Here is a quick (somewhat mathematical) puzzle for your inner munchkin:

Consider two abilities, A and B, which are the same in all regards except for the damage they do.  Ability A does a flat 100 damage.  Ability B is more complicated.  By default it does 1 point of damage.  But when it hits you get to flip a coin.  If it is tails you do nothing.  If it is heads you double the amount of damage the ability would have done and repeat the process.  For example: You get three heads followed by a tails: the ability does 8 points of damage.  You get a tails: the ability does 1 point of damage.  You get two heads followed by a tails: the ability does 4 points of damage.

Which ability does the min-maxer choose for their character, assuming they have a choice between just these two abilities, and assuming they are trying to deal as much damage as possible?  The mathematical answer feels wrong in this case, but is it?

A. It's not even a hard decision. You'd need 7 successes to even get 100 points damage.

Quote from: danbuter;416915A. It's not even a hard decision. You'd need 7 successes to even get 100 points damage.

Oh .. but what is the average damage for B?  This is where the fun comes in.  (hint, it's greater than 100, by a lot)

For B, you'll be lucky to get 3 successes in a row, so I bet the average is around 4. Since you stop at the first tails (unless you are changing the rules).

The average for B is infinity.  1/2*1 + 1/4*2 + 1/8*4 + 1/16*8 + ... = 1/2 + 1/2 + 1/2 + 1/2 + ... = inifinity.

For an explanation see: http://plato.stanford.edu/entries/paradox-stpetersburg/  it's a classic paradox.  But I thought some of the more mathematical minds here might have an opinion on it.

I thought you said you do nothing (implying you stop flipping coins) on the first tails?

Quote from: danbuter;416928I thought you said you do nothing (implying you stop flipping coins) on the first tails?

That's correct.

Well the math looks correct, though initially I didn't believe it either.
100 damage vs. the doubling damage, I'd have to take the 100. The chances of getting more than that are sufficiently small that I'd say 100 reliably is better. If you're actually trying to calculate relative worth of possible infinite damage - the maximum 'real value' of the damage is the hit points of the monster you're trying to kill, the overkill is irrelevant.

Quote from: Bloody Stupid Johnson;416938Well the math looks correct, though initially I didn't believe it either.
100 damage vs. the doubling damage, I'd have to take the 100. The chances of getting more than that are sufficiently small that I'd say 100 reliably is better. If you're actually trying to calculate relative worth of possible infinite damage - the maximum 'real value' of the damage is the hit points of the monster you're trying to kill, the overkill is irrelevant.

Perhaps this is the real trick, isn't it?  The question looks complete, but you actually need to know the typical monster hp to have an answer.  Does this show that there is something wrong with the way we typically analyze the worth of abilities though?  I, for one, always simply consider chance*damage to work out what is better than what.  But if we should be considering the possibility of wasted damage from overkill the math would become much more complicated ...

Well, as you say, there may be more to min/maxing that just average damage. This is kind of a weird 'edge case' though there may be other cases where again, overkill is a factor; minions in some systems being the first that comes to mind.

Replacing the continually overflowing values with a fixed HP rating that's 'monster maximum' for a system seems to give a much better estimates, though.  I don't know exactly how to calculate the sum mathematically (there's probably a shortcut formula) but if you limit maximum value to even 50,000 (e.g. World of Synnibarr HPs), the unlimited roll up seems to be converging at an "average" value of only about 9 points. Drop the HP limit to 1024 (D&D esque), and it drops to a value of about 6.

It depends on the number of hit points enemies have. If enemies have one hundred hit points or even one thousand hit points, you'll kill them faster hitting them for one hundred at a time. Back of the envelope, it looks to me like you on average take out enemies faster with a breakpoint at about twenty five thousand hit points. Which very few games are going to give to any enemy.

-Frank

I would go for 100 since if I sit there flipping a coin all night the other players will get pissed off, quit the campaign, and then it won't matter how awesome my character is, I won't be playing them any more.

That's the sort of thing missing from your analysis, what actually happens at the game table. If you're going to ignore what actually happens at the game table you may as well head over to the Forge.

No Kyle, it is you who is overlooking what happens at the game table.  Because if flipping the coin takes any appreciable amount of time that means someone has hit a long sequence of heads, which would probably be the most exciting thing to happen all night.  Thus we can conclude that for the sake of entertaining the table with the hope of statistically improbable successes that turn the tide of battle in an instant (which is more dramatically exciting as well, especially compared to constant damage) the only reasonable choice is to pick ability B.  That is, if we ignore the mathematics, as you suggest.

Quote from: 837204563;416957No Kyle, it is you who is overlooking what happens at the game table.  Because if flipping the coin takes any appreciable amount of time that means someone has hit a long sequence of heads, which would probably be the most exciting thing to happen all night.  Thus we can conclude that for the sake of entertaining the table with the hope of statistically improbable successes that turn the tide of battle in an instant (which is more dramatically exciting as well, especially compared to constant damage) the only reasonable choice is to pick ability B.  That is, if we ignore the mathematics, as you suggest.

In my opinion more than two or three flips is going to feel like a frustratingly long time, and if a hot streak of coin flips is the most exciting thing to happen in your game session, that is in and of itself a problem.

Quote from: Cole;416958In my opinion more than two or three flips is going to feel like a frustratingly long time, and if a hot streak of coin flips is the most exciting thing to happen in your game session, that is in and of itself a problem.

Hey, I share your feelings, but in my experience people get (unjustifiably) exited by crits.  And once when we were playing with a three-consecutive-twenties-is-a-kill rule, and when that extremely improbable event happened people were talking about it for a while.  But really I was just trolling Kyle who was willfully missing the point of a silly mathematical puzzle.