## Saturday, April 22, 2006

### Team and Individual Skill in PvP

In the comments to my last post on PvP, Thoma writes:
Again, it all comes down to you assuming that team skill/number of players = player skill.

This isn't quite what I'm assuming (though if you amend it to *average* player skill, it is correct). I'm assuming that team skill is equal to the sum of the individual players skills.

Is this a good assumption for WoW Battlegrounds? I think it is. For example, if Alice and Beth beat Joe and Harry, but Beth and Carol lose to Joe and Harry, we can infer that Alice is more skilled than Carol. That seems like a reasonable result to me.

Now, there are other games where this assumption does not hold. For example, you could have a game where team skill is equal to the skill of the worst player. Is this a good model for BGs? I do not think so, because it implies that a team with 14 people will always lose to a team of 15 people. (The missing person is essentially a player with zero skill.) And this is not true for BGs. A team with 14 people is at a disadvantage, but the skill of the remaining players can make up for it. So this model is not good.

How about the opposite model? Is team skill equal to the skill of the best player in the Battleground? Again, I don't think this model holds, because it would imply that in a 1v15 match, if the one person was slightly better than each player on the 15-man team, she would win. And I don't think this is true in BGs. The most likely result is that the one person is going to lose, regardless of her skill.

So I think it's pretty clear that team skill in BGs is some combination of the skills of the individual players. And that means that we can infer individual skill by looking at how different teams with that individual perform. Which is what a system like TrueSkill™ does.

Is this model 100% accurate? No. For one thing we occasionally see teams which are greater or less than the sum of their individual parts. A classic example is the 2004 USA Olympic Men's Basketball Team. But for the most part, the model is pretty accurate, especially when players change teams reasonably often, as is the case with WoW Battlegrounds.

As an aside, I think a more accurate model of team skill would be to say that each pair of players on a team has a bond, and that bond may have some skill, positive or negative, associated with it. Two players may play especially well with each other, or two players may play poorly with each other. Then team skill becomes the sum of the skill of the individual players plus the sum of the skill of all bonds formed. In graph theory terms, under the first model only the vertices in a team have weights, but in the second model, some of the weight of the vertices shifts to the edges in the graph.

And individual skill in my first model is not just individual skill, but individual skill plus the sum of the skills in all bonds with that player multiplied by the probability of each bond occurring. But the first model, where only the vertices have weight, is a *lot* easier to deal with, and is probably accurate enough for the purpose of PvP ranks.