## Tuesday, May 22, 2007

### D&D Statistics

I put this question on yesterday's exam for my Econ-Stat students. I'm curious to see how many people can figure out the answer without having heard the corresponding lesson in class.
The King of Wakovia has two orders of knights at his service: the Order of the Stone and the Order of the Temple. The Stone Knights suffer a higher death rate when fighting dragons that do the Temple Knights. However, when fighting red dragons, the Temple Knights have the higher death rate; and when fighting green dragons, the Temple Knights also have the higher death rate. There are no other kinds of dragons. What do we call this phenomenon, and how is it possible? Be specific.
And if you're an RPG geek, I expect your answer to be even more specific.

August said...

Can the two orders choose their battles? Are the Stone knights willing to fight all the time, while the Temple Knights refuse to come out except on Tuesdays?

I'm out of practice with sort of thing.

Are the populations of the two orders equal?

Andy W said...

Ran said...

(SPOILER WITHIN!)

Clearly the Stone Knights must have a greater tendency to fight the more dangerous kind of dragon, so even though given a specific dragon they're less likely to die fighting it, they have overall a greater likelihood of dying.

Dunno what the technical term for this phenomenon is, but I'd call it "a hypothetical situation where you'd think you can assume independence, but actually the king likes to send better knights to die at the hands of the more dangerous dragons."

Incidentally, depending on the cost of losing a Stone Knight vs. the cost of losing a Temple Knight (Stone Knights presumably being worth more than Temple Knights), the king might be making the wrong choice; he might be better off sending wave after wave of Temple Knights after each dragon, death rate be damned. :-P

Lee said...

This sounds an awful lot like that comparative advantage lesson we all got in Econ where the developing country specialized in certain goods and the developed country in others even though the developed country was more productive at everything... ;-)

Tim Lee said...

I got the same answer ran did. For example, 10 stone nights might have battled red dragons, with 9 of them dying, while only one temple night battled a red dragon and he died. (temples more likely than stones to die against red dragons) Meanwhile, only two knights fought against green dragons. The stone knight lived while the temple knight died. (again, temple knights have a higher death rate) However, overall, 9 out of 11 stone knights have died in total (81%), while 2 out of 3 temple knights have done so (67%). So in the aggregate, stone knights die more.

I think that when I first heard about this paradox, it was presented in the context of medical conditions and hospitals, in which one hospital was more likely to get patients with the deadliest diseases.

Tim Lee said...

Oops, I screwed that up. It should have been:

Stone: 9/10 against red, 1/1 against green, 9/11 total
Temple: 0/1 against red, 1/2 against green, 2/3 total

Tim Lee said...

Crap, I mean:

Stone: 9/10 against red, 0/1 against green, 9/11 total
Temple: 1/1 against red, 1/2 against green, 2/3 total

Glen Whitman said...

Yup, you guys got it. Andy W named the paradox (but is it really paradox once you understand how it works?), Ran explained the principle, and Tim gave a specific example.

And to finish up (although Tim sort of said this), the Stone Knights are obviously fighting proportionately more red dragons than the Temple knights are.

Glen Whitman said...

And Jadagul gets the bonus nerd points for (apparently) knowing that red dragons have more hit dice than green dragons!

Anonymous said...

An analogous line of reasoning explains why a particular surgeon may have a lower than average rate of patient survival and still be the best doctor available. The rate at which his patients successfully recover from any specific type of surgery he performs is above average, but he is the only doctor taking on the high risk cases.

Thomas Blair said...

I'd call it an aggregation error, but I'm sure there's a technical name for it.

Anonymous said...

Isn't this also known as the ecological fallacy? That is, the assumption that within-group correlations should be the same as between-group correlations.

I first encountered it in a real-world example. Someone noticed that Berkeley grad school was rejecting a much higher percentage of female applicants than male applicants. This, naturally, created a stir, and the grad school demanded that every department report its rejection rate for female and male applicants, to identifiy the worst offenders.

Every department reported that it accepted female applicants at a higher rate. But the departments with the highest overall acceptance rates got very few female applicants. The departments with the highest rejection rates got mostly female applicants. And, as far as I know, that was that.

Bob Hawkins