Non-Transitive Dice

[OscarCunningham]

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...

[messe]

Note that by "dice" they are referring to any probability distribution the natural numbers (including those of infinite support), not just 6 sided dice.

[vanderZwan]

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.

[OscarCunningham]

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#...

[freemint]

The term you are looking for is Pareto front.

[AbrahamParangi]

Just because evolution is not monotonic doesn’t mean that progressivism isn’t a good “low resolution” model.

[kiryin]

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."

[Zhyl]

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.

[taylorius]

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.

[boublepop]

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.

[thomasmg]

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.

[james-skemp]

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.)

[dllthomas]

There isn't actually an equal chance of going first, though, is there?

[danw1979]

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.

[saghm]

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!"

[Miiko]

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.

[adrian_b]

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.

[Waterluvian]

This would be brilliant for a board game that implements an elemental (rock paper scissors) damage system.

[klyrs]

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.

[curiousgal]

[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!

[Gravityloss]

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.

[mzl]

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...

[linschn]

That's a great example to teach or learn many aspects of minizinc! Great job :)

[NKosmatos]

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/

[ajuc]

This is so cool, now I want to make an RPG system using this :)

[boublepop]

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.

[spapas82]

Some years ago I had written a quick script in python to simulate the behaviour of these dice: https://gist.github.com/spapas/9b29e1dec752b85fc1d6ad660af7f...

[ineedasername]

I'm surprised this hasn't become some sort of table game in casinos, like 4-card poker.

[aimor]

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%).

[thehappypm]

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?

[haroldship]

This is a cool idea, explained brilliantly. The rock-paper-scissors reference turned on a lightbulb for me.

[wly_cdgr]

Whaaaa! Cool and counterintuitive as heck

[bigger_inside]

finally, Mario Party explained? I wonder whether that went into the design of the game.