Equalizing Probability Distributions of Different Attacks

[Just Joe]

This was inspired in part by the "Killing Attacks, Again" thread, but it's application is broader. Suppose you don't want the size of an attack (in terms of active points, or in terms of the number of dice normally rolled) to effect its probability distribution. You might want a 9d6 normal attack to be as likely to do its maximum (or minimum) damage as a 3d6 normal or 3d6 killing attack are to do their respective maxima or minima. Consider the following approach:

 

1. Determine the standard effect damage of the attack (alternatively, choose 3.5 / die or 4 / die; each has benefits and limitations).

 

2. Roll 4d6, then subtract 4. The result is a number between 0 and 20. Divide by ten and interpret as a %.

 

3. Multiply the value in part 1 by the value in part 2. That is the damage done. It's somewhere between 0 and 2x the average, with a 4d6 curve regardless of the actual size of the attack.

 

Suppose you want to count STUN and BODY separately. You probably want the two positively correlated, but don't want the correlation to be perfect (i.e., 1). No problem. Take the dice for the STUN calculation above. Subtract one (10%) for every even rolled, add one for every odd rolled. Again, you get somewhere between 0% and 200% of the average/standard damage*. Stun and BODY are positively correlated (though admittedly I don't know what the numerical value of the correlation is).

 

Obviously, I do not recommend this as an official change to Hero System rules, but I do think there's a lot going for it, and I'd be interested in hearing what others have to say about it.

 

* On this proposal, whatever you set the standard as becomes the average, whether 3, 3.5, or even 4.

[Solomon]

Re: Equalizing Probability Distributions of Different Attacks

 

It could speed up damage resolution.

 

I'm not sure what kind of effect the proposed system would have on statistical deviation in damage rolls. My gut feeling is that this would increase randomness, and I don't think that would be an enterely desiderable effect. It would also allow for a possible disconnect such as large damage rolls netting zero damage (before defenses). Maybe a 50% to 150% (2d6+3 times 10%) spread instead of a 0% to 200% would suffice.

[Just Joe]

Re: Equalizing Probability Distributions of Different Attacks

 

It could speed up damage resolution.

I was actually worried that it might slow things down. But you might be right, especially for larger attacks (after one becomes comfortable with the system).

 

 

I'm not sure what kind of effect the proposed system would have on statistical deviation in damage rolls. My gut feeling is that this would increase randomness' date=' and I don't think that would be an enterely desiderable effect.[/quote']When I proposed it, I thought it would have the same statistical deviation as a normal 4d6 attack. In that case, it would reduce the deviation of smaller-dice attacks (e.g., 2d6 KA) and increase the deviation of larger-dice attacks (e.g., 12d6 eb). But now it occurs to me that subtracting 4 and converting to a % increases the deviation. For example, consider a roll of 13 on a 4d6 punch. Using official rules, that is 1 below average, which is 1/14 or about 7% below average. Using my proposal, you would get a result 10% below average. (OK, to complicate matters more, once you round, you're back to 1 below average -- if you use 3.5/die as your base; but I think the deviation will be increased anyway, as the rounding up and rounding down should tend to balance out over a large number of rolls).

 

It would also allow for a possible disconnect such as large damage rolls netting zero damage (before defenses). Maybe a 50% to 150% (2d6+3 times 10%) spread instead of a 0% to 200% would suffice.

I actually like the miniscule chance of zero damage before defenses (the glancing blow, or the laser cutting off a few eyebrows). And a 4d6 punch can already come close to this (4 STUN and 0 BODY). Still, all things considered, I think we can make it better. 2d6+3 is an interesting suggestion. It does not give as wide a range of possible results as most attacks in the official rules (or as 4d6-4), but has a lot of deviation.

 

I'll be back later with the three variations of a 3d6 version I came up with overnight, but I really ought to be doing some work right now.

[Vondy]

Re: Equalizing Probability Distributions of Different Attacks

 

I think you would need a pre-figured chart for each character for it to run smoothly during run time. On the other hand, the math looks solid enough and seems to achieve your aims.

[Cancer]

Re: Equalizing Probability Distributions of Different Attacks

 

Suppose you don't want the size of an attack (in terms of active points' date=' or in terms of the number of dice normally rolled) to effect its probability distribution. You might want a 9d6 normal attack to be as likely to do its maximum (or minimum) damage as a 3d6 normal or 3d6 killing attack are to do their respective maxima or minima. [/quote']

A simpler way to do this would be to roll 3d6 and then apply a multiplier, which equals (# of d6 you want)/3, to do the scaling. You suffer more granularity as you scale to larger numbers of dice, but it has the properties you specify. (If your fiducial distribution is, e.g., 4d6 instead of 3d6, swap in "4"'s for "3"'s.)

[Sean Waters]

Re: Equalizing Probability Distributions of Different Attacks

 

Did you mean 'multiply by 10' instead of 'divide by 10'?

 

I think this method would make average rolls likely: the distribution of damage is on a bell curve, a 4d6 bell curve. It is not a bad method.

 

One thing I've flirted with (although it can substantially change the dynamics of a game) is integrating the roll to hit and damage rolls, like this:

 

3d6 percentages	Overall		Exact	DC>>	1	2	4	8	16		1	2	4	8	16		1	2	4	8
3		0.5		0.5		1	2	4	8	16		0	1	1	3	5		3	5	11	21
4		1.9		1.4		1	3	5	11	21		0	1	2	4	7		3	8	13	29
5		4.6		2.8		2	3	7	13	27		1	1	2	4	9		5	8	19	35
6		9.3		4.6		2	4	8	16	32		1	1	3	5	11		5	11	21	43
7		16.2		6.9		2	5	9	19	37		1	2	3	6	12		5	13	24	51
8		25.9		9.7		3	5	11	21	43		1	2	4	7	14		8	13	29	56
9		37.5		11.6		3	6	12	24	48		1	2	4	8	16		8	16	32	64
10		50.0		12.5		3	7	13	27	53		1	2	4	9	18		8	19	35	72
11		62.5		12.5		4	7	15	29	59		1	2	5	10	20		11	19	40	77
12		74.1		11.6		4	8	16	32	64		1	3	5	11	21		11	21	43	85
13		83.8		9.7		4	9	17	35	69		1	3	6	12	23		11	24	45	93
14		90.7		6.9		5	9	19	37	75		2	3	6	12	25		13	24	51	99
15		95.4		4.6		5	10	20	40	80		2	3	7	13	27		13	27	53	107
16		98.1		2.8		5	11	21	43	85		2	4	7	14	28		13	29	56	115
17		99.5		1.4		6	11	23	45	91		2	4	8	15	30		16	29	61	120
18		100.0		0.5		6	12	24	48	96		2	4	8	16	32		16	32	64	128

		                         Stun of normal/BODY of killing			BODY of normal attacks				Stun of killing damage			




In effect you use the rollxDC/3 as the damage.  Technically you should have it based on the percentage chance of rolling that number, but this gives a																					
more even distribution and works pretty well.  OK some of the BODY figures are not possible, but look at me and tell me if you think I care.																					

 

Then the roll to hit determines the damage done - you need to roll as high as you can but still hit for maximum damage, so you will tend to do less damage to characters who are hard to hit than characters who are easy to hit, which makes sense.

 

Probably.

[Hugh Neilson]

Re: Equalizing Probability Distributions of Different Attacks

 

This was inspired in part by the "Killing Attacks, Again" thread, but it's application is broader. Suppose you don't want the size of an attack (in terms of active points, or in terms of the number of dice normally rolled) to effect its probability distribution. You might want a 9d6 normal attack to be as likely to do its maximum (or minimum) damage as a 3d6 normal or 3d6 killing attack are to do their respective maxima or minima. Consider the following approach:

 

1. Determine the standard effect damage of the attack (alternatively, choose 3.5 / die or 4 / die; each has benefits and limitations).

 

2. Roll 4d6, then subtract 4. The result is a number between 0 and 20. Divide by ten and interpret as a %.

 

3. Multiply the value in part 1 by the value in part 2. That is the damage done. It's somewhere between 0 and 2x the average, with a 4d6 curve regardless of the actual size of the attack.

 

If I roll a 14 (average), I do average damage (say 30 for a 10d6 attack, and 10 BOD). If I roll 4 (1 chance in 1,296) I do no damage and a roll of 24 (same odds) will do max damage (60 STUN, 20 BOD).

 

The bell curve smooths out some volatility, but the odds of max damage on anything rolling more than 4 dice damage under the standard goes up. In my experience most attacks roll more than 4d6 (KA's and Transforms are the exception). The odds of 60 STUN, 20 BOD rolling 10d6 is 1 in over 60 million.

 

You're also going to see a lot more volatility in BOD of normal attacks than would be the case rolling normally.

 

Intuitively, I think this will slow down combat. You have the same number of rolls, and less dice in most cases. However, you'll get chart reference and multiplication to be done. In the end, I suspect it won't change speed by a huge amount.

 

Oh, and if you use 3.5 per die, you effectively increase maximum damage. Using 4 will also boost average damage, so increased defenses will be needed. As this system increases the chance of max damage, I suspect you'll see a rise in defenses on average to avoid being one punched.

[Sean Waters]

Re: Equalizing Probability Distributions of Different Attacks

 

OK, here's a table for you: 4d6 roll, first lot of numbers is stun result by DC, second lot is BODY result by DC.

 

Works off standard damage (so that max damage does not exceed theoretical max) which means that average damage will be lower. I can re-work it for average (3.5) damage if you like.

 

The DCs go 1 then 2 then 4 then 8 etc, so it should be easy to add columns to get any intermediate value.

 

[b]4d6 roll    	1	2	4	8	16		1	2	4	8	16[/b]
4		0	0	0	0	0		0	0	0	0	0
5		0	1	1	2	5		0	0	0	1	2
6		1	1	2	5	10		0	0	1	2	3
7		1	2	4	7	14		0	1	1	2	5
8		1	2	5	10	19		0	1	2	3	6
9		2	3	6	12	24		1	1	2	4	8
10		2	4	7	14	29		1	1	2	5	10
11		2	4	8	17	34		1	1	3	6	11
12		2	5	10	19	38		1	2	3	6	13
13		3	5	11	22	43		1	2	4	7	14
14		3	6	12	24	48		1	2	4	8	16
15		3	7	13	26	53		1	2	4	9	18
16		4	7	14	29	58		1	2	5	10	19
17		4	8	16	31	62		1	3	5	10	21
18		4	8	17	34	67		1	3	6	11	22
19		5	9	18	36	72		2	3	6	12	24
20		5	10	19	38	77		2	3	6	13	26
21		5	10	20	41	82		2	3	7	14	27
22		5	11	22	43	86		2	4	7	14	29
23		6	11	23	46	91		2	4	8	15	30
24		6	12	24	48	96		2	4	8	16	32

[prestidigitator]

Re: Equalizing Probability Distributions of Different Attacks

 

There is an easier (IMO) way to do this. Instead of doing all the math, just count the dice multiple times (e.g. if the dice roll 3, 6, 1 and the Damage/Effect Roll calls for 7d6, simply count this as a roll of 3, 6, 1, 3, 6, 1, 3). If the attack is smaller than the fixed number of dice you choose to roll, you do have to make a decision: take an average or roll fewer dice in this situation? It also calls for some method of distinguishing dice, but that's usually not a problem for a gamer ( ), especially if the number of distinguishable dice doesn't have to be large.

 

Note that even the die to roll for the Stun Multiple can just be treated as another die to add to the series, as can half dice (you just repeat natural die rolls rather than the interpreted results); so in the above example, we could have also take a Stun Multiple from a roll of 6 (the next roll in the repeated progression), resulting in 23 Body and 23x5=115 Stun.

 

I created a thread with the details quite a while back. Maybe I'll try to find it and provide a link.

[Just Joe]

Re: Equalizing Probability Distributions of Different Attacks

 

A simpler way to do this would be to roll 3d6 and then apply a multiplier' date=' which equals (# of d6 you want)/3, to do the scaling. You suffer more granularity as you scale to larger numbers of dice, but it has the properties you specify. (If your fiducial distribution is, e.g., 4d6 instead of 3d6, swap in "4"'s for "3"'s.)[/quote']Good point. I think the calculations would be easier with my method, but the difference might be small enough to be trivial. Your and others' responses made me realize that with a computer or good programmable calculator handy, you could choose the curve you want and still allow for any possible result within the given range.

[Just Joe]

Re: Equalizing Probability Distributions of Different Attacks

 

Did you mean 'multiply by 10' instead of 'divide by 10'?

Well' date=' I [i']shouldn't[/i] have said to interpret the result as a %. I should have just made it a damage multiplier between 0 and 2. But given that I did say to interpret the result as a %, yes you should multiply by 10.

 

One thing I've flirted with (although it can substantially change the dynamics of a game) is integrating the roll to hit and damage rolls' date=' like this: [chart snipped']

 

Then the roll to hit determines the damage done - you need to roll as high as you can but still hit for maximum damage, so you will tend to do less damage to characters who are hard to hit than characters who are easy to hit, which makes sense.

 

Probably.

I like the positive correlation between probability of hitting a damage done, but I don't like the correlation being so strong.

[Just Joe]

Re: Equalizing Probability Distributions of Different Attacks

 

There is an easier (IMO) way to do this. Instead of doing all the math, just count the dice multiple times (e.g. if the dice roll 3, 6, 1 and the Damage/Effect Roll calls for 7d6, simply count this as a roll of 3, 6, 1, 3, 6, 1, 3). If the attack is smaller than the fixed number of dice you choose to roll, you do have to make a decision: take an average or roll fewer dice in this situation? It also calls for some method of distinguishing dice, but that's usually not a problem for a gamer ( ), especially if the number of distinguishable dice doesn't have to be large.

 

Note that even the die to roll for the Stun Multiple can just be treated as another die to add to the series, as can half dice (you just repeat natural die rolls rather than the interpreted results); so in the above example, we could have also take a Stun Multiple from a roll of 6 (the next roll in the repeated progression), resulting in 23 Body and 23x5=115 Stun.

 

I created a thread with the details quite a while back. Maybe I'll try to find it and provide a link.

Neat idea. More elegant than mine, and easier to apply, at least for attacks that normally roll more than 3 dice. I would be a bit concerned about the importance of the first die, particularly for 4d6 and 7d6 attacks. I also would not want to have to distinguish dice; it's easy in principle, but can be a hassle if, for example, one player has a red, a white and a blue die while another has a black, a yellow, and a green. Here's how I would tweak your idea. When you divide the # of dice by three, you'll get a remainder of 0, 1, or 2. If 0, then you use each die an equal # of times -- no problem. If one, then use the middle die an extra time. If two, then use the lowest and the highest each an extra time.

 

However, I don't really like using one of these three dice for STUN X, and I'm not particularly fond of it for normal BODY. I don't think your method makes the STUN of KA's or the BODY of normal attacks distribute nearly as nicely as it does the BODY of large KA's and the STUN of normal attacks. So unless and until you (or someone else) show(s) me a way to generalize the method further, I wouldn't be tempted to try it.

 

BTW, this idea seems vaguely familiar to me. I wonder if I read that earlier thread you mentioned way back when . . .