## Tuesday, March 22, 2011

### Tourney Projections–How Do I Do That?

Ok, I’ve been asked about my tourney projections. I’m not going to go into all the detail on how my ratings are created. When Jeff Sagarin gives you the details on his, come back to me. But, I can explain how the tournament ratings are done.

Last year, one of the features I added to my ratings was the ability to run Monte Carlo simulations of the games. It’s surprisingly fast, actually, only slowing down my ratings calculations by a minute or two.

Anyway, for the tournament, I simulate every possible game 10,000 times. Right now, for the Sweet 16, that’s pretty easy. 16 teams means that there’s 120 possible games, so I do 1.2 million simulations. It was a little more time consuming when there were 64 teams as that had 2016 possible games, so I had to do over 20 million simulations.

Once that’s done, the probability calculations are fairly straightforward. A team’s chance to win in a given round is the product of their probability of making that round and the sum of the probabilities of beating each possible opponent multiplied by the probability of the opponent making that round.

A couple of examples will make that clearer.

In the next round, Duke plays Arizona. The probability of Duke advancing to Elite Eight is (from above) the probability of Duke making the Sweet 16 multiplied by the product of the probability of Duke beating Arizona and the probability of Arizona making the Sweet 16. Or:

D(EE) = D(S16) * (D(A) * A(S16))

where:

D(EE) = Duke advances to Elite Eight
D(S16) = Duke makes it to Sweet 16
D(A) = Duke beats Arizona
A(S16) = Arizona makes it to Sweet 16

Since we’re already at the Sweet 16 and the opponents are set, everyone’s chances of making this round is 100%. So, simplified:

D(EE) = 1 * (D(A) * 1)
or
D(EE) = D(A)

I ran the simulation 10,000 times. Duke won 7,952 of them, or 79.52%.

So, D(EE) = 79.52%

So, that was easy. But what about Duke’s chances of making it to the Final Four? They will play either Connecticut or San Diego State.

The formula is:

D(FF) = D(EE) * (D(C) * C(EE) + D(SD) * SD(EE))

where
D(FF) = Duke advances to Final Four
D(EE) = Duke makes it to Elite Eight
D(C) = Duke beats UConn
C(EE) = UConn makes it to Elite Eight (beats SDSU)
D(SD) = Duke beats SDSU
SD(EE) = SDSU makes it to Elite Eight (beats UConn)

I won’t bore you with the how to on these again, but
D(C) = .8249
C(EE) = .4672
D(SD) = .7775
SD(EE) = .5328

so

D(FF) = .7952 * (.8249 * .4672 + .7775 * .5328)
D(FF) = .7952 * (.3854 + .4143)
D(FF) = .6359

So, Duke has a 63.59% chance of advancing to the Final Four.

This process is repeated for each team all the way out to the championship.