I thought this was kind of interesting:

# non-transitive dice

Someone should make a game with a mechanic that includes Non-Transitive dice.

But so that it's not a 'solvable' situation, it would be cool to have to earn access to the dice seperately, so when you challenge me, you choose a die which you have access to and I choose a die that I have access to and we roll off. If I don't have access to the die that beats you 2/3 of the time, then tough for me.

In this case though, the 662222 die is a little better than the rest because it beats the others a little more often while the 444400 die is a little worse. the 555111 and 333333 dice are the same.

Note that having the 'better die' does not guarantee victory by any means. Your chances are either 50-50, 1/3, 2/3, or 55-45. I think that could make for interesting decisions to make based on who has what dice available. The 'better' die could be somehow more expensive to gain access to, and the 'worst' die could be cheaper.

What do you think?

I've actually made 2 games that use non-transitive dice.

One of them, Zoopermarket, is a very simple children's game.

The other one is called Medieval Medicine, that has a great theme IMO. I need to get the mechanics up to speed with the theme, but I've had it on the shelf for a while while working on another project.

They've both hit the table at different Albany Playtests, so a few people here have gotten to try them out.

**fanaka66 wrote:**

I've actually made 2 games that use non-transitive dice.

Care to share??

- Seth

In Zoopermarket, there is a deck of cards made up of animals. There is a mouse, a cat, a dog, and an elephant. They are each represented by a non-transitive die. There is also a monkey that acts as a wild card. Each hand a food item is turned up and the players take turns 'scaring' each other away. If I play a mouse, you would want to play a cat, because you would scare me away 2/3 of the time. A dog usually scares a cat, an elephant usually scares a dog and a mouse usually scares an elephant.

It's OK, but it's tough when my 4-year old nephew plays the right card, but then the cat die doesn't beat the mouse die. He knows he played it right, but doesn't understand why he didn't win. I think it works for a very narrow age range, where they are old enough to understand, but not too old to be bored quickly.

Medieval Medicine is set-up totally different, but I need to work on it a bit more before I can see if it will be worth continuing with.

**sedjtroll wrote:**

Someone should make a game with a mechanic that includes Non-Transitive dice.

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Note that having the 'better die' does not guarantee victory by any means. Your chances are either 50-50, 1/3, 2/3, or 55-45. I think that could make for interesting decisions to make based on who has what dice available. The 'better' die could be somehow more expensive to gain access to, and the 'worst' die could be cheaper.

What do you think?

Well thinking as I type,

Just from your statement, I could see a good fit between a war game, that uses resources and Non-Transitive dice. You would uses your resources to purchase a "power up" for you various units/armies. The "power ups" would come in the form of assigning the purchased, Non-Transitive die to your units.

I think it would add a twist to a combat system, since it would allow the various unit types, equal chance at victory, but based on the available resources to purchase offense/defense.

Thus a game like this would allow one to simulate a grunt farmer unit a chance at overpowering a noble unit. Maybe at the time of combat, players could use the resources to purchase battlefield "power ups". And if your opponent does not have direct knowledge of your available resources, they have to be more careful when considering combat.

Anyways, just a thought from right field!

Yeah Zzzz, that's kinda what I meant. Depending on the game it could be interesting to either know or not know which dice your opponent has access to. It might also be nice to make it so the chances of success are more dispatate than 50-50 vs 67-33, which I'm sure can be done... would that involve dice with more sides?

The basic idea here is putting resources into improving your chances of success. A game that I thought did an admirable job of that is one from these boards, though I don't think you'll find it here- I actually played the game at Hexacon and thought it was right on track with this mechanic. the idea was that there was a target number which must be rolled on 2d6, and you can spend money to add 1 to your roll. Dues to the nature of the 2d6 bell curve, that meant depending on the number there was a different relative value to adding 1 to the die roll. Donovan (the designer) also had it right when successive purchases of +1's cost progressively more each time, so for great costs you could increase your chance a lot. But in the end you could still roll a 2 and not succeed.

In the case of non-transative dice, another way to do it is to assign a particular die to a task, and when you try and accomplish that task you roll a die based on what resources you spend. So if you sepnd the right resources you can maximize your chances, but it's still only a 2/3 chance of success. maybe if you combine that with some way to increase your chances further then it would be really interesting.

P.S. What does adding +1 to a die roll do to non-transitive dice?

- Seth

**sedjtroll wrote:**

P.S. What does adding +1 to a die roll do to non-transitive dice?

The problem is that the dice are nicely designed to not have ties. When you start adding in modifiers, it takes away that nice part. I tried to come up with my own set of non-transitive dice, but couldn't make any major changes to keep the percentages correct.

**fanaka66 wrote:**

The problem is that the dice are nicely designed to not have ties. When you start adding in modifiers, it takes away that nice part.

Couldn't the dice be easily modified to not have ties, even with modifiers? Of course, that defeats the purpose of the modifiers.

I notice that the die with 4 4's and the die with 3 5's actually get better vs the die with 3 5's and the die with 2 6's (respectively) with a +1 modifier. Also, any die against itself gets better with a +1 modifier (78% win 22% lose vs 56% tie, 22% win, 22% lose)

Hey Scurra- I just thought of something. Remember the old RPS combat from All For One? What if the move you played indicated which die to roll... then you don't play cards simultaneously of course. So I play a Thrust and you play a Parry, you have a 2/3 chance of beating me. But if I play a Swing and you play a Dodge than I have a 2/3 chance of beating you.

What do you think? It'd still be nice to be able to spend some resourse to modify that.

- Seth

**SiskNY wrote:**

sedjtroll wrote:Remember the old RPS combat from All For One? What if the move you played indicated which die to roll... then you don't play cards simultaneously of course. So I play a Thrust and you play a Parry, you have a 2/3 chance of beating me. But if I play a Swing and you play a Dodge than I have a 2/3 chance of beating you.

I like the idea of tying action to different non-trnasitive dice! "Should I do this and have a good chance of succeeding against that, what if they do the other and I'm doing this..." Lots of good potential for decision making!

Something like that would even work for Darke's Snow Day...

That it could!

So how then should the mechanic go? Suppose for a moment that it's a swordfighting mechanic, and the moves are Thrust, Parry, Swing, and Block.

Thrust loses to Parry, which loses to Swing, which loses to Block, which in turn loses to Thrust. When I say 'loses to' I mean is at a disadvantage. Each move is associated with 1 of the non-transative dice.

So should it be:

(a) We have cards in hand with moves on them, we each choose one simultaneously and reveal them, then roll the appropriate die- high roll wins the point? Is this as random as Rock Paper Scissors?

(b) One player plays a card (taking the appropriate die), then the other player plays a card and they roll off? This gives the second player an advantage because they know which card to use to get the advantage- if they happen to have that card.

I like (b) better, but there would need to be some way to hedge your bets or guess/keep track of what cards you opponent has. Maybe if you spend resources to get the cards, which are then kept face down so if you notice what cards your opponent has been colledting then you have an idea of what his options are.

The other thing is with the defender having the advantage, there's have to be some reason to want to attack, and it would have to be a good one. but it makes more sense for the attacker to play a card first and the defender to respond. If we assume we're in the middle of a swordfight and that both players are attacking back and forth we could say that the Active player plays his card second, therefore giving him the advantage- if it's a matter of choosing to fight on your turn for some effect...

On the other hand, depending on the game maybe a player should HAVE to be fighting, and at a disadvantage, thereby having to play a card first on their turn.

Thoughts?

- Seth

I like the idea of choosing your actions. I also like option b. I was thinking the same thing about Rock paper, scissors.

I think you could give players fatigue points. Each move has a different amount of fatique that it causes the player to lose. If they ever are down to 0 fatigue they can do nothing. There would have to be some way for fatigue points to increase. Perhaps on your turn, instead of attacking you could take time to renew your fatigue points (see below about attacking)

You should make it like a fencing duel where the attacker gets points for striking his opponent. That way the only way you can earn points is to be on the attack. This should be enough incentive to overcome the disadvantage of havng to play a card first. It would also create some interesting strategies with the above system if you knew your opponent was getting fatigued, attacking them enough to cause them to have to take a fatigue recharge turn, or something. Just thinking with my fingers....

**Trickydicky wrote:**

You should make it like a fencing duel where the attacker gets points for striking his opponent. That way the only way you can earn points is to be on the attack. This should be enough incentive to overcome the disadvantage of havng to play a card first. It would also create some interesting strategies with the above system if you knew your opponent was getting fatigued, attacking them enough to cause them to have to take a fatigue recharge turn, or something.

I like this a lot. But there needs to be more to it... it seems dry all by itself.

So we're fencing. Are we taking turns? I suppose so. Is there a 'pre-duel' phase where we get (draft, buy, choose) our maneuver cards? Is there any way to at least guess what the opponent might be capable of?

Should the Maneuvers have a Fatigue cost, so the better ones could cost more fatigue... only the point is that they're not suppoesed to be 'better' than each other except situationally.

On your turn you could play a card, to which your opponent would have to respond, then you'd roll and see if you score. Only scoring on your own turn should be enough incentive... assuming there's some reason to win the duel (if that's the whole game I think it would be boring).

Or on your turn you could Pass and remove a fatigue counter. I like the idea of the fatigue counter, where having one or two isn't bad but having a certain number of them means you MUST pass, or maybe that you cannot play a card (therefore if your oppoennt can then he automatically scores), or soemthing like that.

It still seems to randm to be the main focus of a game thoughl. What kind of game would have this kind of fencing as a sort of side issue? Something about Zorro? Robin Hood (like Defender of the Crown when you go on a raid and have to swordfight)? Three Musketeers?

Hmm... the cards could be divied up into 2 piles, Offensive cards (Thrust, Swing) and Defensive cards (Dodge, Parry). therefore you can tell by looking SOMETHING about your oppponent's hand- if all the cards are defensive then you know they don't have any Thrusts or Swings... but if you throw a Thrust or a Swing then it's still about 50-50 that they have the right countermove.

Hmm again... maybe the probability thing could be worked in here, where you have 3 cards to choose from to counter their move, and after you choose one another is removed and you get the opportunity to switch...

Crazy probability on top of non-transative dice just to get a better shot at success... is that "too much work?"

- Seth

Cool article!!

**fanaka66 wrote:**

One of them, Zoopermarket, is a very simple children's game.

I remember in this game that the die with all threes represented a relatively bland choice, in that there was no point to rolling (something mentioned in the article linked by doho123). Have you considered using one of the alternate sets of dice to eliminate the one uniform die? Here's the clip from the article:

SET 2: 2, 3, 3, 9, 10, 11; 0, 1, 7, 8, 8, 8; 5, 5, 6, 6, 6, 6; and 4, 4, 4, 4, 12, 12.

SET 3: 1, 2, 3, 9, 10, 11; 0, 1, 7, 8, 8, 9; 5, 5, 6, 6, 7, 7; and 3, 4, 4, 5, 11, 12.

Wow! I wish I knew about non-transitive dice a bit sooner. I think I have a game that this would work great for.

Next week I have the GDW floor, and will be posting a gladiator game. After reading about non-transitive dice, they might be just what I was looking for.

As part of the game, each player has 3 gladiators. These gladiators can be a mixture of different weight classes: light, medium or heavy. I wanted to have the medium gladiators have an advantage over light, heavy over the medium, and the light over heavy. When recruiting gladiators you have to decide what the make up of your line (the 3 gladiators) will be. If you specialize (all the same weight class) you gain more VPs at the end of the round scoring, but the other players will know which gladiator to use against you in the arena (VPs are also gained by the victor in the arena).

Currently, I have a combat system that is fun (IMHO), but a tad long. With non-transitive dice I think I could simplify the combat, and speed up the game.

I'll see if I can convert the combat system to use non-transitive dice and post two versions of combat next week.

**emxibus wrote:**

Wow! I wish I knew about non-transitive dice a bit sooner. I think I have a game that this would work great for.

I kinda thought this would be a good fit for Ludos :)

I would recommend keeping the 4 different possibilities, rather than 3 though. Although I would also recommend keeping 3 gladiators on your 'team'. That screws a little with your bonus for varying your team. Maybe the bonus for a varied team is advantage in combat, and the only scoring bonus should be for having a homogenous team.

- Seth

I noticed (as has been noted here) that the relationships between the dice

A: 0,0,4,4,4,4

B: 3,3,3,3,3,3

C: 2,2,2,2,6,6

were not consistent. I fired up my trusty Java IDE and programmed up a genetic algorithm to attempt to find a set of nontransitive dice that satisfied two requirements-

1. the probability of a win for one die against what it's supposed to beat should be far from 0.5

2. all such probabilities should be as close to each other as possible.

That second criterion was weighted 2.5 times as much (in an earlier simulation I found a novel set of dice that displayed a perfectly consistent probability to win of 0.53- so criterion #2, while very important to me, shouldn't be the whole picture, or weighted so much that it overdominates.)

Because I didn't want the computer to bother with the ambiguity of ties (half a victory?) I modified the first set into something equivalent-

A: 1,2,13,14,15,16

B: 7,8,9,10,11,12

C: 3,4,5,6,17,18

used the modified set as the "seed" of the simulation to get it going, and two hundred thousand generations of potential sets of nontransitive dice later, the computer reported back that the set that I originally gave it was the best out of the possibilities it had checked.

Not ready to accept this, I fed it a different seed- a dummy set of dice:

A: 1,2,3,4,5,6

B: 7,8,9,10,11,12

C: 13,14,15,16,17,18

but it still found the same set that it liked before.

I banged my head against the keyboard like that piano-playing muppet from Sesame Street.

And then, sleep-deprived, unable to give up, I told my computer to look among the possible sets of 12-sided dice for a better set than what the mathematicians had given us. It's still thinking about that one (12 sided dice take 4 times as long to evaluate)- I'm leaving for the weekend and I'll just let it ponder until I get back.

**jpfed wrote:**

I noticed (as has been noted here) that the relationships between the dice

A: 0,0,4,4,4,4

B: 3,3,3,3,3,3

C: 2,2,2,2,6,6

were not consistent. I fired up my trusty Java IDE and programmed up a genetic algorithm to attempt to find a set of nontransitive dice that satisfied two requirements-

1. the probability of a win for one die against what it's supposed to beat should be far from 0.5

2. all such probabilities should be as close to each other as possible.

That second criterion was weighted 2.5 times as much (in an earlier simulation I found a novel set of dice that displayed a perfectly consistent probability to win of 0.53- so criterion #2, while very important to me, shouldn't be the whole picture, or weighted so much that it overdominates.)

Fiddling about, the set

A: 0,0,3,6,6,6

B: 2,2,2,5,5,5

C: 1,1,1,4,7,7

has a perfectly consistent probablitiy to win of 58%.

This is the sort of math-based game design subject I love! I really like designing new dice with mathematical novelties to them. Non-transitive dice are a nifty thing. I'm going to design a few non-transiative dice sets right now!

Funny that I haven't designed a game that uses dice for years, though...

Now that the bugs in my program have been stomped out, it has given me the following results:

6-sider set of 3, consistent 58% relationship:

A: 3,4,5,10,17,18

B: 6,7,8,11,12,13

C: 1,2,9,14,15,16

12-sider set of 3, consistent 61% relationship:

A: 7,8,9,10,11,12,13,14,33,34,35,36

B: 1,15,16,17,18,19,20,21,22,23,24,26

C: 2,3,4,5,6,25,27,28,29,30,31,32

I mention 12 siders because their relationships are slightly more powerful (farther from 50/50) and they should be just about as easy to alter/customize with their big fat pentagonal faces.

--------------

Here is a set of four six-siders, where adjacent pairings have a consistent 58% relationship, and non-adjacent pairings have a 50/50 relationship:

A: 4,5,7,13,22,24

B: 8,9,10,15,16,17

C: 1,11,12,14,18,19

D: 2,3,6,20,21,23

--------------

Lastly, here is a set of five 12-siders, where adjacent pairings have a consistent 63% relationship, and non-adjacent pairings have a 57% relationship.

A: 56,13,16,57,19,12,8,58,52,55,10,9

B: 26,54,15,24,17,18,59,60,20,22,23,28

C: 21,32,27,35,33,30,31,45,25,34,14,39

D: 36,50,1,2,37,38,40,29,3,42,47,41

E: 4,48,46,6,7,53,5,49,11,51,44,43

Wow. These dice sets are really great. Can I use them in a game, giving you full credit, of course?

As far as I'm concerned, they're in the public domain now. That's why I posted them :)

Since there seems to be continuing interest in the subject, here are a few interesting sets I found, searching through sets of five 3-siders (presumably to be used twice each on 6-siders). Each of the dice in these sets have the same total over their sides. The dice are presented in lexical order rather than the order of dominance, since that's how I had them stored...apologies if this is inconvenient. Each of these sets is 'balanced', that is, starting from a scenario where both players uniformly randomly select which die to use (allowing or reselecting duplicates), neither player can gain an advantage by skewing their selection to prefer any of the dice.

A->B means A wins against B with probability 5/9, or the specified probability.

Class I: (A->B,C; B->C,D; C->D,E; D->E,A; E->A,B)

(1,8,15),(2,9,13),(3,10,11),(4,6,14),(5,7,12)

(1,8,15),(2,10,12),(3,7,14),(4,9,11),(5,6,13)

(1,9,14),(2,7,15),(3,10,11),(4,8,12),(5,6,13)

(1,9,14),(2,10,12),(3,6,15),(4,7,13),(5,8,11)

(1,10,13),(2,7,15),(3,9,12),(4,6,14),(5,8,11)

(1,10,13),(2,8,14),(3,6,15),(4,9,11),(5,7,12)

Class II: (A->B,D; B->C,D; C->A,D; D->E (2/3): E->A,B,C)

(1,9,14),(2,10,12),(3,8,13),(4,5,15),(6,7,11)

(1,10,13),(2,7,15),(3,9,12),(4,6,14),(5,8,11)

(1,10,13),(2,8,14),(3,6,15),(4,9,11),(5,7,12)

(1,10,13),(2,8,14),(3,9,12),(4,5,15),(6,7,11)

(1,11,12),(2,7,15),(3,8,13),(4,6,14),(5,9,10)

(1,11,12),(2,8,14),(3,6,15),(4,7,13),(5,9,10)

Class III: (A->B,C,D; B->C,D, C->D,E; D->E (2/3); E->A(2/3),B(5/9)

(1,11,12),(2,9,13),(3,7,14),(4,5,15),(6,8,10)

Share and enjoy, but if you get published, throw in a good word, eh? ^_^

In other news, I can post again! *YAY!*

sedjtroll wrote:I like the idea of tying action to different non-trnasitive dice! "Should I do this and have a good chance of succeeding against that, what if they do the other and I'm doing this..." Lots of good potential for decision making!

Something like that would even work for Darke's Snow Day...