Mathematics of Combat

[tleeuwenburg@gmail.com]

Hi all,

I'm trying to understand the mathematics of combat to a greater extent. So far, I've taken into consideration:
  -- Recalculate damage as the percentage of the opponents health, taking armour damage-reduction into account
  -- Recalculate chance to hit taking opponent's armour into account as necessary
  -- Factor in probability of going first as relevant
  -- Average-turns-to-victory if the opponent does not fight back
  -- The standard deviation of a dice pool

I was considering a combat in WHRP between a starter character and a goblin. I calculated that the character has a 1/3 chance each turn of killing the goblin outright, while the goblin has a 1/5 chance of killing the player. If you had a game where this meant that 2/3 of the time, the character did no damage (and 4/5 times the goblin did no damage), the calculation could be done pretty easily as the sum of a sequence.

The odds of the character winning if they go first is a / (1-r), in this case being
=    (1/3) / ( 1 - (2/3 * 4/5))
=   (1/3) / (1 - 8/15)
=   (1/3) / (7/15)
=   (1/3) * (15/7)
=   15 / 21
=   71.4%

However, characters that *don't* kill eachother are in fact likely to inflict *some* damage some proportion of the time. It strikes me that I should be able to work out the average-time-to-victory which will give me a kind of a ratio which I could use in the equation. But does this make enough sense?

I could use these number to balance all kinds of combat encounters. For example, it would be relatively easy to calculate what game effects a combat-enhancing spell should have relative to the fighting abilities of a non-magical character of the same level. Keeping things mathematically balanced in this way would prevent any one class from becoming massively more effective in game terms. However I really feel that I am still missing some key relationships.

For example, I'm not really sure that I can use the simple calculation I outlined above. I think maybe I need to run some simulations. It's easy (enough) to run a simulation based on average rolls, rolls consistently one std dev below average versus one above from the enemy, etc. But where possible, if it's possible to find hard relationships, that's preferable.

Then there's the issue of optimal strategies and choice.

These thoughts are obviously only half finished, but I figure that is the time to be talking to others. I'd love to hear about any ideas that others have, read any links on the topic so I can learn from those who have gone before, or try to present more information for others.

Cheers,
-Tennessee

[dindenver]

Tennesse,
  I came up with a stat called Combat Threat Level in Lanasia. The idea was, you could use the CTL to figure out if characters were equally matched.
  To come up with the values for this stat, I did the following:
First I laid out all the stats in a spreadsheet. And developed a formula that calculated how long it would take for an average character to defeat an average character.
Then, I made a new row for each stat and copied the original row into each new row.
Then, I increased one stat by one for each row
Then, I made a formula that compared the original row to the current row
Then I used that data to produce the value for each stat on a character's CTL.

  It sounds complicated (and I guess it is a little). but if you want the combats to be balanced, this is a good way to ge close. Of course, there are something that are hard to put into numbers that affect the turns until victory, like initiative and movement. But, with a little work, you can get there.

  Bear in mind that you cannot quantify combos easily. Meaning if two abilities have any sort of synergy, that may not show up in your calculations. An example of this might be things like pun-pun or the old school bag of rats fighter. But most things you can put a value on if you think it through.

  Good luck and let me know if you need any help working out your system.