# Non-transitive dice and "rock-paper-scissors"

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Blake
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Joined: 01/05/2009

Basically, non-transitive dice allow you to model a rock-paper-scissors like asymmetry where one die can be better than a second (statistically), the second can be better than the third, but the third can nonetheless be better than the first! Here's a concrete example of a set of three six sided non-transitive dice:

Die A: 1,4,4,4,4,4
Die B: 3,3,3,3,3,6
Die C: 2,2,2,5,5,5
(A beats B 25 out of 36 times, B beats C 21 out of 36 times, and C beats A 21 out of 36 times)

The most obvious application that comes to my mind is in modeling unit asymmetries in wargames, but my guess is that's just the beginning. Any thoughts on other applications? Also, do people think that incorporating modifiers into non-transitive dice (for example adding or subtracting 1 from a roll) would diminish their elegance?

Here's a link to an article on wikipedia about non-transitive dice (with in depth examples!), as well as a link to a store that sells two different sets of them:

http://en.wikipedia.org/wiki/Nontransitive_dice

http://www.grand-illusions.com/acatalog/Non_Transitive_Dice_-_Set_1.html

hoywolf
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Joined: 01/27/2009
Impressed

Wow, I'm impressed I personally dont like dice game as it is too random, but this idea seems to work out, i guess in wargame type boardgame you can assign each dice type of damage, and if your anti-tank infantry attacks a vehicle you get to roll A and the defense will roll B, if your anti-tank attacks other infantry then you roll A and the defense will roll C. Something to this sense is what your trying to get at right? I would think this is an interesting way to deal with combat, that does not involve advantage but rolling more dice.

I very interesting idea. I have yet to see this in a dice board game.

kodarr
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Joined: 08/04/2008
I second that this is a very

I second that this is a very impressive concept. I really like the 1 out of 6 shot at beating the harder combatant. And one die vs one die would definately speed up combat.

brisingre
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Joined: 01/21/2009
Interesting

I'd all this to my mental list of dice mechanics. It's a neat one.

fecundity
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Joined: 07/28/2008
This is a neat trick.

This is a neat trick. However, the probability of the advantaged side winning is close to 2/3. You could get similar odds by just rolling one ordinary six-sider: Let the advantaged side win on 1-4, and let the the disadvantaged side win on 5-6.

On a somewhat related note. I am wondering now whether there is some gaming use for Sicherman Dice.

brisingre
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Joined: 01/21/2009
Class

Frankly, it's about class. Think about the dice in War of the Ring. Those are perfectly normal six-siders. There is no reason that those symbols couldn't be numbers from 1 to 6. You have to look them up on a table until you memorize what they are. The symbols serve no purpose, really. Except that they make War of the Ring what it is. Unique components are memorable. Peculiar dice make things cool. (I guess I'm just a fan...)

Blake
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Joined: 01/05/2009
More than two dice?

fecundity wrote:
This is a neat trick. However, the probability of the advantaged side winning is close to 2/3. You could get similar odds by just rolling one ordinary six-sider: Let the advantaged side win on 1-4, and let the the disadvantaged side win on 5-6.

What you're saying makes a lot of sense, but I believe the real complexities of these non-transitive dice become more apparent when you look at situations that are not one-on-one. For example, given the same set of dice referenced above:

Die A: 1,4,4,4,4,4
Die B: 3,3,3,3,3,6
Die C: 2,2,2,5,5,5

Imagine 1,000 As vs. 1 B. There would be at least a 1/6 chance of the B die surviving as all it has to do is roll a 6 and it doesn't matter what those 1,000 As roll! (Note: this would not be the case with 1,000 Cs vs. 1 A.) Conversely, if you have 1,000 Bs vs. 1 A it would be statistically amazing if those Bs didn't destroy that A as all they would need is to roll a single 6! In a strange way it could be argued that the B dice become more powerful than the A dice in this situation in contrast to the one-on-one situation where the A dice are clearly superior.

My statistical and computational abilities are limited, but it seems like there is a lot of room for the unexpected to evolve out of these dice. Of course if we're going to talk about situations that are not one-on-one, we'll need to get clear just how the dice interact. For example, if two players each roll ten dice, how do we figure out how the dice are assigned to one another? For example, does the defender (if it's a wargame) simply assign the dice, or do you match highest number to highest number and then work your way down (kind of like Risk), or should some other totally different method be used?

Taavet
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Joined: 08/15/2008
Nice concept!

I really like this concept as well. It opens the door to new designs and uses for dice.

Personally I like assemmetry in games which maintain 'balance'. Haven't played StarCraft the boardgame but the video game definately carries that assemmetry.

Thanks, for sharing.

Blake
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Joined: 01/05/2009
Update!

Wow, just found this really cool website which not only includes a great discussion of non-transitive dice, but even includes a video.

http://singingbanana.com/dice/article.htm

Apparently (among many things pointed out on this webpage) if you roll two of each of the dice mentioned above, the chain of which die beats which is actually reversed if you add the sum of the dice! I'm impressed!

Sorry for dredging this up after so long, I just thought it was quite cool.

Best,
Blake

drktron
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Joined: 07/18/2010
non-transitive dice

This is a very cool idea with a lot of potential applications. Thanks for sharing.

Blake
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Joined: 01/05/2009
My own design?

Bored last night, I decided to try and design my own set of non-transitive dice. Here's what I came up with:

A1: {1, 1, 18, 18, 23, 23}
A2: {2, 2, 16, 16, 24, 24}
A3: {3, 3, 17, 17, 22, 22}

B1: {4, 4, 11, 11, 27, 27}
B2: {5, 5, 12, 12, 25, 25}
B3: {6, 6, 10, 10, 26, 26}

C1: {7, 7, 14, 14, 21, 21}
C2: {8, 8, 15, 15, 19, 19}
C3: {9, 9, 13, 13, 20, 20}

Given any two (different) dice from this list, one will always have a 5/9 probability of rolling a higher number than the other one. Additionally, any C die will (on average) beat any B die, any B die will beat any A die, and any A die will beat any C die. Within any of these three sets, the 3 die will (on average beat the 2 die, the 2 die beat the 1 die, and the 1 die beat the 3 die.

I have no idea what this particular set of nine dice would be useful for, but it feels somewhat good to know that I could taylor a set to my own design with only a fairly minimal amount of work (one to two hours). Now I just have to design a dice game!