Basketball – A Game of Geometry

In a previous post, I described the most optimal offensive strategy for the Knicks based on developing relevant joint probability density functions.

In this post, I attempt a solution to the following problem:

Given 5 players on the court, how can one determine (x,y) coordinates for each player such that the spacing / distance between each player is maximized. Thus, mathematically providing a solution in which the arrangement of these 5 players is optimal from an offensive strategy standpoint. The idea is that such an arrangement of these 5 players will always stretch the defense to the maximum.

The problem is then stated as follows. Let (x_i, y_i) be the x and y coordinates of player i on the court. We wish to solve:


Problems of this type are known as multi-objective optimization problems, and in general are quite difficult to solve. Note that in setting up the coordinate system for this problem, we have for convenience placed the basket at (x,y) = (0,0), i.e., at the origin.

Now, for solving this problem we used the Non-dominated Sorting Genetic Algorithm-II (NSGA-II) in the MCO package in R.

In general, what I found were that there are many possible solutions to this problem, all of which are Pareto optimal. Here are some of these results.





Here are some more plots of of player coordinates clearly showing the origin point (which as mentioned earlier, is the location of the basket):

Each plot above shows the x-y coordinates of players on the floor such that the distance between them is a maximum. Thus, these are some possible configurations of 5 players on the floor where the defense of the opposing team would be stretched to a maximum. What is even more interesting is that in each solution displayed above, and indeed, each numerical solution we found that is not displayed here, there is at least one triangle formation. It can therefore be said that the triangle offense is amongst the most optimal offensive strategies that produces maximum spacing of offensive players while simultaneously stretching the defense to a maximum as well. Here is more on the unpredictability of the triangle offense and its structure. 

Based on these coordinates, we obtained the following distance matrices showing the maximum / optimal possible distance between player i and player j:


Above, we show 5 possible distance matrices out of the several generated for brevity. So, one can see that looking at the fifth matrix for example, players are at a maximum and optimal distance from each other if for example the distance between player 1 and 2 is 9.96 feet, while the distance between player 3 and 4 is 18.703 feet, while the distance between player 4 and 5 is 4.96 feet, and so on.

The Most Optimal Strategy for the Knicks

In a previous article, I showed how one could use data in combination with advanced probability techniques to determine the optimal shot / court positions for LeBron James. I decided to use this algorithm on the Knicks’ starting 5, and obtained the following joint probability density contour plots:

One sees that the Knicks offensive strategy is optimal if and only if players gets shots as close to the basket as possible. If this is the case, the players have a high probability of making shots even if defenders are playing them tightly. This means that the Knicks would be served best by driving in the paint, posting up, and Porzingis NOT attempting a multitude of three point shots.

By the way, a lot of people are convinced nowadays that someone like Porzingis attempting 3’s is a sign of a good offense, as it is an optimal way to space the floor. I am not convinced of this. Spacing the floor geometrically translates to a multi-objective nonlinear optimization problem. In particular, let (x_i, y_i) represent the (x-y)-coordinates of a player on the floor. Spreading the floor means one must maximize (simultaneously) each element of the following distance metric:


subject to -14 \leq x_i \leq 14, 0 \leq y_i \leq 23.75. While a player attempting 3-point shots may be one way to solve this problem, I am not convinced that it is a unique solution to this optimization problem. In fact, I am convinced that there are a multiple of solutions to this optimization problem.

This solution is slightly simpler if one realizes that the metric above is symmetric, so that there are only 11 independent components.

Analyzing Lebron James’ Offensive Play

Where is Lebron James most effective on the court?

Based on 2015-2016 data, we obtained from the following data which tracks Lebron’s FG% based on defender distance:


From, we then obtained data of Lebron’s FG% based on his shot distance from the basket:


Based on this data, we generated tens of thousands of sample data points to perform a Monte Carlo simulation to obtain relevant probability density functions. We found that the joint PDF was a very lengthy expression(!):


Graphically, this is:


A contour plot of the joint PDF was computed to be:


From this information, we can compute where/when LeBron has the highest probability of making a shot. Numerically, we found that the maximum probability occurs when Lebron’s defender is 0.829988 feet away, while Lebron is 1.59378 feet away from the basket. What is interesting is that this analysis shows that defending Lebron tightly doesn’t seem to be an effective strategy if his shot distance is within 5 feet of the basket. It is only an effective strategy further than 5 feet away from the basket. Therefore, opposing teams have the best chance at stopping Lebron from scoring by playing him tightly and forcing him as far away from the basket as possible.


The Relationship Between The Electoral College and Popular Vote

An interesting machine learning problem: Can one figure out the relationship between the popular vote margin, voter turnout, and the percentage of electoral college votes a candidate wins? Going back to the election of John Quincy Adams, the raw data looks like this:

Electoral College Party Popular vote  Margin (%)


Percentage of EC

John Quincy Adams D.-R. -0.1044 0.27 0.3218
Andrew Jackson Dem. 0.1225 0.58 0.68
Andrew Jackson Dem. 0.1781 0.55 0.7657
Martin Van Buren Dem. 0.14 0.58 0.5782
William Henry Harrison Whig 0.0605 0.80 0.7959
James Polk Dem. 0.0145 0.79 0.6182
Zachary Taylor Whig 0.0479 0.73 0.5621
Franklin Pierce Dem. 0.0695 0.70 0.8581
James Buchanan Dem. 0.12 0.79 0.5878
Abraham Lincoln Rep. 0.1013 0.81 0.5941
Abraham Lincoln Rep. 0.1008 0.74 0.9099
Ulysses Grant Rep. 0.0532 0.78 0.7279
Ulysses Grant Rep. 0.12 0.71 0.8195
Rutherford Hayes Rep. -0.03 0.82 0.5014
James Garfield Rep. 0.0009 0.79 0.5799
Grover Cleveland Dem. 0.0057 0.78 0.5461
Benjamin Harrison Rep. -0.0083 0.79 0.58
Grover Cleveland Dem. 0.0301 0.75 0.6239
William McKinley Rep. 0.0431 0.79 0.6063
William McKinley Rep. 0.0612 0.73 0.6532
Theodore Roosevelt Rep. 0.1883 0.65 0.7059
William Taft Rep. 0.0853 0.65 0.6646
Woodrow Wilson Dem. 0.1444 0.59 0.8192
Woodrow Wilson Dem. 0.0312 0.62 0.5217
Warren Harding Rep. 0.2617 0.49 0.7608
Calvin Coolidge Rep. 0.2522 0.49 0.7194
Herbert Hoover Rep. 0.1741 0.57 0.8362
Franklin Roosevelt Dem. 0.1776 0.57 0.8889
Franklin Roosevelt Dem. 0.2426 0.61 0.9849
Franklin Roosevelt Dem. 0.0996 0.63 0.8456
Franklin Roosevelt Dem. 0.08 0.56 0.8136
Harry Truman Dem. 0.0448 0.53 0.5706
Dwight Eisenhower Rep. 0.1085 0.63 0.8324
Dwight Eisenhower Rep. 0.15 0.61 0.8606
John Kennedy Dem. 0.0017 0.6277 0.5642
Lyndon Johnson Dem. 0.2258 0.6192 0.9033
Richard Nixon Rep. 0.01 0.6084 0.5595
Richard Nixon Rep. 0.2315 0.5521 0.9665
Jimmy Carter Dem. 0.0206 0.5355 0.55
Ronald Reagan Rep. 0.0974 0.5256 0.9089
Ronald Reagan Rep. 0.1821 0.5311 0.9758
George H. W. Bush Rep. 0.0772 0.5015 0.7918
Bill Clinton Dem. 0.0556 0.5523 0.6877
Bill Clinton Dem. 0.0851 0.4908 0.7045
George W. Bush Rep. -0.0051 0.51 0.5037
George W. Bush Rep. 0.0246 0.5527 0.5316
Barack Obama Dem. 0.0727 0.5823 0.6784
Barack Obama Dem. 0.0386 0.5487 0.6171

Clearly, the percentage of electoral college votes a candidate depends nonlinearly on the voter turnout percentage and popular vote margin (%) as this non-parametric regression shows:


We therefore chose to perform a nonlinear regression using neural networks, for which our structure was:


As is turns out, this simple neural network structure with one hidden layer gave the lowest test error, which was 0.002496419 in this case.

Now, looking at the most recent national polls for the upcoming election, we see that Hillary Clinton has a 6.1% lead in the popular vote. Our neural network model then predicts the following:

Simulation Popular Vote Margin Percentage of Voter Turnout Predicted Percentage of Electoral College Votes (+/- 0.04996417)
1 0.061 0.30 0.6607371
2 0.061 0.35 0.6647464
3 0.061 0.40 0.6687115
4 0.061 0.45 0.6726314
5 0.061 0.50 0.6765048
6 0.061 0.55 0.6803307
7 0.061 0.60 0.6841083
8 0.061 0.65 0.6878366
9 0.061 0.70 0.6915149
10 0.061 0.75 0.6951424

One sees that even for an extremely low voter turnout (30%), at this point Hillary Clinton can expect to win the Electoral College by a margin of 61.078% to 71.07013%, or 328 to 382 electoral college votes. Therefore, what seems like a relatively small lead in the popular vote (6.1%) translates according to this neural network model into a large margin of victory in the electoral college.

One can see that the predicted percentage of electoral college votes really depends on popular vote margin and voter turnout. For example, if we reduce the popular vote margin to 1%, the results are less promising for the leading candidate:

Pop.Vote Margin Voter Turnout % E.C. % Win E.C% Win Best Case E.C.% Win Worst Case
0.01 0.30 0.5182854 0.4675000 0.5690708
0.01 0.35 0.5244157 0.4736303 0.5752011
0.01 0.40 0.5305820 0.4797967 0.5813674
0.01 0.45 0.5367790 0.4859937 0.5875644
0.01 0.50 0.5430013 0.4922160 0.5937867
0.01 0.55 0.5492434 0.4984580 0.6000287
0.01 0.60 0.5554995 0.5047141 0.6062849
0.01 0.65 0.5617642 0.5109788 0.6125496
0.01 0.70 0.5680317 0.5172463 0.6188171
0.01 0.75 0.5742963 0.5235109 0.6250817

One sees that if the popular vote margin is just 1% for the leading candidate, that candidate is not in the clear unless the popular vote exceeds 60%.