Non-Transitive Dice(singingbanana.com)
singingbanana.com
Non-Transitive Dice
https://singingbanana.com/dice/article.htm
20 comments
It turns out that you can make dice that beat each other according to any directed graph: https://math.stackexchange.com/questions/1624550/can-you-cre...
Note that by "dice" they are referring to any probability distribution the natural numbers (including those of infinite support), not just 6 sided dice.
The actual answer given at that link uses dice with a finite number of sides. In fact they all have an even number of sides, so you could make them physically with bipyramids or trapezohedra.
https://en.wikipedia.org/wiki/Bipyramid
https://en.wikipedia.org/wiki/Trapezohedron
https://en.wikipedia.org/wiki/Bipyramid
https://en.wikipedia.org/wiki/Trapezohedron
Related: I've had this hunch for a while that the way we teach evolution in high-school biology is flawed because it still tends to implicitly be presented in progressivist terms. That is to say: as if things evolve towards something "better", and will eventually reach some kind of global optimum. And I think a part of that is that people aren't used to think in non-transitive properties, outside of Rock-Paper-Scissors and Pokemon perhaps. Maybe these dice could be used as an abstract demonstration of ecological niches being context-dependent.
There is this species of lizard whose forms use different mating strategies which compete in a rock-paper-scissors style. https://en.m.wikipedia.org/wiki/Common_side-blotched_lizard#...
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The term you are looking for is Pareto front.
Thank you! Knowing what to search for helps a lot :)
Just because evolution is not monotonic doesn’t mean that progressivism isn’t a good “low resolution” model.
It's not usually taught as a "low resolution" model though, it's taught as a global model.
Low resolution meaning coarse, like if you had to choose to believe if evolution was progressive or completely directionless, progressive would be a better model.
> if you had to choose to believe if evolution was progressive or completely directionless
We don't have to limit our choices to those two options, and there is a better third one, which undermines the argument before it even got started.
But even if we did limit ourselves to these two options (and I cannot stress enough that there is absolutely no reason to do so), one can still argue that "completely directionless" is a better approximation: evolution has no agency or goal to work towards, it just selects for whatever happens work in any given moment/context. It is the stability of the context that is providing the selection criteria that causes a consistent "direction" to emerge, but as soon as you introduce this explanation we've basically already picked our third option that I mentioned.
We don't have to limit our choices to those two options, and there is a better third one, which undermines the argument before it even got started.
But even if we did limit ourselves to these two options (and I cannot stress enough that there is absolutely no reason to do so), one can still argue that "completely directionless" is a better approximation: evolution has no agency or goal to work towards, it just selects for whatever happens work in any given moment/context. It is the stability of the context that is providing the selection criteria that causes a consistent "direction" to emerge, but as soon as you introduce this explanation we've basically already picked our third option that I mentioned.
A little bit off-topic perhaps but I have to take the opportunity to mention how much I love James Grime and his videos on numberphile. In my opinion, he has the most entertaining and compelling way of explaining and teaching the subject at hand, out of all the "hosts."
Given that we're talking about the author I would say it's tangential rather than completely OT.
I agree, I feel like James is the one that gave Numberphile its personality to me, more so than other mainstays like Matt Parker (I feel like Matt's stamps is more clearly seen in his other projects).
He adds an intrigue and an enthusiasm to Maths that reminds me of how XKCD got me into the subject circa 2006. If anyone is looking for the secret of how to make Maths 'interesting', they should definitely use James and Numberphile as a case study.
I agree, I feel like James is the one that gave Numberphile its personality to me, more so than other mainstays like Matt Parker (I feel like Matt's stamps is more clearly seen in his other projects).
He adds an intrigue and an enthusiasm to Maths that reminds me of how XKCD got me into the subject circa 2006. If anyone is looking for the secret of how to make Maths 'interesting', they should definitely use James and Numberphile as a case study.
Looking at it, the trick seems to be in tweaking the variance of the distribution of possible outcomes for the dice.
That, coupled with the fact that it doesn't matter how MUCH you win or lose by. A loss by 1 and a loss by 4 counts for the same. This nonlinearity is what allows the trick to work, supposedly defying the probabilities.
That, coupled with the fact that it doesn't matter how MUCH you win or lose by. A loss by 1 and a loss by 4 counts for the same. This nonlinearity is what allows the trick to work, supposedly defying the probabilities.
There is no nonlinearity or defiance of probabilities. It’s just simply the fact that comparing the face values between the dice outcomes has dice each winning over the next, because that operation is not transitive in general.
The “trick” here is only that people assume that dice A winning over dice B and dice B winning over dice C should mean that dice A should also beat C, which it doesn’t simply because dice outcome comparisons aren’t transitive.
So there’s no tricks or secrets, it’s just peoples intuition being wrong.
I personally prefer the intransitive set of D3s to explain this. Assume you have three dice R,B,G. They all have 1,2,3. But on ties we have tie breaker R>B>G on 1s, B>G>R on 2s and G>R>B on 3s. So it’s easy to see that for each dice there is 1/3 of times when it wins the ties, 1/3 where if losses, and the last 1/3 it either wins or loses based on which dice it’s facing. That might at face value seem like cheating because “I put the non transitive property into the tie breakers” but that’s just to make it easier to grok, you can change the dice to have faces (1,6,8),(2,4,9),(3,5,8) and you get the same without needing to handle ties. And since 1,2,3 are all equivalent in comparison to anything above 3, it’s practically the same as the first scenario.
The “trick” here is only that people assume that dice A winning over dice B and dice B winning over dice C should mean that dice A should also beat C, which it doesn’t simply because dice outcome comparisons aren’t transitive.
So there’s no tricks or secrets, it’s just peoples intuition being wrong.
I personally prefer the intransitive set of D3s to explain this. Assume you have three dice R,B,G. They all have 1,2,3. But on ties we have tie breaker R>B>G on 1s, B>G>R on 2s and G>R>B on 3s. So it’s easy to see that for each dice there is 1/3 of times when it wins the ties, 1/3 where if losses, and the last 1/3 it either wins or loses based on which dice it’s facing. That might at face value seem like cheating because “I put the non transitive property into the tie breakers” but that’s just to make it easier to grok, you can change the dice to have faces (1,6,8),(2,4,9),(3,5,8) and you get the same without needing to handle ties. And since 1,2,3 are all equivalent in comparison to anything above 3, it’s practically the same as the first scenario.
I think the 'nonlinearity' they are referring to is the difference between expected value and probability. It's possible that A beats B most of the time, and yet the average amount by which A beats B is negative, because when A does lose it loses big.
That’s mis-understanding the effect. It has nothing to do with “losing big but rarely” you could put 20s instead of 6s and it wouldn’t change a thing. It’s just that when you roll A against B it will win more than half, but B against C will have B winning more than half, and finally C vs A will have C winning more than half.
> (1,6,8),(2,4,9),(3,5,8)
I don't think it works, (1,6,8) seems balanced with (3,5,8).
I don't think it works, (1,6,8) seems balanced with (3,5,8).
And that’s why it works (as an example of non-transitivity). P(1 beats 2) = P(2 beats 3) = 4/9, but P(1 beats 3) = 2/9.
That’s a typo, it should be (3,5,7) every number should appear only once. My bad.
I think we're saying the same thing. I described the dice outcome comparison as a nonlinearity.
> (4,9,2), (3,5,7), (8,1,6)
These are the rows of a magic square.
These are the rows of a magic square.
This seem similar to "Penney's game" and the "Humble-Nishiyama Randomness Game" (https://en.wikipedia.org/wiki/Penney%27s_game) - betting on which coin-toss sequence appears first. For the 3-bit game there is also always a better pattern.
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Grimes' appearances on Numberphile are some of my favorites to watch.
I picked up a set of these dice from the official site a while back and they were fun to play with.
As board gamers, we also picked up some of the dice that you can roll to determine who goes first. (Dice have various numbers, each die is unique so no ties, but equal chance to have a higher number than the other.)
I picked up a set of these dice from the official site a while back and they were fun to play with.
As board gamers, we also picked up some of the dice that you can roll to determine who goes first. (Dice have various numbers, each die is unique so no ties, but equal chance to have a higher number than the other.)
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There isn't actually an equal chance of going first, though, is there?
For n = 2, a trivial solution is to have one die with three 1s and three 3s and one die with six 2s.
In general there are 6^n different outcomes and n! different orderings, so a necessary condition is that 6^n is a multiple of n! That means any n ≥ 5 won’t work.
In general there are 6^n different outcomes and n! different orderings, so a necessary condition is that 6^n is a multiple of n! That means any n ≥ 5 won’t work.
For equal chance of all orderings, yes. "Equal chance of going first" is more flexible, though I expect at some point we hit a limit in expressiveness (at least) if we're still using 6 sided dice.
I don't know whether any of those can be non-transitive, though. I'd guess that it's sometimes possible but it's only a guess.
I don't know whether any of those can be non-transitive, though. I'd guess that it's sometimes possible but it's only a guess.
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The math behind it is a little beyond me, but http://www.ericharshbarger.org/dice/go_first_dice.html
The dice they have for sale: https://mathsgear.co.uk/collections/dice/products/go-first-d...
(No affiliation, just the same site from the original article.)
The dice they have for sale: https://mathsgear.co.uk/collections/dice/products/go-first-d...
(No affiliation, just the same site from the original article.)
Go first dice are cool but I don't think they are non-transitive. In particular they claim that each ordering is equally likely, and for any pair of players (Jack and Jill, say) we can observe that across all orderings we should find Jack preceding Jill exactly half the time. This would mean that no pair of dice should exhibit the kind of edge that exhibits non-transitivity in non-transitive dice.
Sorry, two different lines of thought.
The first was the non-transitive Grime dice, the second was the board game/Go First dice.
My last comment was only in regards to the second set of dice, that I thought they asked about.
The first was the non-transitive Grime dice, the second was the board game/Go First dice.
My last comment was only in regards to the second set of dice, that I thought they asked about.
Why not. If all players/dice have an equal chance of rolling highest, then...?
If that, then yes, of course. But does that hold?
For the Grimes dice, unless I've made an error, it would seem not. I get `(Sum {getSum = 1416},Sum {getSum = 2160},Sum {getSum = 2160},Sum {getSum = 1440},Sum {getSum = 600})` from the following program:
For the Grimes dice, unless I've made an error, it would seem not. I get `(Sum {getSum = 1416},Sum {getSum = 2160},Sum {getSum = 2160},Sum {getSum = 1440},Sum {getSum = 600})` from the following program:
{-# LANGUAGE LambdaCase #-}
import Data.Monoid
oneIf x = Sum $ if x then 1 else 0
toRed = \case
1 -> 4
2 -> 4
3 -> 4
4 -> 4
5 -> 4
6 -> 9
toYellow = \case
1 -> 3
2 -> 3
3 -> 3
4 -> 3
5 -> 8
6 -> 8
toBlue = \case
1 -> 2
2 -> 2
3 -> 2
4 -> 7
5 -> 7
6 -> 7
toMagenta = \case
1 -> 1
2 -> 1
3 -> 6
4 -> 6
5 -> 6
6 -> 6
toOlive = \case
1 -> 0
2 -> 5
3 -> 5
4 -> 5
5 -> 5
6 -> 5
main = print $ mconcat $ do
red <- toRed <$> [1..6]
yellow <- toYellow <$> [1..6]
blue <- toBlue <$> [1..6]
magenta <- toMagenta <$> [1..6]
olive <- toOlive <$> [1..6]
let first = maximum [red, yellow, blue, magenta, olive]
[ ( oneIf $ first == red
, oneIf $ first == yellow
, oneIf $ first == blue
, oneIf $ first == magenta
, oneIf $ first == olive
) ]Always grateful for a YouTube channel recommendation, but it would have been awesome if it really was the Grimes I thought you were referring to.
Am I the only one who found it noticeable that they called the third die "olive"? It's definitely a color I've heard before, but in this context I would have just expected them to say "green" since that's far more common. Even if they used the same exact graphics, I find it hard to believe anyone would have objected by saying "that's olive, not green!"
That was the first thing I noticed too. Apparently, the reason for that is to have these nice properties:
> This set of five dice is similar to other sets of dice we have seen, in that we have a chain where Blue>Magenta>Olive>Red>Yellow>Blue.
> However, we also have a second chain, where Red>Blue>Olive>Yellow>Magenta>Red.
> Notice the first chain is ordered alphabetically, while the second chain is ordered by word-length.
> This set of five dice is similar to other sets of dice we have seen, in that we have a chain where Blue>Magenta>Olive>Red>Yellow>Blue.
> However, we also have a second chain, where Red>Blue>Olive>Yellow>Magenta>Red.
> Notice the first chain is ordered alphabetically, while the second chain is ordered by word-length.
Olive normally means yellow-green or yellowish green, not green.
However on that page I see indeed green, not olive.
Another poster explained the likely reason for this choice of word, but the discrepancy between the shown color and its name is slightly confusing.
However on that page I see indeed green, not olive.
Another poster explained the likely reason for this choice of word, but the discrepancy between the shown color and its name is slightly confusing.
It becomes even more confusing when you realise actual olives aren't even that colour. Olives are just _green_ green (or very dark brown to blue/black).
The plant parts which will eventually become green, when they will contain enough chlorophyll, pass through an initial stage that is either light green when they were initially white or yellow-green when they were initially yellowish (due to carotenoids).
Immature olives pass usually through a stage when they are yellow-green, but that stage may not last long.
I agree that using as a color term the name of something that may have a variety of colors, is not a good choice.
Nevertheless, the use of "olive" as yellow-green is entrenched in various domains, e.g. the yellowish green mineral that was called "topaz" by the Ancient Greeks (now topaz means something else) and which is called now "peridot" when in gem form, is also called "olivine", due to its color.
Immature olives pass usually through a stage when they are yellow-green, but that stage may not last long.
I agree that using as a color term the name of something that may have a variety of colors, is not a good choice.
Nevertheless, the use of "olive" as yellow-green is entrenched in various domains, e.g. the yellowish green mineral that was called "topaz" by the Ancient Greeks (now topaz means something else) and which is called now "peridot" when in gem form, is also called "olivine", due to its color.
This would be brilliant for a board game that implements an elemental (rock paper scissors) damage system.
Shout out for my favorite nontransitive elemental damage system, from the kingdom of loathing. Hot, Cold, Stench, Sleaze, Spooky. I haven't played in nearly a decade, but I still see those words in color.
KoL got me through some of the most interminably boring stretches of time at an early job where there was, literally, nothing to do. I was hired to help fill out a team for a project that was delayed for a year, and in the meantime fielded a rare request on the current system. And did maintenance on it-- once a week-- for half a day.
I would count down to the next KoL action point I'd get. IIRC, I was a saucerer.
I would count down to the next KoL action point I'd get. IIRC, I was a saucerer.
[tangent]
If you are interested in dice and probability I recommend this new game with a very SEO-unfriendly name: Roll https://store.steampowered.com/app/1585910/Roll/
I have no connection whatsoever, just a sad individual who sank in so many hours!
If you are interested in dice and probability I recommend this new game with a very SEO-unfriendly name: Roll https://store.steampowered.com/app/1585910/Roll/
I have no connection whatsoever, just a sad individual who sank in so many hours!
The same principle applies how voting district borders are fought over in some countries.
You try to make the borders so for your party that you will lose by a big margin in one district while winning with a small margin in many other districts.
You try to make the borders so for your party that you will lose by a big margin in one district while winning with a small margin in many other districts.
I've previously written a model that generates sets of non-transitive dice using MiniZinc. It can be seen in this Gist: https://gist.github.com/zayenz/4fa2d5ca8f96f7ce939275be25bdd...
That's a great example to teach or learn many aspects of minizinc! Great job :)
A small typo on the text next to the tree diagram: “the probability of rolling a 5 with the Red die and a 2 with the Blue die is 5/6 x 1/2 = 5/12” should be:”the probability of rolling a 3 with the Red die and a 2 with the Blue die is 5/6 x 1/2 = 5/12”.
James is great for explaining these kinds of topics, especially with his enthusiasm and way of talking to the camera, something that is true for other Numberphile hosts/presenters. Definitely go check out https://www.youtube.com/numberphile and https://www.numberphile.com/
James is great for explaining these kinds of topics, especially with his enthusiasm and way of talking to the camera, something that is true for other Numberphile hosts/presenters. Definitely go check out https://www.youtube.com/numberphile and https://www.numberphile.com/
This is so cool, now I want to make an RPG system using this :)
I felt exactly the same way the first time I read about them, but for boardgame design. But I found that in practice the probability of the effect just isn’t really enough to make it fun, at least in my attempts, and I ended up just using rock>paper>Scissor mechanics instead, because it’s both more noticeable, and has the benefit of being instantly recognized and understood intuitively by players.
I think it would be pretty intuitive if we had red, green and blue dies for attack >> feint >> parry.
And in rpg d20, d12 and d10s are pretty common, so I think it could be tweaked for larger effect.
The challenge would be to make attributes matter in a non-binary and fair way (in the obvious d6+attribute vs d6+attribute system attributes only matter when they overcome the threshold).
And in rpg d20, d12 and d10s are pretty common, so I think it could be tweaked for larger effect.
The challenge would be to make attributes matter in a non-binary and fair way (in the obvious d6+attribute vs d6+attribute system attributes only matter when they overcome the threshold).
I have used genetic algorithms to search for non-transitive die triplets among the standard set of platonic dice, and exhaustively searched d6 and d8 numberings. If one insists that the advantage of A over B be the same as the advantage of B over C and C over A, then 61% is the best (farthest from 50%) that can be achieved for d6 and d8, and the best that I have been able to achieve for any larger dice.
Some years ago I had written a quick script in python to simulate the behaviour of these dice: https://gist.github.com/spapas/9b29e1dec752b85fc1d6ad660af7f...
I'm surprised this hasn't become some sort of table game in casinos, like 4-card poker.
It kind of gets mentioned in footnote 2, but if you both pick at the same time Olive is the strongest die with 40/72 (56%) chance of winning against a different color, and Red is the weakest with 32/72 (44%).
What's interesting is that there does seem to be a "best" die if you roll all three based on a Python script I ran. Can anyone intuit which one it would be?
This is a cool idea, explained brilliantly. The rock-paper-scissors reference turned on a lightbulb for me.
Whaaaa! Cool and counterintuitive as heck
finally, Mario Party explained? I wonder whether that went into the design of the game.