March Madness Simulations

My March Madness NCAA basketball playoff simulation this year followed the schema of the Playoff Fantasy Football simulations.

I used the Sagarin ratings to get information on the teams' expected performance.

The difference between the ratings for two teams is the expected spread and I used the normal distribution with a standard deviation of 8.83 points to get the probability of a team beating another team.

I then simulate the whole tournament as many times as I like using a random number generator and the probability generated above to pick the winner of each matchup.

To make my picks I then compared a particular set of picks to the simulations, typically 1000, to determine the points for that set of picks.

Recall that in the bracket pool in which I participate not only are the rounds weighted 1,2,4,8,16,32 for rounds 1 through 6, but you multiply those points by the seed of the winning team. If Utah State makes it to the final four, for that round you get 8 points times its 12 seed or 96 points.

I compared several selections to each other to try to find the one that had the most points in competition with other selections in my simulations.

Top seed advances wins - 6% of simulations
Top Sagarin points advances wins - 23% of simulations
Top Pomeroy rating advances wins - 25% of simulations
Secret winning picks wins - 47% of simulations


Thus I was at least able to find a set of picks that outperformed Sagarin and Pomeroy in my simulations. I suspect it has something to do with the seed rating. The results reflect the expected outcomes before the start of the tournament but after the play-in games.

A chart (click for larger) showing the expected outcome of a given seed helps explain why lower seeds might yield more potential points than higher seeds. using the simulated results, the top plot shows the expected round that a team will advance to. Round 6 is the championship. The middle plot shows the expected points for a given seed assuming a standard point scheme of 1,2,4,8,16, and 32 points for winning a round. The bottom plot is the one of interest here, it includes the seed rating in the points.

Upper plot: First seeds are expected to do well advancing to the elite eight on average. Next seeds two through four generally make it to the sweet sixteen. Seeds from about 5 through twelve generally make it one round, and that uncertainty is where the fun comes in.

Middle plot: The standard point assignments don't change the expected value of a team much.

Bottom plot: Including the seeds in the expected points really shows how three and four seeds can be worth much more than a one seed. Even more interestingly, the correct 10 through 13th seed can be worth more points than a five through 9 seed, because they do about as well in the tournament, but have more points due to the seed multiplier. That is what makes the seed multiplier a fun bracket pool game.

Beautiful Bracketology

Leonardo Aranda's Bracketology - NCAA 1985 - 2009 is a beautiful rendition of the results of the NCAA March Madness men's basketball tournament for the 25+ years of the 64 team format.



I myself have compiled compiled these statistics over the years (2009, 2009, 2008, 2007, 2007, 2006, 2006) in an attempt to win the NCAA March madness bracket pools in which I have participated and just for the fun of studying the statistics. But I am green with envy as well as another appropriate color with admiration when someone takes data that I have kicked around for years and makes a striking visualization from it.

Again I am Salieri* to some Mozart. I recognize genius and beauty, but I can only produce mediocrity. (*the Amadeus movie version of this story, not the real one)

(via Castro's Favorite Color)

March Madness Simulations - Game winning probability schemes

After my recent success with simulating Playoff Fantasy Football, I wanted to apply that success to a simulation of the NCAA Basketball playoffs known as March Madness. Given the amount of data analysis that I have done over the years (2009, 2009, 2008, 2007, 2007, 2006, 2006) that even enabled me to win one year, I figured that a simulation might help.

My simulation matches up the teams that play in the NCAA bracket and uses one of the schmes below to generate a probability for a Monte Carlo simulation of games between the teams.

Probability scheme 1: Sagarin ratings only

The first simply uses the Sagarin ratings to create a probability of the team 1 winning. Probability = team 1 Sagarin /( Team 1 Sagarin + Team 2 Sagarin). I use the Predictor Sagarin Rating because that is what he suggests for predicting the score and outcome of a game. A random number from 0 to 1 which is less than the probability above means that team 1 wins, otherwise its team 2.



I calculated every team's probability of winning vs every other team and then plotted this vs the difference in seeds. A -15 means a 1 seed played a 16 seed. This scheme results in probabilities that only vary from 58% to about 50% for matchups between seeds with up to 15 difference to even. Unfortunately no 16 seed team has even beaten a number 1 seed so this scheme leave the games too evenly matched and does not reflect the history of outcomes in the tournament.

Simulation results with this scheme show the number of simulations out of 1000 that a given seed was the champion. The actual history is here. The results in the chart show far too high a probability that low seeds are the champion in the tournament in these simulations.

A histogram of the teams with seeds and the number of times they are champions in 10,000 simulations, shows that Kansas is the most likely winner, but the spread of the data even includes the unlikely play in winner at 16 seed as a champion. This simulation is unrealistic.

Probability scheme 2: Seed difference and tournament history only

Another approach is to use the seeds of the team in the tournament. With 25 years or so of data I captured the number of times a favorite beat an underdog based on the seed difference. For instance, never has a 16 seed beaten a 1 seed, while 8 vs. 9 seeds are almost 50/50. I use the data from 25 years of round of 64, round of 32 and round of 16 and then fit a line assuming that even seeds are 50/50 and that a seed difference of 15 (1 vs. 16) will result in a favorite win 99.07% of the time. That represents 1 in 108, though this upset has never occurred in 26 years of data, it will happen someday, and that could be as soon as 1 this year. Thus (26*4+3) wins/(27*4) attempts is 99.07%.

I did not use the fitted line in the curve above because of its unrealistic probabilities at high seed difference. While this approach captures the history, I feel this approach neglects the variation between similar seeded teams as reflected in the Sagarin ratings. Additionally the history shows pretty wide variations in outcome.

Simulation results with this scheme show the number of simulations out of 1000 that a given seed was the champion. These results are more similar to the historical outcomes, but the matchups between evenly seeded teams will be tossups that ignore the differences as determined by the Sagarin ratings.

A histogram of the teams with seeds and the number of times they are champions in 10,000 simulations, shows that Kentucky is the most likely winner, with low seeds favored to be champions, but I fear that it neglects the difference in teams as represented by the Sagarin ratings. This simulation is unrealistic.


Probability scheme 3: Sagarin ratings scaled by seed difference and tournament history

The final approach combines the two by scaling the average of the Sagarin ratings probability by the expected probability due to seeds as predicted by historical performance. Thus we make sure the average for teams. In practice I add the residuals of the line fitted through the Sagarin rating probabilities to the line fitted by setting the 15 difference probability to 99.07% and the even difference to 50%.

Thus the probabilities reflect the historical data with a more realistic and very rare chance of 16 seeds beating 1 seeds but with the Sagarin ratings to sort between evenly matched teams.

Simulation results with this scheme show the number of simulations out of 1000 that a given seed was the champion. The results is similar to the seed difference with history scheme above, but now the Sagarin ratings are included.

A histogram of the teams with seeds and the number of times they are champions in 10,000 simulations, shows that Duke is the most likely winner, and low seeds are still favored as is true historically. This is the simulation scheme we will proceed with.