uniformly better than any alternative. That die is C. If your opponent is playing correctly, C will give you the optimal outcome. And if your opponent isn't playing correctly, C will also give you the optimal outcome. This is not true of the other two dice.
  - kadoban 4y ago
    There is an optimal choice of die, yes. But you still lose more often than you win if you pick first. The counterintuitive part is that the first player cannot win more often than they lose.
    - brookst 4y ago
      Perhaps better framed as “there is no optimal die until after the first player picks”.
  - 4y ago

bbstats 4y ago

It appears on the surface that this phenomenon is related to (if not exactly equal to) Simpson's paradox.

pxx 4y ago
what? no, it has nothing to do with simpson's paradox, which is a property that comes from conditional probabilities (of which there are none in this problem; each die roll is independently distributed).

zoklet-enjoyer 4y ago

I have a set of these https://mathsgear.co.uk/products/go-first-dice https://mathsgear.co.uk/products/go-first-dice

082349872349872 4y ago
https://mathsgear.co.uk/collections/games/products/jigsaw-jigsaw https://mathsgear.co.uk/collections/games/products/jigsaw-ji... is brilliant

ramraj07 4y ago

I’m trying to understand what the implication of the Gates-Buffet story is: is it supposed to be an exaggerated story like a Mathematician joke? Or is it being implied here that Gates instantly deduced this intransitive property just merely looking at the dice?

throw_pm23 4y ago
Or maybe he just suspected there was something funny about the game, and that the second player had a winning play. Playing second he would also buy more time to figure out what's going on exactly.
082349872349872 4y ago
It might just imply that Gates reads Martin Gardner (who had a column in a popular periodical) or the equivalent, but I don't see why he wouldn't have been able to figure expectation values in his head?
mensetmanusman 4y ago
It’s to offer substantial proof that Bill Gates is a God among men and Windows 11 should in fact have advertisements in the start menu.
kadoban 4y ago
It's really not that hard to work out that it's true when given 3 dice with this property, especially if you already know that it's possible or if your suspicion level is high due to the framing. All you need to do is look at each pair of dice and compare. Especially for some sets of dice the comparison is trivial. Like look at the ones in the article with five 3s and five 4s, you don't even need to think to figure out which one wins more.
bntyhntr 4y ago
People like apocryphal stories about successful people. Celebrities, billionaires, professors. imo the story serves as a way to connect the reader to the article. The HN crowd tends to care less about these kinds of things, which is fine, but the audience is much broader. See for example: http://www.borrett.id.au/computing/petals-bg.htm http://www.borrett.id.au/computing/petals-bg.htm or the umpteen versions of http://math.bme.hu/~petz/vnsumming.html http://math.bme.hu/~petz/vnsumming.html

mabbo 4y ago

I love my little set of transitive dice. I cannot wait until my child is old enough to be fooled by them.

pxx 4y ago

Wow that figure is bad. The table compares only the situation where both dice roll the same labeled side, which isn't meaningful (you would be able to conclude that a d6 + 1 is a worse die than a d6 this way...). But at least the numbers on the bottom are right.