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Probability and Cumulative Dice Sums

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A Stupid and Strange Way of Looking at Sports Power Ratings that could be Smart and Useful

As I've mentioned previously, a common method used in sports for estimating game outcomes known as log5 can be written \[p = \frac{p_1 q_2}{p_1 q_2+q_1 p_2}\] where \(p_i\) is the fraction of games won by team \(i\) and \(q_i\) is the fraction of games lost by team \(i\). We're assuming that there are no ties. What's the easiest way to derive this estimate? Here's one argument. Assume team \(i\) has a probability \(p_i\) of beating an average team (a team that wins half its games). Now imagine that this means for any given game the team has some "strength" sampled from [0,1] with median \(p_i\) and that the stronger team always wins. Thus, the probability that team 1 beats team 2 is \[ p = \int_0^1 \int_0^1 \! \mathrm{Pr}(p_1 > p_2) \, \mathrm{d} p_1 \mathrm{d} p_2 .\] This looks complicated, but but with probability \(p_1\) team 1 is stronger than an average team and with probability \(p_2\) team 2 is stronger than an average team. From this perspective the log5 estimate is just the Bayesian probability that team 1 will be stronger than an average team while team 2 will be weaker than an average team, conditional on either team 1 being stronger than an average team and team 2 weaker than an average team, or team 1 weaker than an average team and team 2 stronger than an average team. In these cases it's unambiguous which team is stronger. The cases where the strength of both teams is stronger or weaker than an average team (the ambiguous cases) are thus discarded.

How could this be useful? Instead of ignoring the ambiguous outcomes when estimating the outcome probabilities under this "latent strength" model, we could instead determine which probability distributions best fit the outcome distributions for a given league! Furthermore, this allows us to cohesively put a power rating system into a Bayesian framework by assigning to each team a Bayesian prior strength distribution. These priors could either be uninformative or informative using e.g. preseason rankings.

Comments

  1. merrittsDecember 31, 2013 at 5:36 AM

    This is interesting and seems to be closely related to the Bradley-Terry ranking model. I think the missing link in either of these approaches is connecting p_i with underlying cumulative strength of the team's individual players.

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    1. Christopher D. LongDecember 31, 2013 at 11:30 AM

      Yes, a Bayesian strength Bradley-Terry-Luce model would have \[ p= \frac{S_1}{S_1+S_2},\] where \(S_i\) is sampled from some probability distribution. As I've mentioned in a previous article, these approaches may be related by setting \(S_i = \frac{p_i}{q_i}\).

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