MCM-2024 Problem C
The probability of a player winning in the game is not entirely determined by the intrinsic ability of the player, otherwise, it would be hard to explain why the results of the game between two players who sometimes seem to have a disparity in strength will be unexpected, and why the results often reverse in the game.
We introduce the Psychological Angular Momentum Model (PAMM) to address the problem. In the model, we make an analogy between the competition of two players and the interaction between two spinning tops. We quantify players’ psychological momentum by the actual angular momentum of the spinning tops, and their intrinsic abilities by their moment of inertia.
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For task 1, our core aim is to visualize and explain which player has a better performance and how the flow changes. We first construct the momentum curve for both players by calculating moment increments and summing them up. Then we use** rotational kinetic energy** to represent players’ possibility of winning. By calculating the Performance Coefficient we get to know who is at better performance, which is correlated to match data.
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For task 2, our goal is to analyze the Random hypothesis of PAM. After calculating the Pearson’s Correlation Coefficient, we reject this hypothesis by visualizing the Correlation Matrix The result turns out that the momentum is not random, instead, is highly related to match features.
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For task 3, By adding or subtracting variables in the model and observing the changes in momentum fluctuations, we obtain the degree of correlation between different variables and momentum. In order to find the indicator of the swing change during games, we compare the points where influential parameters occur and the parameter curve with the momentum curve of the player to see if the density of points or changes in parameter curve will affect the the flow of game, so as find that in general, serving speed and number of unforced errors are most related to momentum flow and can be used as indicators of the swings'
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For task 4, we migrate the pre-trained parameters from other men’s tennis matches and find that our model is still effective, regardless of slight differences in different players’ sensitivity to different parameters. To examine the ability of generalization, we first process the dataset of 2011 ITF Women’s World Tennis Tour, sort out the available parameters (missing four out of nine parameters we used before), and bring in the data. The result shows that the sum of the two players’ momentum is nearly zero. The differences in results suggest that the missing parameters are also essential.
Keywords: Psychological momentum; Angular momentum; Linear regression.