## Stat for Comparing Teams at the Same Tournament

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cvdwightw
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### Stat for Comparing Teams at the Same Tournament

So, thinking through some of the arguments in the Fred's ranking thread, I came up with the following statistic to describe the performance of teams at the same tournament.
From the stats we compute the average points per correct tossup (APCT) as follows: TP_avg+BC, where TP_avg is the weighted average of a team's tossup points when accounting only for correct buzzes (for instance, (15*powers+10*tens)/(powers+tens)).
We also compute the expected per-packet loss due to negs as 100*neg% (which follows from 20 tossups*neg% negs for -5 points each).

Our statistic is thus: suppose that team 1 and team 2 play a 20-tossup game in which they both perform at their expected performance for the tournament. Team 2 scores one more tossup than team 1, and the game ends in a tie. How many tossups must team 2 score?

This leads to the equation (a-1)*APCT1-(100*Neg%1) = a*APCT2-(100*Neg%2).

We then find a as: a = ((100*deltaNeg%)+(APCT1))/(deltaAPCT), where delta means find the stat for team 1 and subtract that for team 2.

A higher a-value means that teams performed more closely to each other. For instance, using the CBCT stats, Rancho Bernardo vs. North Hollywood had an a-value of 11.43, while RB vs. Uni A had an a-value of 19.39 and Uni A vs NH had an a-value of 19.79. Obviously, if two teams had the exact same APCT, then they would have an infinite a-value.

If we take out negs, then we can simplify the equation and get a different stat, using some algebra. In this case, we assume that all 20 questions in a game between the two teams are answered correctly, and the game ends in a tie. How many more tossups must team 2 score? This is computed as:

a = 20/((2*APCT1/deltaAPCT)-1)

Here, the closer the a-value is to zero, the closer the two teams are to each other.

Essentially, what this stat does in ACF-format is scale bonus conversion quasi-logarithmically. Bonus conversion ratios of 25:18, 20:14, 15:10, 10:6, and 5:2 all yield equivalent adjusted a-values - this means that a team with a bonus conversion of 15 is doing just as well relative to a team with a bonus conversion of 10 that second team is relative to a third team with a bonus conversion of 6. Similarly, a team with a bonus conversion of 20 is doing just as well relative to a team with a bonus conversion of 19 as a team with a bonus conversion of 10 is relative to a team with a bonus conversion of 9.33.

All tossups not converted by a team are either converted by the opponent or not answered correctly at all. This means that every team's tossup performance can be quantified as a vector [a b 1-a-b], where a is the percentage of tossups converted by the team and b is the percentage of tossups converted by the opponent. A team PATH then is akin to a/(1-b). We can compute the percentage of tossups we might expect team 1 to answer as a1 = p1(1-p2)/(1-p1p2) and team 2 as a2 = p2(1-p1)/(1-p1p2). This ought to mean that there is some amount of dead tossups.

Using this method, we can presumably use the formulas TP1 = a1*(APCT1) and TP2 = a2*(APCT2) to predict what might happen if teams 1 and 2 played each other and remained at performance. In fact, logically, we should be able to do this for teams playing on the same packet set with different field strengths, though the predictive power is probably much less in that case. The caveat is that this does not include powers (that's a much more complicated equation that I don't feel like deriving right now) and doesn't fully take into account negs (only negs that aren't rebounded).
Dwight Wynne
socalquizbowl.org
UC Irvine 2008-2013; UCLA 2004-2007; Capistrano Valley High School 2000-2003

"It's a competition, but it's not a sport. On a scale, if football is a 10, then rowing would be a two. One would be Quiz Bowl." --Matt Birk on rowing, SI On Campus, 10/21/03

"If you were my teammate, I would have tossed your ass out the door so fast you'd be emitting Cerenkov radiation, but I'm not classy like Dwight." --Jerry

cvdwightw
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### Re: Stat for Comparing Teams at the Same Tournament

I am still in office hours and everyone who had a question has gone already, so I derived the formula for doing this with powers:

Let p1,P and p2,P be the PATH-equivalent for powers for teams 1 and 2. This is calculated as Team_Powers/(1-Opponent_Powers).
Let p1,T and p2,T be the PATH-equivalent for regular tossups for teams 1 and 2. This is calculated as Team_Tens/(1-Opponent_Tens-Opponent_Powers-Team_Powers).

Note that dead tossups and tossups answered by opponent are treated the same regardless of if a team negged.

Let a1,P and a2,P be the expected percentage of powers earned by team 1 and team 2 if they played each other. This is given by:

a1,P = p1,P*(1-p2,P)/(1-p1,P*p2,P) and a2,P = p2,P*(1-p1,P)/(1-p1,P*p2,P)

Let a1,T and a2,T be the expected percentage of ten-point tossups earned by team 1 and team 2 if they played each other. This is given by:

a1,T = p1,T*Q*(1-p2,T)/(1-p1,T*p2,T) and a2,T = p2,T*Q*(1-p1,T)/(1-p1,T*p2,T)

where Q = (1-p1,P-p2,P)/(1-p1,P*p2,P).

We can do a similar thing for negs (since, in all of my quizbowl models, negs are independent of answering a tossup, and in this one only the end result - dead or answered - matters, not whether it was negged) with

a1,N = p1,N*(1-p2,N)/(1-p1,N*p2,N) and a2,N = p2,N*(1-p1,N)/(1-p1,N*p2,N)

Then we can estimate the expected score per tossup heard to be:

Team 1: 15*a1,P+10*a1,T-5*a1,N+BC1*(a1,P+a1,T)
Team 2: 15*a2,P+10*a2,T-5*a2,N+BC2*(a2,P+a2,T)
[we can multiply team 1 and team 2's scores by some number of tossups to get an expected game score]

The obvious next step is to figure out what to do with this statistic. If it's just a cool trick to idly pass the time away, then nothing. If it's intended to be used as modeling for ranking teams playing tournaments with the same packet set and different fields (e.g. yet another S-value), then it needs some model validation. If it's intended to be used as modeling for ranking teams playing tournaments with wildly different packet sets and fields that sometimes overlap (e.g. Fred's rankings), then it needs some more work to do proper weighting for different tournaments.
Dwight Wynne
socalquizbowl.org
UC Irvine 2008-2013; UCLA 2004-2007; Capistrano Valley High School 2000-2003

"It's a competition, but it's not a sport. On a scale, if football is a 10, then rowing would be a two. One would be Quiz Bowl." --Matt Birk on rowing, SI On Campus, 10/21/03

"If you were my teammate, I would have tossed your ass out the door so fast you'd be emitting Cerenkov radiation, but I'm not classy like Dwight." --Jerry