## Wednesday, April 20, 2011

### Infinitely Deep RPI

In the previous posting, we looked at extending the "depth" of RPI to an additional level -- that is, including OOOWP in our RPI calculation.  In this posting, we'll look at extending RPI to "infinite" depth.

We can view the RPI for a team as having two components.  The first is a measure of the team's strength, and the second is a measure of it's opponent's strength:
RPI = (Team's Strength) + (Team's Opponents' Strengths)
The standard RPI formula estimates the first term with WP and the second term with OWP and OOWP.  But a better estimate of a team's strength is the RPI itself, so we could use that instead, at least for the opponents:
RPI = (Team's Strength) + (Average of Team's Opponents RPIs)
But now we have a weirdly self-referential equation:  Calculating Duke's RPI will involve calculating UNC's RPI which will involve calculating Duke's RPI, ad infinitum.

We can solve this sort of self-referential formula iteratively -- that is, we can calculate an initial RPI estimate for all the teams and then repeat that process using the latest RPI estimates at each step until we reach an answer.  There are two requirements for this to be successful.  First, we need "bootstrap" values to get ourselves started, and secondly, we need to structure our formula so that the answers converge.

Conveniently for us, Andrew Dolphin has already done the hard work of determining a formula that meets those criteria:
RPI = (WP-0.5) + (Average of Team's Opponents RPIs)
so all we have to do is calculate.  Plugging this formula into our framework and testing gives us:

Predictor    % Correct    MOV Error
1-Bit62.6%14.17
RPI (unw,15+15+70)75.4%11.49
RPI (infinite)74.6%11.33

This gives us slightly better performance (in MOV Error) than the RPI that stops at OOOWP, suggesting that the standard RPI formula depth (to OOWP) is probably sufficient for our purposes.

Next we'll look at one of the most perplexing oddities of RPI.

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