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.
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