March Madness Optimization

This is the full code used for the Shiny march madness app. You can run the code yourself to customize more 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. The data for the shiny app is in the Shiny folder but it is just less customizable. Below I explain the methodology.

1. Projection and Simulation

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

2. Bracket-Pool Simulation

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, while others seem to be undervalued in the pool relative to their projection.

3. Optimization

Finally, you can apply your scoring to get finishes for each bracket in the pool compared to eachother across the simulations. 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

Using this system allows you optimize the brackets you enter for march madness. In addition, it allows you to change the scoring system, pool sizes, number of brackets entered, and projection system. By testing out the projected finish based on different strategies like point maximization, or point maximization in R1-2 only, or other ideas, you can get an idea of how you should balance expected point maximization with being contrarian.