This graph is the chance for a character to win a 3e combat (the y-axis) against an opponent of the same PL, as I vary the relative attack tradeoff of the character whose chance to win the combat the graph shows (the x-axis) and the relative attack tradeoff of his opponent (the colors of the dots). E.g., if a defender has Dodge=Parry=Toughness, then a "relative attack tradeoff" of 2 means the attacker has Attack at PL +2 and Damage at PL -2. Attackers are just using damage, and no one has any hero points. I assume that dazed/hindered/staggered conditions don't impact a character's ability to fight.

Throughout, I'll use "defense bonus" to mean Dodge/Parry bonus, which are assumed to be identical as both are limited to the same amount by Toughness. The linear nature of M&M means that I performed one calculation, which is accurate at any PL. This is based on 500,000 simulations.

Note: relative attack tradeoff means that you take the attacker’s attack tradeoff (where a negative number denotes a damage tradeoff) and subtract the defender’s defense tradeoff (where a negative number denotes a Toughness tradeoff). The term tradeoff should be very familiar to those who played 2e.

Formally, an attack tradeoff is (Attack bonus-Effect rank)/2; a defense tradeoff is (defense bonus - Toughness)/2. So 4 relative attack tradeoff could be an attacker with +14 attack/6 damage and a defender with +10 defense/+10 Toughness, or it could be an attacker with +10 attack/10 damage and a defender with +6 defense/+14 Toughness, and so on. A -2 point attack shift is what we'd also call a 2 point damage shift; a -3 point defense shift we'd also call a 3 point Toughness shift, and so on.

A character wins more often when he has more of a relative damage-shift and his opponent has more of a relative attack shift, over the whole range I’ve shown. Having a 3 or 4 point damage shift relative to the opponent tends to work the best. A 3 point damage shift character wins against a 6-point attack shift opponent about 88% of the time. E.g., this could be a +7 attack/13 damage/+10 Defense/+10 Toughness character against a +16 attack/4 damage/+10 Defense/+10 Toughness opponent.

The intuition for why attack shifts are bad is that when you are substantially attack-shifted, it decreases the chance you hit with the attack and the defender fails the resistance check, and if the defender does fail the resistance check it tends to be by less, as you can see in these results (the link is for 2e, but the principles are analogous in 3e).

I haven't tested extensive results between characters at different PLs. The chance to win, between two characters with no relative tradeoffs (e.g., two "vanilla" characters), for the higher PL character, is as follows (500,000 simulations)

0 PLs up: 50% exactly (A good test case. See below for more)

1 PL up: 72.0%

2 PLs up: 87.3%

3 PLs up: 95.6%

4 PLs up: 99.1%

**Math note:** This wasn't actually 500,000 simulated combats between each of these combatants. I used a method that is easier to do computationally and intuitively should have a lower margin of error. 500,000 actual simulated combats would have a +-99% confidence interval margin of error of no higher than +-0.4%.

Since I assume that dazed/hindered/staggered conditions don't impact a character's ability to fight, I can simply calculate the cumulative distribution function (explanation of this term from a post on 2e) of the number of rounds it takes one character to incapacitate another. From there, each combatant essentially gets a draw from this distribution and you just have to calculate the chance that one is lower than the other, which means winning the fight by incapacitating the other character first (assign each to go first half of the time to resolve ties).

One way to see the advantage of this method in yielding a lower standard error is that it ensures that an identical attacker (the even PLs/no relative tradeoffs case) is predicted to win exactly 50% of the time. So no matter how many simulations I had run, the answer should come out to exactly 50% above (and it does). If you simulated some number of fights between two identical combatants, though, you'd expect to get something a little different from 50%, even with thousands of simulations.

## Bookmarks