Nontransitive dice

January 24, 2010 2:04 PM

Nontransitive dice are sets of dice (A, B, C, etc.) with counterintuitive properties: die A beats die B and die B beats die C, but die C beats die A.

   A        B        C    
  +-+      +-+      +-+   
  |1|      |3|      |2|   
+-+-+-+  +-+-+-+  +-+-+-+ 
|4|4|4|  |3|3|3|  |2|5|2| 
+-+-+-+  +-+-+-+  +-+-+-+  
  |4|      |3|      |5|   
  +-+      +-+      +-+   
  |4|      |6|      |5|   
  +-+      +-+      +-+

(For this set of dice, each die beats the next at least 7 out of every 12 games.) Sets of nontransitive dice exist for more than 3 dice, and for games with more than 2 players; you can try playing with some yourself with this (vintage-style) game.

posted by Upton O'Good (54 comments total)

Isn't that the rules to rock, paper, scissors?
posted by TheJoven at 2:16 PM on January 24, 2010

TheJoven: "Isn't that the rules to rock, paper, scissors?"

No. Nothing beats rock.
posted by Joe Beese at 2:17 PM on January 24, 2010 [3 favorites]

But with dice!
posted by TwelveTwo at 2:17 PM on January 24, 2010 [1 favorite]

[Did the best I could to unfuck the formatting on the ASCII art; seems not to want to play well, pre tags or no, possibly a result of weird widths on the graphics characters.]
posted by cortex at 2:18 PM on January 24, 2010 [1 favorite]

[And I'm probably just making it worse. Post looks three different ways in three different browsers and was probably composed in, and fine in, IE in the first place. Sorry about the mess, Upton, I'll see if there's some way to get it to behave after all.]
posted by cortex at 2:21 PM on January 24, 2010

die A beats die B and die B beats die C, but die C beats die A

That would depend on the rules of your dice game, wouldn't it?
posted by Faint of Butt at 2:21 PM on January 24, 2010

The dice would be better represented without the line-drawing characters, which seem like they mess up in fixed-width.

  +-+     +-+     +-+
  |1|     |3|     |2|
+-+-+-+ +-+-+-+ +-+-+-+
|4|4|4| |3|3|3| |2|5|2|
+-+-+-+ +-+-+-+ +-+-+-+
  |4|     |3|     |5|
  +-+     +-+     +-+
  |4|     |6|     |5|
  +-+     +-+     +-+

Non-transitive dice are amazing and wonderful, and thinking about them can be mind-expanding. Martin Gardner once had a great Mathematical Games column about them in Scientific America, which is collected in multiple books I think, including the recent collection The Colossal Book of Mathematics.
posted by JHarris at 2:25 PM on January 24, 2010

   A             B             C    
  ┌-┐           ┌-┐           ┌-┐   
  │1│           │3│           │2│   
┌─┼─┼─┐       ┌─┼─┼─┐       ┌─┼─┼─┐
│4│4│4│       │3|3│3│       │2│5│2│ 
└─┼─┼─┘       └─┼─┼─┘       └─┼─┼─┘
  │4│           │3│           │5│
  ├─┤           ├─┤           ├─┤
  │4│           │6│           │5│   
  └-┘           └-┘           └-┘

Well, the above doesn't work in preview, but I can't help giving it a try anyway....
posted by Chuckles at 2:29 PM on January 24, 2010 [3 favorites]

Ha, victory!!!
posted by Chuckles at 2:29 PM on January 24, 2010

It even works in IE, but blink doesn't :(
posted by Chuckles at 2:31 PM on January 24, 2010

Oh, and there is that accidental pipe character, much better if it was replaced with the correct graphic character...

The weirdest thing is that JHarris' version looks different in preview vs. the regular page, but mine looks the same in both...
posted by Chuckles at 2:34 PM on January 24, 2010

Interesting concept...the question is how to apply it to a game. How about three colors of dice, and the win goes to a tie such that :

Red->Yellow->Blue->Red?

Might make for some interesting changes in several multi-die games.

Too bad I'm too lazy to chart it out right now...
posted by Jimmy Havok at 2:41 PM on January 24, 2010

If anyone is computer science inclined, then it's an interesting problem to generate nontransitive dice given the number of dice and the number of sides per dice.
posted by esprit de l'escalier at 2:49 PM on January 24, 2010 [1 favorite]

One thing to keep in mind with these dice is that, although each die beats the one before it a good percentage of the time, that doesn't mean there isn't a "best" die in terms of its average score. Saying that a die beating another one doesn't take into account the margin of victory. That's at the heart of why non-transitive dice don't seem intuitive to us, I think.
posted by JHarris at 3:08 PM on January 24, 2010

Two-person game: each player has all three dice. Each chooses a die in secret, and throws it. Result is a simple win/lose matrix, with no concern for numerical scores.
posted by Jimmy Havok at 3:21 PM on January 24, 2010 [1 favorite]

JHarris, dice A B and C all have the same average score of 3.5, just like a regular die.
posted by aspo at 3:39 PM on January 24, 2010

"Isn't that the rules to rock, paper, scissors?"

No. Nothing beats rock."

Obviously, but when you choose paper, rock or scissors, you aren't actually choosing to fight with a piece of paper, a rock or a pair of scissors. You're choosing what banner your rock will fight under.
posted by scope the lobe at 4:00 PM on January 24, 2010 [6 favorites]

So how do I build me some? Any suggestions on how to home-make some decent-quality dice?
posted by Mister Moofoo at 4:02 PM on January 24, 2010

You people are a bunch of geeks.

And I love you all. Well, some of you anyways.
posted by marxchivist at 4:12 PM on January 24, 2010

Doesn't paper beat rock?
posted by Marisa Stole the Precious Thing at 4:18 PM on January 24, 2010

Doesn't paper beat rock?

How could a measly piece of paper defeat the mighty rock? That just doesn't make any physical sense.

Also, this post is great!
posted by albrecht at 4:29 PM on January 24, 2010 [1 favorite]

On a related note... you don't even have to have your dice particularly cooked up to end up with non-transitive strategies in a probabilistic game. See Knock 'Em Down (pdf).
posted by weston at 4:48 PM on January 24, 2010

Good ol' rock. Nothing beats that.
posted by autopilot at 4:49 PM on January 24, 2010 [1 favorite]

So how do I build me some?
Blank dice.
posted by MtDewd at 4:58 PM on January 24, 2010 [2 favorites]

I'm sorely tempted to buy some.
posted by Johnny Assay at 5:00 PM on January 24, 2010 [1 favorite]

But who will stand up for the untiring scissors?
posted by localhuman at 5:10 PM on January 24, 2010

The World RPS (rock paper scissors) Society (than which, surely, no higher authority exists) game rules say:
Rock wins against Scissors,
Scissors wins against Paper
Paper wins against Rock
("Basics" link in left nav)

As I recall, the rationale for the third rule is that Paper "covers Rock".
posted by beagle at 5:21 PM on January 24, 2010

Let us not forget RPS-101
posted by milnak at 5:45 PM on January 24, 2010 [3 favorites]

I bought some blank dice and used those cheapy colored sticker dots to set up for more intuitive dice for a couple of games (Mechaton, Burning Wheel), and I could see something like that working here as well.
posted by yeloson at 5:53 PM on January 24, 2010

Rock even beats Spock.
For those of you not in on the "good ol' rock joke, please see this FAQ.
posted by autopilot at 5:54 PM on January 24, 2010

JHarris, dice A B and C all have the same average score of 3.5, just like a regular die.

Er, that is true. I was thinking about the example Gardner presented in his column. Hm.
posted by JHarris at 5:55 PM on January 24, 2010

I was just reading in a Warren Buffet biography that he kept some of these on his desk as an ice breaker, so whoever rolled first would lose.
Apparently Bill Gates and Saul Kripke worked it out, but nobody else.
posted by bystander at 6:00 PM on January 24, 2010

These work because when your die wins the margin doesn't matter. If the margin mattered (i.e., you got [your value] - [their value] points) then the best die would just be the die with the highest expected value, and since expected values are transitive real numbers, the dice would have to be transitive.

Consider this case (three sided dice):
Die 1: {-1, -1, 3*10^6}, E ~= 10^6
Die 2: {0,0,0}, E = 0

2/3rds of the time : -1 vs 0, and die 2 wins with a margin of 1
1/3rd of the time : 3*10^6 vs 0, and die 1 wins with a margin of 3*10^6
posted by Pyry at 6:13 PM on January 24, 2010

They all have the same average score; the idea is that the die that's favored in any pair usually wins by a little when it wins and loses by a lot when it loses. For example, with A versus B: 25 times out of 36 A wins by 1, 10 times out of 36 A loses by 2, and 1 time out of 36 A loses by 5.
posted by madcaptenor at 6:15 PM on January 24, 2010

Good ol' rock. Nothing beats that.

Not even tigers.
posted by weston at 6:16 PM on January 24, 2010

Tiger eats rock. But then rock makes tiger sick, tiger throws rock up.

I'm not sure who wins here.
posted by madcaptenor at 6:26 PM on January 24, 2010

The obvious reference:

KRAMER & MICKEY: Rock, paper, scissors match.

MICKEY: all right, rock beats paper.

(Mickey smacks Kramer on the hand for losing)

KRAMER: I thought paper covered rock?

MICKEY: Nah, rock flies right through paper.

KRAMER: What beats rock?

MICKEY: (looks at his hand) Nothing beats rock.

KRAMER: all right come on.

KRAMER & MICKEY: Rock, paper, scissors match.

KRAMER: Rock.

MICKEY: Rock

KRAMER & MICKEY: Rock, paper, scissors match.

KRAMER: Rock.

MICKEY: Rock.

[YouTube]
posted by A-Train at 6:44 PM on January 24, 2010

This is why paper wins against rock.
posted by Night_owl at 9:33 PM on January 24, 2010

This messes with my hippocampus.
posted by twoleftfeet at 10:16 PM on January 24, 2010

These dice are blowing peoples' minds? This is pretty much the model for all games.

Spearmen > Cavalry > Swordsmen > Spearmen

Bomber > Infantry > Flak gun > Bomber

etc. etc.
posted by Meatbomb at 11:39 PM on January 24, 2010

These dice are blowing peoples' minds? This is pretty much the model for all games.

It (R-P-S) is a strong design element, yes, but not much of a game in itself. Since all of the possible throws are isomorphic with each other there is no in-game strategy for picking one over another. This still works for RPS because it becomes a front for the psychology of the opponents, just like Poker is most interesting when it isn't about the cards themselves. As in Lisa and Bart's game, Bart thinks Rock always wins, so Lisa, knowing Bart, can pick Paper.

I read somewhere a long time ago that human beings have a hard time doing something arbitrarily. If you sat down with a sheet of paper and a pencil and wrote a list of random numbers from 1 to 10 as fas as you could, you would find that you settled into a pattern along the way. If you recognized this and changed the pattern, you would settle into another pattern.

In RPS against a completely random opponent, the best you could hope for over the long run is a one-third win rate. But because humans have difficulty simulating randomness, you could potentially do better against a person. Of course, being a person yourself, you would have your own patterns, and in fact trying to take advantage of an opponent's pattern is itself a pattern, one that could be taken advantage of by that opponent.

RPS gets a little more interesting when you expand it with throws that are not perfectly symmetrical with the others. If you had a RPS-style game in which one throw is slightly better than the others, then the best straight strategy would be to always pick it. But other players, knowing this, could pick the throw that beats the "better" strategy. And that tactic too could be recognized and taken advantage of.
posted by JHarris at 2:05 AM on January 25, 2010 [1 favorite]

Wow another post where not replying is the only winning move. Oh dammit!
posted by vicx at 2:31 AM on January 25, 2010

In RPS against a completely random opponent, the best you could hope for over the long run is a one-third win rate.

Because of the house odds?
posted by Obscure Reference at 4:18 AM on January 25, 2010

For some reason, these dice remind me of a game I read about many years ago called "Lucky Louie". I think it was in a magazine article about something to do with game theory, and it was given as an example of a game where there could be no strictly logical way to play it.

The game is very simple. It requires three players. Each player picks an arbitrary positive number and writes it down. Then the players reveal their number choices, and whichever player's number is between the other two wins that round. If two people pick the same number, the player whose number differs from the other two wins. If all three pick the same number, that round is a tie.

The weird part about it is there's no upper limit to the range of numbers that you can pick from. Any positive real number will do. It quickly turns into a game of guessing the order of magnitude your opponents will be thinking about in the next round, and there don't seem to be any game-theoretical ways to come up with a winning strategy.
posted by FishBike at 5:42 AM on January 25, 2010

Please, please. We all know there's only one way to settle this.
posted by griphus at 6:21 AM on January 25, 2010 [1 favorite]

Because of the house odds?

Because the other two thirds of the games would be split between losses and ties.

If you had a RPS-style game in which one throw is slightly better than the others, then the best straight strategy would be to always pick it.

Well, the best naive strategy would be to do an appropriately weighed random throw, really. Chuck out Rock+ a bit more often in proportion with its relatively higher expected payout. If your opponents come to the same conclusion, you get back to parity; if they try either of a the unmodified even-odds approach or a Rock+ Only strategy, you get a leg up.
posted by cortex at 7:15 AM on January 25, 2010 [1 favorite]

These dice are blowing peoples' minds? This is pretty much the model for all games.

I think what's mind-blowing about this particular setup (at least what blows my mind) is not so much the RPS-style nontransitivity but rather the fact that you can have that nontransitivity using dice (which are basically just a convenient representation of a discrete probability distribution). Before I saw these dice, if you had asked me to guess whether the property "wins more than half of the time" was transitive in the space of probability distributions, I would have guessed that it probably was, because my intuition would have been derived from identifying "winning" with "having a higher expected value." This is a great tangible example of the difference between those concepts, and it's part of what makes probability so damn counterintuitive and fascinating. (See also: Simpson's Paradox, Benford's Law, the Base Rate Fallacy, or the Monty Hall Problem.) I've been studying this stuff professionally for most of a decade and I still get tripped up by things like that.

It's like those optical illusions with two lines that look like they're different lengths but actually aren't. You could say, "What's the big deal? You can just measure them!" but that's kind of missing the whole point.
posted by albrecht at 7:36 AM on January 25, 2010 [6 favorites]

and it's part of what makes probability so damn counterintuitive and fascinating.

Word.

(See also: Simpson's Paradox, Benford's Law, the Base Rate Fallacy, or the Monty Hall Problem.)

And, my favorite, regression to the mean.
posted by Mental Wimp at 10:30 AM on January 25, 2010

cortex:

Well, using a weighted random choice goes back to picking randomly in RPS. In the long run, you won't lose any more often than usual, but neither will you win. The idea is to anticipate what opponents who don't play randomly will do, since there is nothing you can do, for or against, to counter random players.

Eh, but it is a complicated thing, and I could be missing something. I've thought a lot about this in conjunction with a game idea (actually, multiple game ideas) and I still don't have a good handle on all the implications for a well-designed, unbalanced RPS game.
posted by JHarris at 5:57 PM on January 25, 2010

Well, using a weighted random choice goes back to picking randomly in RPS. In the long run, you won't lose any more often than usual, but neither will you win. The idea is to anticipate what opponents who don't play randomly will do, since there is nothing you can do, for or against, to counter random players.

Granted, yeah; my argument towards a weighted mixed-strategy isn't applicable to all situations, it's more about establishing a safe baseline. Whether it'd be an effective strategy for carving out winnings against your opponent is an open question that has mostly to do with that opponent.

Against a random-play opponent who doesn't recognize the uneven weighting, it'd be effective, but that's about as far as it goes. It certainly wouldn't be as effective as exploiting a significant weakness in an opponent's non-random play.

Always Play Rock+ would be a lousy strategy, however, because even a lousy opponent would pick up on it pretty quickly, and adopt an Always Play Paper strategy even if Rock+ beats paper one times out of ten or whatever—the obviousness and viability of that strategy is just too accessible to be missed by any thinking player.

And so Always Play Rock+ has to evolve into something like Lead With Rock+ And Watch For Reactions, and at that point you're on your way to a mixed strategy after all.

I remember reading, years ago, an interesting writeup of a Rock, Paper, Scissors bot tourney—dozens or maybe hundreds of entrants put relatively lightweight RPS programs into a round-robin competition to see what came out on top. As you'd expect, there were some straight-random bots that came out in the middle, but there were also a variety of non-random approaches in evidence, some of which did worse and others of which did better than straight odds.

I haven't been able to track it down since, maybe just bad googling on my part, but it was a nice bit of applied game theory and made fore some good examples of how zero-sum games with stable strategies can be played aggressively when opponents aren't random.
posted by cortex at 6:14 PM on January 25, 2010 [1 favorite]

No better way to find something I've been missing than by embarrassing myself by talking about not being able to find it. Here we go:

International RoShamBo Programming Competition. I probably noticed it via Slashdot in 2000.
posted by cortex at 6:21 PM on January 25, 2010

Blank dice.
Okay, but how do I make those?
posted by Mister Moofoo at 8:00 PM on January 25, 2010 [1 favorite]

I kid. But I'm trying to avoid buying dice at all, if I can just make some at home, that aren't little paper boxes, or poorly-balanced wood, or something.
posted by Mister Moofoo at 8:02 PM on January 25, 2010

AH, I may have seen that myself back in the day, I read a lot of Slashdot back then. I remember a lot of what I've read, but not all of it.
Always Play Rock+ would be a lousy strategy, however, because even a lousy opponent would pick up on it pretty quickly

Always Play Rock+ would be a lousy strategy, however, because even a lousy opponent would pick up on it pretty quickly

Well, it wasn't intended to be a viable thing, just an example of how one could not treat the game as basic Rock-Paper-Scissors. That losing the symmetrical relationship between the throws need not make the game into something hugely different.
posted by JHarris at 9:11 PM on January 25, 2010