This is the full code used for my march madness
optimization app. The app calculates a set of diversified brackets which gives you the best chance of winning in march madness. You can run the code
yourself to customize things such as increasing the number of
simulations, increasing the bracket pool size, changing the projection
model, and more. To run, the order of files is: 1-simulate-tournament,
2-simulate-brackets, 3-calculate payouts, 4-optimize-brackets in that
order. Or you can just use the optimization app. Below I explain the methodology.
The algorithm starts by projecting all the tournament matchups and then simulating the tournament:
head(probs, 20)
## R1 R2 R3 R4 R5 R6
## 1 Villanova 0.984 0.861 0.660 0.464 0.290 0.184
## 1 Virginia 0.969 0.798 0.621 0.425 0.286 0.158
## 2 Duke 0.976 0.824 0.499 0.350 0.190 0.120
## 2 Purdue 0.961 0.711 0.501 0.229 0.127 0.071
## 2 Cincinnati 0.903 0.700 0.472 0.241 0.148 0.065
## 1 Kansas 0.912 0.695 0.511 0.243 0.103 0.059
## 3 Michigan State 0.901 0.705 0.355 0.225 0.117 0.057
## 2 North Carolina 0.975 0.762 0.467 0.250 0.129 0.052
## 4 Gonzaga 0.895 0.646 0.387 0.221 0.107 0.044
## 1 Xavier 0.975 0.674 0.349 0.192 0.080 0.038
## 3 Texas Tech 0.877 0.552 0.231 0.101 0.041 0.018
## 5 Ohio State 0.752 0.296 0.147 0.070 0.028 0.018
## 3 Michigan 0.820 0.492 0.246 0.124 0.051 0.017
## 3 Tennessee 0.861 0.556 0.280 0.109 0.040 0.015
## 5 West Virginia 0.868 0.468 0.155 0.065 0.031 0.013
## 4 Arizona 0.821 0.480 0.135 0.060 0.030 0.011
## 6 Houston 0.662 0.343 0.153 0.054 0.017 0.007
## 5 Kentucky 0.685 0.369 0.120 0.055 0.023 0.006
## 4 Wichita State 0.877 0.497 0.121 0.054 0.019 0.005
## 4 Auburn 0.809 0.459 0.169 0.049 0.013 0.005
Above are the probabilities of teams reaching each round for 2018, 1000
simulations
Then, using ESPN Pick Percentages, you can simulate a pool of brackets.
head(ownership, 20)
## Team_Full R1 R2 R3 R4 R5 R6
## 59 1 Virginia 0.986 0.938 0.645 0.532 0.334 0.172
## 58 1 Villanova 0.992 0.946 0.817 0.624 0.301 0.156
## 29 3 Michigan State 0.975 0.892 0.476 0.330 0.199 0.093
## 14 2 Duke 0.979 0.880 0.472 0.297 0.167 0.093
## 21 1 Kansas 0.976 0.909 0.759 0.279 0.152 0.081
## 36 2 North Carolina 0.986 0.899 0.561 0.347 0.164 0.078
## 28 3 Michigan 0.967 0.794 0.331 0.202 0.085 0.044
## 2 4 Arizona 0.939 0.598 0.220 0.170 0.095 0.043
## 64 1 Xavier 0.979 0.859 0.477 0.187 0.080 0.033
## 9 2 Cincinnati 0.976 0.842 0.549 0.111 0.059 0.030
## 18 4 Gonzaga 0.961 0.727 0.359 0.156 0.053 0.028
## 41 2 Purdue 0.979 0.848 0.602 0.160 0.053 0.022
## 23 5 Kentucky 0.830 0.339 0.096 0.078 0.038 0.014
## 51 3 Tennessee 0.945 0.681 0.267 0.037 0.018 0.011
## 55 3 Texas Tech 0.919 0.562 0.183 0.047 0.018 0.010
## 62 4 Wichita State 0.904 0.455 0.063 0.042 0.018 0.008
## 61 5 West Virginia 0.864 0.469 0.073 0.050 0.015 0.008
## 37 5 Ohio State 0.789 0.220 0.082 0.026 0.013 0.005
## 15 6 Florida 0.803 0.349 0.100 0.021 0.008 0.005
## 57 13 Unc Greensboro 0.039 0.019 0.009 0.006 0.005 0.005
Above are the ownership percentages by round for the pool of brackets, 1000 brackets. You can start to see that certain teams are overvalued in the pool of brackets compared to their projections (ex: Mich St), while others seem to be undervalued in the pool relative to their projection (ex: Villanova).
Finally, you can apply your scoring to get finishes for each bracket in the pool compared to eachother across the simulations. For example, the 2nd bracket generated in the pool of brackets will have different finishes for each simulation, and so by summarizing its finishes I can get its P(90th percentile) or P(95th percentile). Then you can set up an optimization to return the optimal bracket(s) for any number of specifications, Ex: Return 1 Bracket to maximize P(90th percentile). 3 brackets to maximize P(97th), 1 bracket to maximize points, etc. I can also test out other brackets against the pool of 1000 brackets in order to get alternative brackets which may do well but weren’t included in the pool.
Below is the bracket which maximized P(90th percentile) for 2018:
brackets$Prob90<-apply(brackets[, grepl("Percentile", colnames(brackets)) & !grepl("Actual", colnames(brackets))], 1, function(x) sum(x>=.90)/sims)
max(brackets$Prob90) #projected P(90th percentile)
## [1] 0.306
plotBracket(brackets[which.max(brackets$Prob90), 1:63], text.size = .8)
brackets[which.max(brackets$Prob90), c("Percentile.Actual", "Score.Actual")]
## Percentile.Actual Score.Actual
## 261 0.925 1070
The bracket ended up getting in the 92.5th percentile (getting the chamption right will usually get you there). I calculated the 92.5 by comparing the bracket's actual score to the actual scores of the brackets in the bracket pool.
In conclusion, using this system allows you to optimize the brackets you enter for march madness. With the app or the code you can change the scoring system, pool sizes, number of brackets entered, and projection system. Using this system gives you an analytical way to balance expected point maximization with being contrarian.