Dice probabilities in the Pool

[Paganini]

I finall got around to running those numbers:

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Dice    Percentage
----    ----------
 1     16.67% (17%)
 2     30.56% (31%)
 3     42.13% (42%)
 4     51.77% (52%)
 5     59.81% (60%)
 6     66.51% (67%)
 7     72.09% (72%)
 8     76.74% (77%)
 9     80.62% (81%)
10     83.85% (84%)
11     86.54% (87%)
12     88.78% (89%)
13     90.65% (91%)
14     92.21% (92%)
15     93.51% (94%)


Not actually as bad as I thought. (Cue 'I-told-you-so' from Mike.) Although, it is a bit worse than some made it out to be. (Cue 'I-told-you-so' from Pag. :)

Assuming you do what I said you did (9 dice in the Pool, gamble them all every time). Worst case: you roll 10 dice (if the GM only gives you 1, and you don't have any traits). You can figure that you'll make about 8 rolls before you fail. When you fail, you lose your entire pool. Now, I'm assuming that you're not going to take a MoV on every one of those 8 successes (anyone who is aware of this line of reasoning will make sure that he doesn't!). Let's say that you take 4 MoVs (which means that you're exercising directoral power 1/2 the time that you make rolls). This seems like plenty to cover personally important scenes. So, when you fail your roll, you've built up around 8 reserve dice in your pool to gamble. The odds are that you will get *another* 7 or 8 rolls before you fail. Let's say you fail on the 7th roll, and you have 6 reserve dice in your pool. You're still not going to fail for another 6 or 7 rolls.

It evens out right around here. With 6 dice in your pool you can expect to make around 6 rolls before you fail. If you take dice from three of those rolls, you've maintained your status. This is assuming that the GM *never* gives you more than a single die, and it's assuming that you have *no* traits to use. So, in other words, these are worst case numbers.

OTOH, if you decide to take a MoV every time, then it gets very bad. After around 6 rolls you start to thrash.

So, basically, this post is a line of reasoning with this conclusion: never have fewer than 6 extra dice in your pool, if you can help it. People have been saying "yeah, but this isn't how the Pool is played." The point I'm making is that, if you understand the numbers, this is *exactly* how the Pool *must* be played. There's no reason to ever play it differently. The Pool is primarily a resource that determines *player* effectiveness... it's designed that way. The more you have, the more effective you are. If said resource was unimportant, people wouldn't complain about thrashing. So, if you don't want to thrash, keep an eye on the numbers. It's easy to keep from thrashing, all you have to do is leave a certain amount of directoral control with the GM.

[Valamir]

definitely a case for going back to 1 die per passed up MOV instead of 2.

[Paganini]

definitely a case for going back to 1 die per passed up MOV instead of 2.

Good point, Valamir. That would change things drasticaly. That might just be the solution.

[Ron Edwards]

Hey,

I always liked the one-die rule better.

Best,
Ron

[hardcoremoose]

No fair to bust on James when he's out to lunch, but I liked the one-die rule better as well.  There was a simple elegance to it, which was the hallmark of The Pool in general.

And is one of the reasons I never really dug the MoD.  But that's just my personal cross...

- Scott

[Mike Holmes]

Assuming you do what I said you did (9 dice in the Pool, gamble them all every time). Worst case: you roll 10 dice (if the GM only gives you 1, and you don't have any traits). You can figure that you'll make about 8 rolls before you fail.

No, no, no, no, no. I refuted this before. Everyone makes that superstitious assumption that you'll roll on average 8 times before you Fail. Actually there are two problems here. Even if you were exepcted to always get through a full set of rolls before failing (which you are not) your math would be off. At 84% that logic would give you one in 6 failure. One in 8 would be 87.5%.

In any case, the fallacy is that your assumption is about expected value. That is, yes, it is true that if you roll the ten dice six times that you'll likely get one failure somewhere in that set of rolls. But it says nothing about where you are likely to fail. It is as likely that the failure will occur on the first roll as the last. So what are the odds? The chart below shows the chance that you have starting out of getting past a certain number of rolls using a certain number of dice.

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     Dice
Rolls 10       11       12       13       14
1     83.85%   86.54%   88.78%   90.65%   92.21%
2     70.31%   74.89%   78.82%   82.17%   85.03%
3     58.95%   64.81%   69.98%   74.49%   78.40%
4     49.43%   56.09%   62.12%   67.53%   72.30%
5     41.45%   48.54%   55.15%   61.21%   66.66%
6     34.76%   42.01%   48.97%   55.49%   61.47%
7     29.14%   36.35%   43.47%   50.30%   56.68%
8     24.44%   31.46%   38.59%   45.60%   52.27%
9     20.49%   27.22%   34.26%   41.33%   48.20%
10    17.18%   23.56%   30.42%   37.47%   44.44%

You'll note that even at 14 dice that it's even odds that the character will fail at some point before 9 rolls. At ten dice the same can be said of just 4 rolls. And only a 24.44% chance of actually getting to 8 rolls. Not something I'd count on.

Do you see the problem? If you don't believe me try rolling a d6 and try to get to six rolls without getting a 1. Very similar to the 10 chart above. You'll find you only do it about one in three times. The rest of the time you fail at some point earlier than that. Not what I'd call a winning strategy. Then again, as I point out, the only winning strategy is to risk no dice.

I can get much more in depth with this too. This all says nothing about the game theory of continuing odds, and expected rewards. For example, if you look at it from an expected reward view point, then one die is your best bet in all cases. That is, assuming that a win is worth 2 dice at least (as you can use it as that), then the risk of 1 die to 2 dice won tips the scale well versus 9 to 2 (from between a two to one ratio with few additional dice to a four to one ratio). It's not even close; of course by that ratio, risking no dice is a huge advantage. Many gamblers play these sorts of odds.

In any case, I'd suggest playing it and seeing what happens. No use speculating what sorts of strategies it will produce until you actually play.

Mike

[Jeffrey Straszheim]

There are two situations that I want to avoid

That a player will flounder around too long with an empty pool
after a big hose

That a player will find a "winning" strategy that will allow them to
bet lots of dice, but win dice enough to keep growing their pool.
This would reward an "always bet lots of dice" strategy that I think
would be bad.
[/list:u]

Now, I ran a simple simulation that rolled dice.  I found that if a
player followed a strategy of always betting their entire pool (or 9
dice), and always took the bonus dice, the following happened:



If they received 3 dice from gifts and/or traits, then with a two dice
reward their pool would eventually grow unbounded.  With only one
reward die, it would fluctuate, but often grew into triple digits
before it came back down.

If they consistently received 5 gift/trait dice (assume an experienced
character with large traits), then both the single and two dice
options quickly grew unbounded.
[/list:u]

I think this is undesirable because it can reward an uninteresting
strategy (always bet hight).  Obviously a character's taking MOV's
would lessen the effect, but not eliminate it for a character skilled
at probability.

I would suggest a cap on the size of the pool to counteract this problem.

As a side note, a tried a strategy that rewards dice based on the
current size of the pool.  I used this scheme

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empty pool   :  3 dice
1 - 9 dice   : 2 dice
10 - 15 dice : 1 die
16 + dice    : none


The advantage here is that the pool stays capped, so that it cannot
run away, but also players who are hosed will not flounder around with
zero dice for too long.

[Paganini]

Mike, you're right... darn it! (Note to self: always use calculator!)

However, the main point I was making still stands... it doesn't matter how soon you fail if you can replenish your pool before doing so.

[edit: That is, if you know you're going to fail sometime in the first 8 rolls, then you're going to do your darndest to make sure that you minimize the consequences of that failure. You're going to take dice instead of MoVs until you have enough backup dice in your pool not to worry about it. If you can get as many or more dice into your pool by doing so failure is trivialized. Standard thinking assumes that the failure will come in the middle of the tries you make. (That is, if you're brute-forcing a crypto system with 100 keys in a random order, you assume that you'll have the correct key after about 50 tries.) So, if you've got 8 to 1 odds, statisticaly, yes, that one failure could occur on any of your nine rolls. But for the purpose of strategy you can assume that you'll get at least 3, possibly 4 rolls before failing. With the current rules, that means 6 - 8 dice added to your pool. Like I outlined below, this ability to regenerate is not trivial. It will effect your approach to deciding whether to take dice or MoVs. So, summary: I like the one-die option better than the two-die option. :)]

Stimuli, You pretty much nailed it from my perspective. I don't think your proposed fix fits with the philosophy of the Pool though. Two points: The number of dice that you can gamble is already limited to 9. From your post I'm not sure you took that into account. The other thing is, if you can only get one die back from a successful roll, things are much different.

So, anyway, Mike, I disagree with you when you talk about the pool not being Gamist. Or rather, I see it as a bit of a non sequitir. The Pool gives you a resource, and it makes it valuable. This encourages skillful management of the resource, *without* implying any sort of inter-player competition.  If a way exists to maximize returns, then there is absolutely no reason not to take advantage of it. The design of the system encourages this sort of thinking: "How can I use my dice most effectively?" Not "more effectively than everyone else," just "how can I get the most out of them?" They're a resource... that's what they're for. :)

What it comes down to, IMO, is how much you want to risk on a "put up or shut up" basis. How much are you willing to risk in order to gain an extra die or a MoV? In order for the game to work this has to be meaningful - you have to be able to evaluate the worth of the rewards and weigh their value against the value of the stakes.

This is why the system is so cool... on the one hand you have a resource (dice) that can be quantitatively evaluated. On the other hand, you have a resource (directoral control in the form of MoVs) that can not be objectively quantified. The player has to decide how much risk a MoV is worth in terms of the quantifiable component of the system.[\i] You can't just say "oh, I think a MoV here is worth three dice to me." You have to decide what your reward dice are worth in terms of success chances (which correspond directly to the reward dice you would get by forfeiting a MoV). Consider this:

For example, if you look at it from an expected reward view point, then one die is your best bet in all cases. That is, assuming that a win is worth 2 dice at least (as you can use it as that), then the risk of 1 die to 2 dice won tips the scale well versus 9 to 2 (from between a two to one ratio with few additional dice to a four to one ratio).

This isn't true *unless* your chance of success is 50%. If you have an 80% chance of success (8 dice from the pool, plus 1 die from the GM), then the odds are 4 to 1 in favor of success. So risking 8 dice against 2 dice is fair odds. So the question then becomes, how does the MoV you stand to get compare to the 2 dice that you'll have to give up. It will depend on your reserve dice, and on how badly you want the MoV. :)

[Jeffrey Straszheim]

Stimuli, You pretty much nailed it from my perspective. I don't think your proposed fix fits with the philosophy of the Pool though. Two points: The number of dice that you can gamble is already limited to 9. From your post I'm not sure you took that into account. The other thing is, if you can only get one die back from a successful roll, things are much different.

I did, in fact, use the 9 die limit in my tests; and using 1 die did help.  But it was still possible to build up pools in the triple digits, which would allow for gratuitous trait purchases, which in turn would only worsen the problem.

[Paganini]

I did, in fact, use the 9 die limit in my tests; and using 1 die did help.  But it was still possible to build up pools in the triple digits, which would allow for gratuitous trait purchases, which in turn would only worsen the problem.

Hmm. Next question. Did you take the reward die after every success, or did you account for some MoVs in there? I suggest taking the reward die every other time.

[Mike Holmes]

OK, first Nathan, using your strategy, you will not often have MOVs. Which would be really boring. Sure, you can do it, but since you can't spend these points on a porsche, it seems silly to hoard them. In play, you are probably going to want to take an MoV from time to time. If you want to play it as a Gamist exercise where you are just trying to accumulate points, you can "win" going home with the most points at the end of the night (heck, you can just roll your base dice and gamble nothing, and most certainly increase in points). But that's hardly the point of play in The Pool. Ron pointed this out before, and you have yet to rebut in any way.

The converse argument is also true. That is, given that you cannot buy a porsche with the points, what's the downside to having a small pool? The character has a period of negative results that he has to sort of climb back from. Isn't that good for making a story? Isn't failure an important part of Protagonism? Do you really want your character to succeed all the time? If so, then, once again, you miss the point of the game. What changing strategies does for you is to alter the flow of the game back and forth. Which could be fun (a lot more fun than a game about accumulating dice from rolling them).

Both you and Jeff are also assuming an enormous number of rolls, as well. What if there are six rolls per game for your character, and the game goes for only six sessions. Sounds about right. You'll never grow to triple digits because the story won't last that long. If you won every roll and never took an MoV you'd only have 72 dice; at which point the story would be over. You are going to squander the vast majority of your rolls on successes without MoVs. Let's say you take one in eight as MoVs to keep your pool high. That gives you maybe one a session, and only four in the character's whole existence. And in order to keep the strategy you must at times forgo the MoV when you might really want to take one.

Your arguments about when you expect to come across a random event are just wrong, Nathan. You do not assume half. You run a limit summation based on the odds of the original chance that it happens each try. Again, this is more voodoo statistics. In any case, such an argument does not allow for waht happens if the odds happen to go against you. Wherin you have to resort to other strategies. So you can start out with such strategies, but the game might just not allow you to use them (the obvious example is what do you do if you fail that first roll, and go straight to the bottom?).

Again, you refuse to actually play and see what happens. If you did it might change your opinion. I defer to Ron who has played. This is not D&D, its not about acumulating dice. It's a Narrativist system. Yes, if you play it Gamist, it will suck. So don't do that.

Mike

[Jeffrey Straszheim]

Hmm. Next question. Did you take the reward die after every success, or did you account for some MoVs in there? I suggest taking the reward die every other time.

No, I assumed they always took the reward.  What I wanted to find out was if it was possible for a pool explosion.  Now, assuming the cusp between bounded and unbounded behavior is rather narrow (which I haven't checked, but is the norm for such recursive functions) there will be some fraction where the player can take MoV's and still get the unbounded behavior.  I'm not sure where this lies, but it wouldn't be too hard to find.

[Mike Holmes]

No, I assumed they always took the reward.  What I wanted to find out was if it was possible for a pool explosion.  Now, assuming the cusp between bounded and unbounded behavior is rather narrow (which I haven't checked, but is the norm for such recursive functions) there will be some fraction where the player can take MoV's and still get the unbounded behavior.  I'm not sure where this lies, but it wouldn't be too hard to find.

I totally agree. The point is that the point of taking MoVs is far enough apart that it's not a very interesting stratey to play. It assumes that charcter success is the only thing a player wants, and that the Gm will not be able to tempt them into taking MoVs more often. Something that's impossible to calculate.

Mike

[Jeffrey Straszheim]

OK, first Nathan, using your strategy, you will not often have MOVs. Which would be really boring. Sure, you can do it, but since you can't spend these points on a porsche, it seems silly to hoard them. In play, you are probably going to want to take an MoV from time to time. If you want to play it as a Gamist exercise where you are just trying to accumulate points, you can "win" going home with the most points at the end of the night (heck, you can just roll your base dice and gamble nothing, and most certainly increase in points). But that's hardly the point of play in The Pool. Ron pointed this out before, and you have yet to rebut in any way.

Speaking for myself, my only goal was to discover whether is was possible to abuse the system.  Also, I'm interested in finding a good balance between slow pool growth, and a statistical explosion.  I doubt there is a perfect balance, but it is worth knowing the tradeoffs.

Both you and Jeff are also assuming an enormous number of rolls, as well. What if there are six rolls per game for your character, and the game goes for only six sessions. Sounds about right. You'll never grow to triple digits because the story won't last that long. If you won every roll and never took an MoV you'd only have 72 dice; at which point the story would be over. You are going to squander the vast majority of your rolls on successes without MoVs. Let's say you take one in eight as MoVs to keep your pool high. That gives you maybe one a session, and only four in the character's whole existence. And in order to keep the strategy you must at times forgo the MoV when you might really want to take one.

But who says the game should only go for six sessions?  Who says there will only be a few rolls?  It is certainly possible for a player to choose to roll very often.  And it might all seem perfectly valid while it's happening.

Again, you refuse to actually play and see what happens. If you did it might change your opinion. I defer to Ron who has played. This is not D&D, its not about acumulating dice. It's a Narrativist system. Yes, if you play it Gamist, it will suck. So don't do that.

I know you weren't talking to me here; however, I'll respond anyhow.  I have played The Pool and found it a profoundly interesting system.  We played a very short game, and never ran into any pool explosions.  In fact, we had quite the opposite problem; players who'd lost their pool ended up floundering around with no dice for too long.  This was not fun.  We were, at the time, using the single die reward system.

I am still concerned, however.  I think the system can, over longer term play, reward some undesirable strategies.  The fact that we all agree (I think we do) that these strategies aren't the best way to play doesn't stop that they are rewarded.  I wonder if there aren't some small tweaks that can be made to mitigate this problem?

It occurs to me that there, perhaps, shouldn't be one single According-to-Hoyle The Pool.  Just as James tweaked the resolution mechanic in QB to make the flow more predictable, I believe that perhaps some aspects of The Pool (as a generic entity) can be tweaked by various play groups to match the flow they desire.  I'm thinking of such aspect as

The number of reward dice
The presence or absence of a MoD
The cost for traits
... and so forth ...
[/list:u]
but it seems I'm drifting from the topic.

[Paganini]

[Note: This message was originaly posted by accident... so I'm clearing it out and using to append something I forgot in my other post. :)]

Mike, I forgot to mention that my comments about expecting results were not "just plain wrong." As I said, what I described is a *common method.* I've seen it in many texts, including several of notable authorship (for example, Bruce Schnier's Applied Cryptography). I'm sure that there are more accurate and more complex methods of prediction. Citing the exitence of such, however, in no way detracts from the validity of my point.