Quote from: 837204563;416922The 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.

The average isn't the number you want.  This is a highly skewed distribution, you'll likely want the mean as a more valid measure, and that won't be anywhere near infinity, or a hundred I bet.  The average is a worthless number here.

Quote from: 837204563;417153This is literally the classic gambler's fallacy A high result is not more likely after a large amount of attacks than it is on the first attack.  You are, of course, more likely to see a high result in a large sample set than in a small sample set, but this is irrelevant to the expected value.  And so how many times you use the ability is irrelevant to the value of the "St. Petersburg power".  (Buying more lottery tickets means you are more likely to get a winner, but since buying another ticket doesn't affect the expected results of the other tickets it doesn't raise the value of buying tickets.  Similarly, one use of the power doesn't affect any of the others, so how many times you get to use it doesn't change its expected value.)  

No. That's not Gambler's Fallacy. The larger the number of attempts you take, the larger the chances of getting a very large pile of doublings. The random damage pays off if and only if you would get a very large number of doublings on a single attack within the projected number of attacks you would have to make anyway.

Literally and specifically with the flipping method, you get higher damage expectations per attack the more of them you are having to make. So for example, if your opponent has only 3 hit points you have a 1 in 4 chance of eliminating them in 1 attack and a 1 in 4 chance of eliminating them in 3. The effective average damage thus is 1.75 per attack. If your opponent has 5 hit points, you have a 1 in 8 to do it in one hit, a 7 in 16 to finish the job in 2, a 3 in 16 to do it in 3, a 3 in 16 to complete in 4, and a 1 in 16 to finish in 5 hits. For an average real damage of 2.33 per attack. And so on.

The more attacks you expect to make, the more damage you average per attack. Your expected damage per attack is only actually infinite when your target has infinite hit points and you have to attack them an infinite number of times. At any real number of hit points you actually have chances of delivering the goods in various numbers of attacks, meaning that you have an actual finite average effective damage. And the more attacks you'd be expected to make, the higher that average is per attack.

If you were playing a video game and fighting some boss with millions of hit points such that you'd have to click the attack button tens of thousands of times either way, the random damage would pay off. Most of the attacks would seemingly have no effect on the enemy health bar, but a few times you'd see a huge dip from some kind of critical hit and you'd on average have to click less times.

Bottom line: it's not gambler's fallacy. The way the limitations of the problem are set up, you actually do get different expectations and break even points depending on how many times you repeat it.

-Frank

Quote from: Xanther;417161The average isn't the number you want.  This is a highly skewed distribution, you'll likely want the mean as a more valid measure, and that won't be anywhere near infinity, or a hundred I bet.  The average is a worthless number here.

I don't think that "average" even has a precise definition, does it? It looks like he's calculating the expected value (which in this case is an infinite series). I don't think the mean will help any either as you don't have a finite data set. Worse, if you just brute force it with a Monte Carlo, you will get potentially radically different results per run. Fact is, this is just a case where the EV is not a good measure.

Here's what I would care about: what are the repercussions of failure? Because with this method you will fail a lot between huge successes, and if failure means death, then it doesn't matter what would have happened next time.

Quote from: 837204563;417153This is literally the classic gambler's fallacy A high result is not more likely after a large amount of attacks than it is on the first attack.  You are, of course, more likely to see a high result in a large sample set than in a small sample set, but this is irrelevant to the expected value.  And so how many times you use the ability is irrelevant to the value of the "St. Petersburg power".  (Buying more lottery tickets means you are more likely to get a winner, but since buying another ticket doesn't affect the expected results of the other tickets it doesn't raise the value of buying tickets.  Similarly, one use of the power doesn't affect any of the others, so how many times you get to use it doesn't change its expected value.)  

Of course if you reason like this you are open to a sorites paradox ...

The St. Petersburg paradox involves a single gamble, the martingale with an infinite series of bets.  The St. Petersburg paradox is paradoxical because a rational gambler should (in some sense) be willing to pay any amount for a single shot at the game regardless of whether further gambles are allowed.

There's massive confusion on terms and applications here, so I'll try again. Common to analysis in a distribution is to look at the mean (which you are), and the standard deviation (or variance, if you prefer), and the latter has been neglected. In the short term, the mean becomes less and less useful as the standard deviation increases. Yes, the probability of rolling 'infinity' on the first hit is the same as on the 1,000th, but that's NOT how a min-maxer makes his decision, he has to consider how likely it is to roll so high, as well.

A high mean for damage is nice, but it might do no good if the standard deviation is high. In the coin tossing damage case, the standard deviation is also infinite, which makes analysis difficult (and, incidentally, most results in probability theory void). The "high" standard deviation means that you're not getting results near the mean (infinity) much at all.

Suppose, for example, we toss in an extra detail: if you 'deal' less than 1,000 points of damage, then the monster is instead retroactively granted 2^128 temporary hit points.

The mean damage you deal is STILL infinite, so by your reasoning coin-tossing damage is still superior. On the other hand, I hope,  it's painfully obvious that the coin-tossing way of dealing damage is a bad deal, so, by contradiction, I hope I've hinted at the problem:

Mathematically, you're still dealing an expected damage of infinite with each hit, but now slaying a single monster could easily take 2^129 rounds of combat...now we're back to utility.

The mean, by itself, in this particular situation, really doesn't tell the story, and this is above and beyond the fact that monsters never have an infinite number of hit points anyway, making the 'tail end' of your damage a moot point in every single applied circumstance.

Quote from: Kyle Aaron;416947I 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.

This.