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Non-Transitive Dice
- Waterluvian 5y agoThis would be brilliant for a board game that implements an elemental (rock paper scissors) damage system.
- klyrs 5y agoShout 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.
- ineedasername 5y agoKoL 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.
- thomasmg 5y agoThis seem similar to "Penney's game" and the "Humble-Nishiyama Randomness Game" (https://en.wikipedia.org/wiki/Penney%27s_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.
- deleted 5y ago[deleted]
- thehappypm 5y agoWhat'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?
- james-skemp 5y agoGrimes' 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.)
- deleted 5y ago[deleted]
- dllthomas 5y agoThere isn't actually an equal chance of going first, though, is there?
- Someone 5y agoFor 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.
- deleted 5y ago[deleted]
- dllthomas 5y agoFor 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.
- verve_rat 5y agoWhy not. If all players/dice have an equal chance of rolling highest, then...?
- wly_cdgr 5y agoWhaaaa! Cool and counterintuitive as heck
- ineedasername 5y agoI'm surprised this hasn't become some sort of table game in casinos, like 4-card poker.
- curiousgal 5y ago[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/ https://store.steampowered.com/app/1585910/Roll/ I have no connection whatsoever, just a sad individual who sank in so many hours!
- haroldship 5y agoThis is a cool idea, explained brilliantly. The rock-paper-scissors reference turned on a lightbulb for me.
- saghm 5y agoAm 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 5y agoThat 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 5y agoOlive 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.
- dafoex 5y agoIt 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).
- adrian_b 5y agoThe 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.
- kiryin 5y agoA 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 5y agoGiven 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.
- mzl 5y agoI'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/4fa2d5ca8f96f7ce939275be25bdd72b https://gist.github.com/zayenz/4fa2d5ca8f96f7ce939275be25bdd...
- linschn 5y agoThat's a great example to teach or learn many aspects of minizinc! Great job :)
- NKosmatos 5y agoA 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 https://www.youtube.com/numberphile and https://www.numberphile.com/ https://www.numberphile.com/
- OscarCunningham 5y agoIt turns out that you can make dice that beat each other according to any directed graph: https://math.stackexchange.com/questions/1624550/can-you-create-non-transitive-dice-for-any-finite-graph https://math.stackexchange.com/questions/1624550/can-you-cre...
- messe 5y agoNote that by "dice" they are referring to any probability distribution the natural numbers (including those of infinite support), not just 6 sided dice.
- OscarCunningham 5y agoThe 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/Bipyramid https://en.wikipedia.org/wiki/Trapezohedron https://en.wikipedia.org/wiki/Trapezohedron
- bigger_inside 5y agofinally, Mario Party explained? I wonder whether that went into the design of the game.
- taylorius 5y agoLooking 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 5y agoThere 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.
- ajuc 5y ago> (1,6,8),(2,4,9),(3,5,8) I don't think it works, (1,6,8) seems balanced with (3,5,8).
- leephillips 5y agoAnd 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.
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- ajuc 5y agoThis is so cool, now I want to make an RPG system using this :)
- boublepop 5y agoI 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.
- ajuc 5y agoI 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).
- jpfed 5y agoI 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.
- spapas82 5y agoSome years ago I had written a quick script in python to simulate the behaviour of these dice: https://gist.github.com/spapas/9b29e1dec752b85fc1d6ad660af7fef3 https://gist.github.com/spapas/9b29e1dec752b85fc1d6ad660af7f...
- vanderZwan 5y agoRelated: 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.
- AbrahamParangi 5y agoJust because evolution is not monotonic doesn’t mean that progressivism isn’t a good “low resolution” model.
- vanderZwan 5y agoIt's not usually taught as a "low resolution" model though, it's taught as a global model.
- AbrahamParangi 5y agoLow resolution meaning coarse, like if you had to choose to believe if evolution was progressive or completely directionless, progressive would be a better model.
- vanderZwan 5y ago> 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.
- Gravityloss 5y agoThe 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.
- aimor 5y agoIt 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%).