Game Theory in Sports:
As an athlete, I have always wondered how game theory can be used to explain athletes’ match strategies. I always wanted to plot the outcomes of two-player games on a pay-off matrix, and try and find a Nash Equilibrium. In this post, I will analyze badminton games through the application of academic authors’ Kovash & Levitt’s theory in on game theory in sports.
In their paper “Professionals Do Not Play Minimax: Evidence from Major League Baseball and the National Football League” published in the National Bureau of Economic Research, Kovash et al examined about 3 million Major League pitches from 2002-2006 to conclude that “In the perfect world of game theory, two players locked in a zero-sum contest always make rational choices.” (1) They also use a “minimax” solution, defined as choosing the moves that minimize a player’s loss. However, before we are able to use Kovash et al’s theories in a two-player game, we first must prove that the players are locked in a zero-sum contest, so as to avail application of the theory. I do this using the following playoff matrix for badminton:
| B: Win | B: Loss | |
| A: Win | 0,0 | 1, -1 |
| A: Loss | -1, 1 | 0,0 |
This matrix represents the events of Player A or B winning/losing points in a match. In badminton, ties are impossible. Thus, the result of a win-win or loss-loss tie is a (0,0) payoff for A & B. If A wins a point, B loses a point to A, creating a payoff of (1,-1), and vice versa if B wins a point. We make it so that when a player wins a point, the other loses a point. This is because any player’s win means the other player’s loss. This, as we can see, creates a zero-sum matrix. Now, having proved that, we can apply Kovash & Levitt’s theory to a badminton match.
In the beginning of the match, the relative stakes are low & each player has low knowledge of the other player’s game. Thus, game theory cannot apply here, due to incomplete information.
However, Kovash & Levitt’s theory can apply to the middle & end of the game. This is because as the game progresses toward the end, players make more rational choices, or less risky decisions.
If we analyze Carolina Marin & PV Sindhu’s 2016 Olympic match, we see fewer rational decisions at the beginning of the match. Silly mistakes, reckless shots & lazy, short rallies are commonly seen by both players. However, as the match goes on, the rallies become longer, with “safer” shots by both players.
. There are 2 reasons for this:
1) Information acquisition: After a decent chunk of the match is over, players are able to understand & process information about one another’s playing styles & play their shots accordingly. Also, a pattern gets created in the strategies used, which is based on rational, calculated choices. Thus, game theory can apply.
2) Rational choices: As we approach the middle of the game, players become more rational. Their opponent also expects this rationality, helping them anticipate the shots one would play. For example, if Marin is pushed to the backhand, the rational choice would be to hit a drop, and the riskier choice would be to hit a smash. Marin, in the beginning of the match, may have chosen to hit a smash, but by the middle of the game, she chooses to play the more rational choice, a drop. Sindhu expects rationality and intercepts this shot. Due to less unexpected shots, players can easily reach the shuttle and return the shot, causing longer rallies.
Thus, this analysis of the ranging rationality of players affects the quality and type of game played by athletes.