Fun with math or Why high variance can be a good thing

[padlock]

One thing that may initially seem counter intuitive to a lot of people is how different distributions can have a large effect on damage calculations even when the expected averages are the same.

To demonstrate this, let's take a look at the amount of damage inflicted on a unit with 4 defense when being attacked by figures who have different ways of achieving an average of 2 skulls.

In order from the least variance to the most it would like this:


1. No Variance - Assume a fictitious unit which didn't have to roll any dice and always got 2 auto skulls

Average damage on a figure with 4 defense: .790123


2. Low Variance - Sharwin - 1 auto skull and 2 dice

Average damage on a figure with 4 defense: .864198


3. Normal Variance - A figure who rolls 4 dice

Average damage on a figure with 4 defense: .932099


4. High Variance - Zetacron or a Minion (2 dice with skulls doubled)

Average damage on a figure with 4 defense: 1.061728


5. Very High Variance - A fictitious figure who rolls 1 die and quadruples skulls

Average damage on a figure with 4 defense: 1.333


In all cases, the average damage is 2 skulls, but the figure with very high variance does a whopping 69% more damage then the one with no variance.

What's the point of all of this? I'm not sure, but I thought it was interesting.

[Devo Hulk]

I'd rather have low variance (and hence high precision) in my attacking damage, at least lower than the variance in my opponent's defense. Let them throw blanks first!

[tcglkn]

Very well thought out. Maybe this is why I prefer figures like Zetacron, the Minions, Snipers and Migol. I love taking chances! I would like to see a figure get one attack die that quadrupled, that'd be fun. Of course that sort of power would quickly become over powered if you added height, glyphs or any other attack bonus.

[odbo255]

Originally Posted by tcglkn

Of course that sort of power would quickly become over powered if you added height, glyphs or any other attack bonus.

Sounds like a perfect candidate for a special attack.

[Killometer]

Originally Posted by padlock

5. Very High Variance - A fictitious figure who rolls 1 die and quadruples skulls

4 skulls 50% of the time? That sounds like just my type of unit!

Of course I'm a sucker for high variance units-D20 powers are my favorite abilites in the game.

[padlock]

Thanks for the responses.

A couple more points:

-The difference in average damage between high variance and low variance goes up when the defense goes up relative to the expected number of skulls. Attacking a Raelin backed Q9 with a figure with 2 average skulls is much more successful with a very high variance figure then a low variance figure. At the other extreme, if attacking a hero with zero defense, there would be no difference in average damage calculations.

So does that mean that higher variance always does at least as much average damage as low variance? No, which brings me to the second point.

-The above calculations all assume that the defending figure has at least as much life left as the highest possible attack. Clearly, if attacking a figure with only 2 life left, rolling 4 skulls 50% of the time is a waste.

To sum up, if attacking a figure who has as much life left as your maximum attack, the higher variance figure will do at least as much, and often much more average damage as the low variance figure. The amount of extra damage goes up in proportion to the strength of the defender versus the expected number of skulls.

One more thought, average damage aside, other strategic considerations come into play. There may be times where you'd rather have a higher chance of doing some damage at the expense of a lower average damage amount.

[TnT2]

Nice analysis. But one nit-pick.

the higher variance figure will do at least as much, and often much more average damage as the low variance figure.

Not true. High variance, by it's very definition has a wide swing in both directions.

For example, lets look at your mythical figure that quadruples skulls rolled.

True, he has a 50% chance to roll 4 skulls. And a figure that rolls 4 normal dice only has a 6.25% chance to achieve the same number of skulls.

However, that same mythical figure has a 50% chance to roll ZERO skulls. And again, with 4 normal dice, there is only a 6.25% chance of no skulls.

Having said that, you do caveat your statement with "higher average damage"

Interesting and thought provoking....

[Rich10]

This discussion illustrates why the Stingers are so good. They have variable variance. Against squad figures, it is best to use the normal attack of 3. Against heroes (especially heroes with high defense), it is better to increase the variance of the attack by using Stinger Drain.

[dok]

Good analysis. Note that the results flip when we talk about defense.

[Quintaton16]

Originally Posted by TnT2

Nice analysis. But one nit-pick.

Not true. High variance, by it's very definition has a wide swing in both directions.

Wrong, because what he does and you do not do is take into account defense. Against a figure with four defense, the fictitious quadrupler has a 50.6% chance of no wounds, 4.9% chance of one wound, 14.8% chance of two wounds, 19.8% chance of 3 wounds, and a 9.9% chance of 4 wounds.

The auto 2-skuller, on the other hand, has a 40.7% chance of no wounds, a 39.5% chance of one wound, and a 19.8% chance of two wounds.

What he was saying is average damage, so the better the defense, the better the high-risk one stacks up. at six defense, the chance to wound is the same for both figures, and higher than that it substantially favors high risk. (In other words, the caveat was there for a reason.)

[Kroc]

Originally Posted by TnT2

Nice analysis. But one nit-pick.

Not true. High variance, by it's very definition has a wide swing in both directions.

By definition. But this is a cool post, +rep to you. I think most people don't like high variance for this reason. The highest variance are all d20 powers and those usually have a low power ranking--because in the right situation they may be your only hope/deadly but usually they are just too high risk.

[GreenLanturn]

Originally Posted by Kroc

The highest variance are all d20 powers...

Actually I would argue that the D20 powers generally are very low variance, just in the opposite direction. Mindshackle comes to mind immediately; there is a 95% chance that you will fail that roll. That to me is low variance. Obviously it changes greatly depending on the power; Isamu has a high variance ability because it could go either way easily.

Vary interesting thread.