# Weird probabilities of non-transitive "Grime Dice"

Michael de Podesta has been doing the math on "Grime Dice" -- six sided cubes whose sides average out to 3.5, but whose face values are all radically different:

The interesting thing about these is that the odds of one die beating another are simple to calculate, but shift radically once you start rolling dice in pairs. It's a beautiful piece of counterintuitive probability math:

The amazing property of these dice is discernible when you use them competitively – i.e. you roll one dice against another. If you roll each of them against a normal dice then as you might expect, each dice will win as often as it will lose. But if you roll them against each other something amazing happens.

• Dice A will systematically beat Dice B
• Dice B will systematically beat Dice C

and amazingly

• Dice C will systematically beat Dice A

So the fact that Dice A beats Dice B, and Dice B beats Dice C does not ensure that Dice A will beat Dice C. Wow!

And how about this: If you ‘double up’ and roll 2 Dice  A‘s against 2 Dice B‘s – the odds change around and now the B‘s will beat the A‘s ! Is that really possible? Well yes, and just to convince myself I wrote a Spreadsheet (.xlsx file) and generated the tables at the bottom of the article. If you download it you can change the numbers to try out other combinations.

### 25

1. This stirs up an old wargaming memory . . . for some reason, old-school miniatures wargamers (Ancients, Napoleonics, etc.)  would occasionally use “averaging dice” with faces marked 2-3-3-4-4-5.

2. EvilSpirit says:

I predict that “dice” will be the proper singular for “die” within my lifetime.

But I don’t concede the point yet.

1. It may become widespread, but it will always be wrong.

I’m going to file singular “one dice” with “an historical,” “but you can’t use less with count nouns!,” “but you can’t split the infinitive!,” “but you can’t put a preposition there!,” and “[verb which takes the accusative] you and I,” as hypercorrections which shall not pass, and where I am willing to use prescriptivism to stop the prescriptivists.

1. Woody Mann-Caruso says:

Look, I know you think you’re terribly clever, charging into a math thread to complain about plurals.  Unfortunately for you, you’re also completely wrong.

English changes.  It always has.  It always will.  Before you start whining that a ‘change’ to ‘dice’ being singular would ‘alwasy be be wrong’, consider that _this use was the fucking norm_ from the 14th to 17th centuries.  (Go on, log into the OED and see for yourself.)  _You’re_ the ones using the evolved English.

Christ, I can imagine you bores sitting around in the Middle English period complaining that, in your day, “somme pleide wyþ des – what’s this ‘played with dice’ rubbish?”

1. morgaen says:

“English changes”
I could of cared less.

1. Antinous / Moderator says:

Hey, I care alot!

2. Andy Simmons says:

Don’t stress out about it.  You could have a heart attack and dice.

3. Even though the average of both dice is 3.5, the “curves” are skewed.
21 out of 36 times (58%) A will beat B
15 out of 36 times (42%) B will beat A

4. SamSam says:

Grime’s site is down, but this does look very clever.

The three dice above show the simple property quite clearly: If I’ve done the math right, Dice A will beat dice B just over half the time of the time (half the time dice B rolls a 2, and 1/6th of the remaining times dice A rolls a 6, so Dice A wins 7/12ths of the time), Dice B will beat Dice C just over half the time (Dice B rolls a 5 half the time, and 1/6th of the remaining time Dice C rolls a 1, so 7/12ths), and Dice C beats Dice A just over two thirds of the time (5/6ths of the time Dice C rolls a 4, but 1/6th of those Dice A wins).

I haven’t tried to work out how the chain goes backwards when the dice are doubled, and how there’s a pentagon in the same direction in both cases. That sounds crazy.

1. Jellodyne says:

It goes backwards if the losing die loses by less and, when it wins, wins by more, then even if it loses more often, when rolled in pairs it still loses on lose+lose and wins on win+win, but it might also win win+lose and lose+win.

1. SamSam says:

I see — that makes sense.

2. Petzl says:

Just because Michael de Podesta is illiterate doesn’t mean we have to follow his lead.
Dice is plural.  Die is singular.  No exceptions. No excuses.

5. “one dice”

Is the singular/plural distinction so difficult here?

6. theophrastvs says:

no where does the frequentist model fit more uncomfortably than where the median bids farewell to the mean – so sayeth the mode

7. SomeDude says:

Reminds me of this interesting post on FutilityCloset:

Dick and Jane are playing a game. Each holds up one or two fingers. If the total number of fingers is odd, then Dick pays Jane that number of dollars. If it’s even, then Jane pays Dick.At first blush this looks fair, but in fact it’s distinctly favorable for Jane. Let p be the proportion of times that Jane holds up one finger. Her average winnings when Dick holds up one finger are -2p + 3(1 – p), and her average winnings when he holds up two fingers are 3p – 4(1 – p). If she sets those equal to one another she gets p = 7/12. This means that if she raises one finger with probability 7/12, then on average she’ll win -2(7/12) + 3(5/12) = 1/12 dollar every round, no matter what Dick does. Dick’s best strategy is also to raise one finger 7/12 of the time, but the best this can do is to restrict his loss to 1/12 dollar on average. It’s not a fair game.

8. GawainLavers says:

Someone needs to integrate these into D&D.

1.  Or Warhammer.

9. dragonfrog says:

Probabilistic rock-paper-scissors.  Nice.

10. Disappointing that they didn’t actually show the bloody things in action. That might’ve made it a little more concrete for us non-maths-heads to understand.

1. SamSam says:

Unfortunately, unless my math is all wrong, most of the time one die is only expected to beat the other 7/12ths of the time. Since this is actually quite close to 50%, they might need to roll ten or twenty times to show that it is statistically more likely to beat the other, and even then it wouldn’t look all that different from random chance to the average person. I don’t think it would have been that amazing to watch.

11. bobcorrigan says:

And now, I shall always make my saving throw, and my fireballs will roast the goblins more reliably.

12. Jordie says:

Is there a use for these besides tricking your friends into buying you beer? (or giving you money or whatever) I mean that’s all well and good for an amusing afternoon, but this is just a rigged game that seems fair at first glance, which is essentially only useful to con artists.

1. I did something similar, testing combinations of all five dice:

http://pastebin.com/VZLexDjc

I’m a bit of a Python newb, but I thought I’d give this a try out of curiosity (not realizing someone else was already on the case).

13. Martin Gardner covered “Circularly transitive”  in 1970. Google “Efron’s Dice” or

Gardner, M. “Mathematical Games: The Paradox of the Nontransitive Dice and the Elusive Principle of Indifference.” Sci. Amer. 223, 110-114, Dec.
1970.

Or Ivars Peterson, “Tricky Dice Revisited,” April 13, 2002. http://se02.xif.com/articles/20020413/mathtrek.asp.