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2015-09-18, 11:14 AM
Idea
I had an idea for a dice pool system with two possible victory conditions. One is to have more big numbers than the other guy, you have three 5s and they have two 2s, so you win. The other, however, is to get a poker straight. They rolled five 6s, but you rolled a 1, 2, 3, 4 and a 5, so you win. Like in poker.

Moreover, you might be able to build straights off your opponent's dice pool, as if it were a flop from Texas hold'em. Normally, the guy with a much larger dice pool is bound to win. But if they use a 2, 3, 4, and a 5, and you have a 1, then you can make a straight. Suddenly, throwing a hundred dice at the problem isn't the best idea.

It forces you to reconsider which dice you use. If you have a 3, 4, 5, and two 6s, you might decide to just use the two 6s and the 5, as that gives you a really strong roll, but it's unlikely your opponent will be able to make a straight out of it. Larger dice pools are still stronger, as it offers your opponent the chance to use twelve 3s or the like, but this allows you to potentially counter them.

With re-rolls, this also creates an interesting strategy, of whether you should re-roll to try and get an even mix of dice for forming straights, or whether to keep re-rolling small numbers to try and get as many big numbers as possible.

Problems and Conclusions
Now, there are some serious problems and questions this system presents. For one thing, 1d6s makes it really easy to get a straight, too easy. Another thing is that with both sides having a large enough dice pool, a straight will become more and more likely, until it is definite. Another is the question of how winning through a straight should be calculated. There are a few possible answers.

Possible Solutions
The essence of the idea is to have two possible victory conditions from comparing dice pools, one of which can allow someone with less dice to come out ahead. So, one idea is to change it from a straight, to something else. The question being, what?

You could have it that the straight only serves to neutralize the dice involved. So, if you make a straight using a 1, 2, 3, 4 and 5 of your own, you only serve to neutralize your own dice. If you make a straight using a 1 of your own, and a 2, 3, 4, and 5 of your opponent's, you got rid of four of their dice at the cost of one of your weakest dice. This seems like the best immediate answer, as it gives the straight some power and makes it more strategic, although it isn't precisely a victory condition so much as a means of making your dice total bigger than the opponent's.

For the ease of getting a straight, the simple answer would be to use 10 or 12 or 20 sided dice. If you needed to make straights easier or harder to get still, you could require longer or shorter straights. Potentially, certain character builds might be able to use longer or shorter straights.

Anyone have thoughts on whether this seems interesting?

Aedilred
2015-09-18, 01:38 PM
If I'm not mistaken, isn't the probability of rolling such a "straight" as you describe actually higher than rolling five sixes? It's certainly no lower. Note that in poker, four of a kind beats a straight (indeed, anything but a straight flush, albeit in poker each card is a 1/13 chance rather than 1/6).

Sure, a die has as much chance of rolling a 1 as a 6, so probability isn't everything, but at least there is a standard mechanism in place for dealing with those sort of ties. I wouldn't necessarily say no to a slightly more nuanced approach to dice rolling, but this seems like adding extra complexity and ambiguity for the sake of it.

Thrudd
2015-09-18, 01:39 PM
Idea
I had an idea for a dice pool system with two possible victory conditions. One is to have more big numbers than the other guy, you have three 5s and they have two 2s, so you win. The other, however, is to get a poker straight. They rolled five 6s, but you rolled a 1, 2, 3, 4 and a 5, so you win. Like in poker.

Moreover, you might be able to build straights off your opponent's dice pool, as if it were a flop from Texas hold'em. Normally, the guy with a much larger dice pool is bound to win. But if they use a 2, 3, 4, and a 5, and you have a 1, then you can make a straight. Suddenly, throwing a hundred dice at the problem isn't the best idea.

It forces you to reconsider which dice you use. If you have a 3, 4, 5, and two 6s, you might decide to just use the two 6s and the 5, as that gives you a really strong roll, but it's unlikely your opponent will be able to make a straight out of it. Larger dice pools are still stronger, as it offers your opponent the chance to use twelve 3s or the like, but this allows you to potentially counter them.

With re-rolls, this also creates an interesting strategy, of whether you should re-roll to try and get an even mix of dice for forming straights, or whether to keep re-rolling small numbers to try and get as many big numbers as possible.

Problems and Conclusions
Now, there are some serious problems and questions this system presents. For one thing, 1d6s makes it really easy to get a straight, too easy. Another thing is that with both sides having a large enough dice pool, a straight will become more and more likely, until it is definite. Another is the question of how winning through a straight should be calculated. There are a few possible answers.

Possible Solutions
The essence of the idea is to have two possible victory conditions from comparing dice pools, one of which can allow someone with less dice to come out ahead. So, one idea is to change it from a straight, to something else. The question being, what?

You could have it that the straight only serves to neutralize the dice involved. So, if you make a straight using a 1, 2, 3, 4 and 5 of your own, you only serve to neutralize your own dice. If you make a straight using a 1 of your own, and a 2, 3, 4, and 5 of your opponent's, you got rid of four of their dice at the cost of one of your weakest dice. This seems like the best immediate answer, as it gives the straight some power and makes it more strategic, although it isn't precisely a victory condition so much as a means of making your dice total bigger than the opponent's.

For the ease of getting a straight, the simple answer would be to use 10 or 12 or 20 sided dice. If you needed to make straights easier or harder to get still, you could require longer or shorter straights. Potentially, certain character builds might be able to use longer or shorter straights.

Anyone have thoughts on whether this seems interesting?

It sounds interesting as a yahtzee-like game with a strategy component. Using the opponent's dice feels similar to cribbage, where you both make your best hand, discard the remainder, and then the dealer gets to see if the leftovers get them any extra points. Add in some type of wagering or bidding component: like you can get more points if you roll less dice, maybe your total score is divided by the number of dice you rolled, or every die rolled subtracts x number of points from your total. Or alternatively, you both secretly bid a number of dice from your original pool, prior to rolling, for the opportunity to roll the leftover pool (made up of those same unrolled/bidded dice). This gives you the chance to greatly increase your score for that round, though the more dice you take from your pool, the less chance you will have many points.

For an RPG, it seems like a game within a game which would detract a lot of attention from the role playing adventure. You would spend a lot of time looking at the dice. Unless every element of the dice strategy game maps to and is dictated by some action or decision that is happening in the narrative, it does not seem like it would be very intuitive. Like, how is a player's decision to keep certain dice or remove certain dice from their opponent dictated by the situation their character is facing? What does getting a straight represent in the game, why does it beat four of a kind? What does having less dice vs more dice represent?

Basically, I would work it from the narrative angle, first. You want characters with less raw power to be able to affect those with more, sometimes? Or you want there to be a drawback to attempting the strongest possible attack every time. Do characters declare their actions simultaneously, are defenses active or passive, is the result of an attack binary (hit/miss) or a range of results. How abstract is the combat meant to be, do the dice model specific movements and techniques or overall performance? All these sorts of considerations should inform how the dice system works. What will best model the scenarios you're hoping to describe, and help players engage with the narrative?

halfeye
2015-09-18, 04:52 PM
I don't really know or play either game, however, in poker (with five cards to a hand) flushes beat straights, but in brag (with three cards to a hand), straights beat flushes. This is because the odds of the condition arising in a hand change with the number of cards in a hand; in both games, the most unlikely wins.

2015-09-19, 02:38 AM
Aedilred: Yes, the comparative probabilities are one of the problems listed. The solution would be to have high requirements for the straight, or larger dice than a d6. There's also the question of what benefit you derive from a straight. Potentially you could get to keep one of the involved dice, so you can steal the biggest one from your opponent. However, neutralizing all involved dice might be enough of a benefit.

The reason I'm trying to work out something to this nature, is to add a secondary victory goal in using dice pools.

Thrudd: I thought what I detailed showed that a straight doesn't really beat four of a kind. Do you disagree?

The narrative role of this system is to represent counters. I wasn't sure how to represent counter attacks in game, so I thought of the idea of trying to get a certain combination of dice. This makes counters strategic to arrange, while attacking with a lot of large dice is a basic, strong attack. It was also to discourage opening yourself up to a counter by being overly aggressive.

halfeye: This is true. The unlikelihood of the straights can be modified, and the effects they cause. The question being how much in which directions would get the best result.

Aedilred
2015-09-19, 10:05 AM
Aedilred: Yes, the comparative probabilities are one of the problems listed. The solution would be to have high requirements for the straight, or larger dice than a d6. There's also the question of what benefit you derive from a straight. Potentially you could get to keep one of the involved dice, so you can steal the biggest one from your opponent. However, neutralizing all involved dice might be enough of a benefit.

The reason I'm trying to work out something to this nature, is to add a secondary victory goal in using dice pools.
Increasing the die size wouldn't help with the probability angle. As with poker, which one could envisage as using a d13 and where the chance of four identical results is less than of four consecutive results.

There would have to be an additional restriction in place to make it work, I think.

2015-09-19, 02:26 PM
My current thinking is it would only neutralize the dice that make up the straight, as I detail in the Possible Solutions. This'd make it easier for the player who unleashed a strong straight to succeed, but it doesn't guarantee success with that system.

meschlum
2015-09-19, 08:27 PM
One issue is simply a matter of symmetry: If the low die player can use the high die player's four results (2, 3, 4, 5), why can't the high die player use th elow die player's single result (1)?

So using the other's dice to complete a straight needs to be better defined!

Past that, asusming d6s, let's run some numbers.

Say you have 10 dice. You are sure to get doubles (that's a given from 7 dice on), so they can't be worth much (and triples from 13 dice on).

To get two straights, you need two each of 2-5, and two of (1 or 6). That's three combinations (two 1s, two 6s, a 1 and a 6), and it happens with probability 0.75%

To get one straight, you just need 2-5 and either 1 or 6, which is much more flexible. Having crunched the math (there are shortcuts), you get a 44% chance of getting a straight.

A double is guaranteed, and you have only pairs 6.75% of the time (6% of the time, you have a straight as well - this is where double straights occur).

A triple happens 52.9% of the time, so it's more common than straights (27.5% is a straight and a triple, so if you have a triple, you have a straight too around half the time).

A set of four happens 31% of the time, so it's less common than straights (9.2% is a straight and quadruple).

A set of five happens 7.8% of the time (with a straight as well being 1.3% of all cases)

A set of six happens 1.3% of the time (and the very rare straight + six of a kind, with one die in common, is 0.08%).

A set of seven happens .15% of the time (and no straights are possible from this set on).

A set of eight happens about once in 9,000 rolls.

A set of nine happens about once in 200,000 rolls.

A set of ten happens about once in ten million rolls.

So with 10 dice as your baseline (and I'm not going to bother to run the math for other numbers, though there are tricks to make approximations easier - Poisson distributions come to mind), you're getting straights all the time, and four of a kind is fairly common too - sadly, more than four of a kind is rather rare, so you can't really set up a sensible heirarchy.

Manipulating straights gives you some flexibility in the results, but (again, with 10 dice) you're not going to be able to slice the odds in a convenient way.

2015-09-19, 09:01 PM
Nice statistics, thanks Meschlum!

I believe I described a method that would give you a good solution to the problems of symmetry and and many dice, as you'd be neutralizing only a 1-point die of the enemy's, while losing a 4, 5, 3, and 2 of your own. Similarly, even as dice scale up, it essentially favours the person with less dice. You can potentially neutralize all of an opponent's dice with straights, if you have enough, but at that level of difference it makes sense the enemy would be defenceless on a good roll.

meschlum
2015-09-19, 11:36 PM
Not quite: if one player has 1, 2, 2 and the other has 1, 2, 2, 3, 4, 5, 6, the second player can cancel his straight using the first's dice and leave the low die player with only a 2 (to his 6). Or the first player can cancel a straight using his 1 and the other's 2, 3, 4, 5, leaving the first player with a pair of 2s to the second player's 6.

So when cancelling, it matters who gets to do it!

If one low die player has 3 dice and the other has 7, there is a 44% chance that a straight will turn up (and a 0.75% chance that two will), since there are 10 dice in all.

If no straight turns up, the 7 dice player has a much better chance of getting a longer series (he's got to have at least a pair, while the 3 die player hasa 44% chance of not getting a pair!).

The result of cancelling out the straight is that there are 5 dice left. The dice removed by the straight could be any of the 10 dice, so:

There are 252 ways in which the 5 dice could be assigned.

In 21 cases, the low die player loses all his dice.
In 105 cases, the low die player has one die to the high die player's four.
In 105 cases, the low die player has two dice to the high die player's three.
In 21 cases, the low die player has three dice to the high die player's two.

So 11 times out of 12, cancelling is to the disadvantage of the low die player.

These are the specific numbers for the case where there are 10 dice in all and one player with three dice, but as a rule the player with more dice can afford to burn more than one die per die if he has the advantage (7/3, the high die player has more than twice as many dice).

If the low die player had four dice (high die player has 1.5 times as many as the low die player, so it's closer to parity), then when there is a straight:

In 6 cases, the low die player loses all his dice
In 60 cases, the low die player has one die to the high die player's four.
In 120 cases, the low die player has two dice to the high die player's three.
In 60 cases, the low die player has three dice to the high die player's two.
In 6 cases, the low die player has four dice to the high die player's one.

So the high die player has the advantage in 31/42 cases (~75%).

All this is assuming that straights get cancelled in a neutral manner, which isn't the case. So you've got extra complications and shifts in the odds linked to that to figure out, and it looks overall like 'neutral' cancellation will advantage the high die player anyway.

2015-09-20, 01:02 PM
Well, that's what I mean. The player with 7 dice is generally assuredly going to beat the player with 3. If you take the situation without straights, a 1, 2 and 2 against 1, 2, 2, 3, 4, 5, 6, obviously the latter player wins. Three dice against seven is a pretty steep disadvantage, normally. Now, with straights, the situation has still improved for the weaker player, even if they ultimately still lose. Either the weaker player spends their 1 (keeping two 2s) and takes away the enemy's 2, 3, 4 and 5, reducing the enemy to a 1, 2, and 6; or, the enemy has to spend three higher number dice, a 3, 4 and 5, to get rid of just two lower number dice, a 1 and 2.

Because the rolls were so steeply weighted in the stronger player's favour, they will still win out regardless, but each of the sequences serves to help the weaker player more than the stronger one.

This does give me an interesting idea, though. You could have an unlimited straight, where you can make it a straight of 1 to 6. Or if you were using a d20, 1 to 20. This'd give more power to the straight to further help weaker players, while being unlikely enough that it won't overpower the straight. Of course, there would be a minimum length for a straight. So you can't make a straight of 1 to 4.

As for the question of which player gets to act, it'd probably be the defender who'd get the choice to make use of straights first, if there is one. You could potentially make it the person who spends less dice has this advantage, but I'm uncertain.

One thing to note is that in this system, I was thinking of having it that the players' pre-roll their pools, then choose which of the rolled dice they want to use in their attack. This would affect the likelihood of a straight, as stronger players will be selecting doubles and a disconnected series of numbers, while weaker ones will be using all their dice in hope of forming a straight against the stronger enemy to weaken them. Of course, it will also depend on the results of the roll. If you get a really poor low with a lot of 1s, even if you have more dice.

All this talk has made me wonder about a dice pool system, where the person who rolls the most 6s wins. If no one rolls a 6, the person with the most 5s wins. If no one rolls a 5, the person with the most 4s, and etc.. The score of your roll is based off how many more Xs you rolled than your opponent. I think this would also sort of favour the weaker person, as there's a chance someone with 1 die will roll a 6, and that their opponent with six dice comes up only with 4s, 5s, 3s and 2s. Of course, in such a system, it wouldn't matter if you made straights unless they took out some 6s.

Lvl 2 Expert
2015-09-20, 04:42 PM
All this talk has made me wonder about a dice pool system, where the person who rolls the most 6s wins.

Reroll everything but your sixes, always.

More common in this kind of game is that the person with the biggest set wins, and if the sets are the same size, that's when 3 sixes beats 3 fours.

Alternatively, any set is worth as much as the total amount of points rolles, two sixes is twelve points, just like four threes. (A straight would be worth 21 points, if allowed, which is a lot for how common they are in about any possible setup.)

If 6's are worth substantially more than 1's it's also no more than reasonable to think about making 6's less common. In a card game you could do that by having more 1's in the stack than 6's, in a dice game you could say that the dice are rolled in pairs, and only the lowest dice of every pair counts. Or maybe you roll say seven dice, and only the lowest five form your hand. (Crap, I just know I'm going to be doing calculations on that scenario this week, with a whole range of numbers of dice.) That way it's harder to have a decent amount of 6's showing, because the first two don't count. (And if you have just one six then one of those fives doesn't count either etc)

(Yes, I am making a mess of numbers as words and as number signs conventions.)

meschlum
2015-09-20, 10:07 PM
So we're down to setting rules for the exceptions to the rules, which is not a good place to be in a roleplaying game.

It's entirely possible to come up with a complex system that deals with all this, but from the player and GM perspective if you need to spend a few minutes figuring out what to do - which straights to cancel, who gets priority, what you think the other person is going to do and how you'll adjust your strategy, whether you want to roll more dice or try for the possible advantages of having fewer, decide how many fewer should be...

Lots of time strategising, not much time having fun.

Also, when you get into a three sided conflict (and you will), things get even messier - whose straihgts do you cancel? How do you coordinate with others? How do you deal with the insane number of dice involved and keeping track of what you may or may not want to cencel depending on what all the different sides are doing or intend?

That said, if you're just taking the highest roll the math is somewhat easier.

The odds of getting a high roll of N with D 6-sided dice are (N^D - (N-1)^D)/6^D.

So with 4 dice, you have:

6: 52% (671 / 1296)
5: 28.5% (369 / 1296)
4: 13.5% (175 / 1296)
3: 5% (65 / 1296)
2: 1% (15 / 1296)
1: 0.1% (1 / 1296)

With 6 dice, you have:

6: 66.5%
5: 24.5%
4: 7%
3: 1.5%
2: 0.1%
1: 0.002%

So you're going to get 6s fairly often (and when you don't more dice means your odds of getting a 5 are noticeably higher), meaning that comparing the number of results you get matters. And, again, more dice are better here.

If this type of mechanics interests you, you could look at Don't Rest Your Head, which has a resolution system with interesting tradeoffs between high rolls and low ones (more dice is always better, but there is a cost to getting more dice that you may hesitate to pay).

2015-09-20, 10:44 PM
Err... no. The very premise of the system requires you to decide who gets to form straights first, it is not exceptional.

Strategizing is at the heart of engagement in a turn based game. I do not consider it much to decide allotment of dice and choosing which to re-roll, that's perfectly common in d6 games. If you don't want to consider what your opponent might do and how to adjust your strategy, competitive games are simply not a desirable venture.

Err, a three-sided conflict? No, I didn't think of that being a common consideration. Systems like DnD allow a third party to apply a bonus to one side in a skill test or contested roll, but I don't know of many examples of three sided rolls. Whether it's tugs of war or climbing or attacks, it's generally more logical to roll for each contest separately--which really was part of the idea of a dice allotment mechanic.

I'm not sure what you mean about coordinating with others or insane numbers of dice. Have you played a game that uses d6 dice pools? I hope I didn't mistakenly indicate there would be more dice involved than is common for such.

Expert: You could have it that a pair of 6s has the same point value as a pair of 1s, but a pair of 6s beat a pair of 1s as a tie-breaker.

Rolling pairs of dice and taking the lower roll sounds like a mechanic you could do something interesting with. It would essentially make rolling 6s into a critical (only a bit less unlikely than a nat 20). Taking the lowest of a bunch of dice for a poker hand reminds me of the Witcher's dice poker, though it's system was a bit different.