Strange math problem

[Urban Ranger]

I got this strange math problem. It seems simple, but I can't think of the solution.

It all starts with a board game called Titan I play once in a while. There are all sorts of monsters in the game, and they fight each other all the time. All monsters have two ratings: power and skill.

Power is the number of dice you roll for it, and skill determines what number do you need on each die to score a hit. An attacker swinging at a defender of the same skill level requires a 4 or higher on a die to hit. Each +/- in the difference in skill level shifts the required number one higher or lower. So a skill level 4 monster attacking one with a skill level 3 needs only a 3 or higher to hit.

All hits are scored against power as damage. When a monster receives damage equal to its power, it dies.

For example, an ogre (6-2) is fighting a centaur (3-4). The ogre rolls 6 dice and needs 6's to hit the centuar (a shift of 2 in favour of the centaur). The centaur rolls 3 dice and needs 2's to hit the ogre. The ogre gets an expected outcome (EO) of one 6 each time, so it will need 3 swings to kill of the centaur. Meanwhile, on 3 rolls (total of 9 dice) the centaur misses 1/6 of the time, so it misses 1 1/2 times, or scoring 7 1/2 hits against the ogre. They both kill each other with the centaur gets a bit of leftover. If you multiply their power and skill, they both are 12. So it seems pretty natural they kill each other.

Strange thing happens when you consider creatures with skill level 1. Consider a hypothetical creature with 12 power and 1 skill level, a 12-1. Pitting it against an ogre, it needs 5's or higher to hit the ogre, while the ogre needs 3 or higher.

This creature can expect 4 hits each turn, so it kills off the ogre in two turns with 2 damage points left over. In two turns, the ogre gets to roll 12 dice with an expected outcome of 8. So this creature kills off the ogre without getting killed.

You can try various combinations with the same total, like a 60-1 against a 30-2 or a 20-3. I realise that any skill level one creature gets a massive bonus against skill level four creatures because they still only need 6's to hit, so don't try those pairings.

I have been trying to find out the cause why skill 1 critters breaks the rule but to no avail. Can you see why?

[MRT144]

god, reminds me of warhammer "to hit" and "to wound" charts

[Wraith]

Looks like it's just a boundary case. Things often act differently on the very edges, especially on the extremely basic algorithms like this one.

Not sure exactly what you're looking for, really. You've got the math all laid out already.
Tying the power and hp together like that is really what causes it. They should have done something extra with the skill to balance things or run a seperate rating for hp.

Wraith
I'm not God. I'm your GM. Consider me your angel of Murphy

[Jaakko]

Looks like to me that Power is simply disproportionate in game effect when compared to Skill, which is aggravated even more by the fact that there are only two stats.

Skill is simply a modifier for a d6 roll, and a difference of +/- 2 is the maximum effect. The system is apparently based on rolling lots of small (d6) dice to even out the spread of results, so there's not a whole lot any restricted modifier can do. Having hit points equal offensive power is another big factor, as Wraith said.

[Urban Ranger]

I am just curious why skill 1 monsters kill other monsters with the same point total. It could be a boundary case, but since rolling a die gives you uniform distribution, I can't see where the boundary is.

[Jaakko]

'Cuz Power always gives you 1 HP and 1/6 of a hit at least, while any Skill in excess of 2 greater than that of the opponent is a waste.

[Urban Ranger]

I thought about that for a bit. It seems to me that you can use a generalised system by using d8's or even d10's.

Consider a case where a 10-1 is fighting a 1-10. Both have the same point total. The 1-10 will need 10 rounds to kill the 10-1, but the 10-1 is expected to get a 10 on 10d10, finishing the combat in 1 round.

Anyway, the system works well enough when the skills are restricted in a narrow range, from 2-4. Is there any sort of mathematical formulation that shows the system breaks with a skill of 1?

[Jaakko]

There might be, but I can't be arsed to think about it.
I should take the probability math course some day...

Just ditch that game and play Carcassonne.

[tinyp3nis]

For example, an ogre (6-2) is fighting a centaur (3-4). The ogre rolls 6 dice and needs 6's to hit the centuar (a shift of 2 in favour of the centaur). The centaur rolls 3 dice and needs 2's to hit the ogre. The ogre gets an expected outcome (EO) of one 6 each time, so it will need 3 swings to kill of the centaur. Meanwhile, on 3 rolls (total of 9 dice) the centaur misses 1/6 of the time, so it misses 1 1/2 times, or scoring 7 1/2 hits against the ogre. They both kill each other with the centaur gets a bit of leftover. If you multiply their power and skill, they both are 12. So it seems pretty natural they kill each other.

Why do you have to multiply? Try add, the mother of all calculations.
From your own example, 60-1 and 30-2. So basically you have 2x 30-1 units, and now you give the other unit 1 skill and the other 30 power... If the +1 skill would double the offensive strenght, they would be equal. But it doesn't double it.

Edit: the add thing doesn't work much better, but the last part of my post makes sense

[Urban Ranger]

Hmm...

You might be onto something. Let me think about it tomorrow.

[SnowFire]

Multipication is misleading. What matters is the difference between skills when comparing two creatures, not their actual values. It's hopefully obvious that a 4-3 vs. a 6-2 is the same match-up as a 4-100 vs. a 6-99, and yet according to your multipication rule, the first matchup is even while the second is hopelessly lopsided towards the second one.

And for those arguing of the importance of power, look at the following pairs which both sum and multiply out to be equal: 13/10 vs. 10/13. The higher skilled one will automatically win each of its rolls, and kill in 1 3/10 of a round (in practice, 2). Even if 1 is an auto-failure, 2 rounds is still an assured kill. The higher powered one will only get 2 1/6 damage per turn, and will thus take 5 turns to kill the other. In this case, skill dominates.

It may seem simple, but the general rule is that if skills are similar (1 or 0 difference), then power dominates. If skills are highly different (2 or more), then it requires ridiculous amounts of power to compete. Anyone who has played Shadowrun can attest to how it works there (you have dice pools and target numbers on d6's. If you roll a 6, you can reroll it and add it to the 6. You can quickly see that for target numbers 5 and less, lots of dice helps a lot. Once the target numbers hit 6 and over, you're lucky to get 2 successes even rolling a lot of dice- and at 12 and over, just forget it, you need 36 dice to average just one 12 or higher).

Fair matchups go something like this (ignoring addition and multipication rules):
If no skill difference: same power.
If skill difference of 1: Say we have power a and power b. Assume their expected times to slaying are the same. We get:
1/3*a*t=b
2/3*b*t=a
Divide on through, we get
a/2b = b/a
a^2 = 2b^2
a = (root 2) b
This can be confirmed rather quickly. A match between (root 2):0 and 1:1 will be finished in 3(root 2)/2 rounds with both of them killing each other at the same time. Obviously, the more you change the ratio in favor of one side, the better the odds.
If skill difference of 2: Same procedure.
1/6*a*t=b
5/6*b*t=a
a^2 = 5b^2
a = (root 5)b
So for fair matches with 2 skill differences, the lesser skilled player should have a little more than double the power of the other. This implies that 10:12 vs. 12:10 will be a walkover for 10:12, but 10:12 vs. 22:10 should be a fair match.

[One_Brow]

As someone who has played Titan a bit, let me add this: within the boudaries of the game (maximal skill four creature is 10, maximal skill three is 12 (IIRC), and maximal skill 2 is 18, no other skill levels exist expect for terrain condition modifiers), the multiplying of skill x power accopmplishes its objective of giving a quick and approximately equal comparison for the purpose of determining experience points. It is not designed to be a rating system per se, and it does not work well in this regard. As the very first post noted, the 3-4 creature (not to menmtion the 4-3 creature) is slightly better than the 6-2. In gerenral, skill four and three creatures yeild slightly fewer experience points for their combat worth when compared to skill two creatures. There are other abilites (range striking, flying, frequency and degree of terrain bonuses) not considered at all.

As for the math "problem", it is the result of using an inherently flawed rating system (since it was never designed to be a rating system).