My Strange Senior Research Project

Hello Everybody,

It has been a while, and I have not even posted yet, so I definitely need to update my progress on my senior research project. My project, as you may know, is studying “Research of Game Theoretic Models in Non-Market Games,” with Dr. John Wooders at the Department of Economics. But I am specifically studying minimax strategy in Table Tennis. Even more specifically, I will be studying whether the server in Table Tennis is playing according to a specific mixed strategy that creates a Nash equilibrium.

However, it can be sort of confusing as to whether I am studying Minimax, mixed strategy Nash equilibrium, or both. I am studying both, though because I would say that they are interrelated. As far as I understand it, there are really two kinds of minimax, although you would not formally define minimax that way. The simple definition of minimax is that in a two player zero sum game, one player is trying to maximize potential gain while minimizing potential loss, while trying to minimize potential gain for the other player and maximizing potential loss for that player. Of course, this is quite obvious as basically all I am trying to say is that is any two player game, one player wants to inherent more utility (a higher payoff) than the other person. For those who were in game theory class, you might remember with the football example in which we studied the payoffs (which was the amount of yards which the offense gained in this example) for a team’s offensive strategy to try and predict the result of the next play (offensively and defensively) if both teams played according to minimax. However, when I am studying minimax, I will not simply be studying one point, and therefore I have to incorporate mixed strategies. This is where this second kind of minimax comes in, where a player plays according to a specific mixed strategy to maximize payoffs while minimizing losses. In this way, minimax is very much connected to mixed strategy Nash equilibrium.

Nash equilibrium in general is a game in which one player cannot benefit him/herself by changing his/her strategy unilaterally. In this same way, mixed strategy Nash equilibrium is when a player cannot benefit him/herself by changing his/her’s mixed strategy unilaterally.

Through all of this, it would make sense that minimax would lead to mixed strategy Nash equilibrium because if one player was to choose a mixed strategy that gave them the highest payoff (assuming a mixed strategy for the other player), that player could not change strategies while inheriting more utility. So, in a way, testing for mixed strategy Nash equilibrium is a way to test for whether players play according to minimax.

Yet how does Table Tennis AKA Ping-Pong connected to any of this? Well, think of a game between a server and a returner. The server has the decision to serve left or right, and that server must mix these serves to give the server an advantage such that the server can win points. The server will try to serve left when the returner is expecting a right serve for example, to try and catch the returner off guard. In this way, the server will try to choose a specific mixed strategy that gives the server the highest chance of winning points, which in this case will be such that the probability of winning for the server will be the same for both serving directions. This is, in a way, a mixed strategy Nash equilibrium.

So, what I am doing is ordering DVDs from the 2010 World Table Tennis Championship Tournament. I will observe every point 28 men’s Table Tennis matches and take of the direction of the serve and who won the point. I am going to find out the winning percentages of serves to find if each serve has equal probabilities for win rates (which could end up being just equal win rates). If this is the case, than Table Tennis players probably play according to minimax. I will, though, have to test the viability of my findings with Karl Pearson’s chi-squared goodness-of-fit test (Pearson Statistic) and find a P-value that is leads to no rejection of findings.

I might even observe recreational play of the Southern Arizona Table Tennis Association if these results pan out to see if the same results are true of recreational ping-pong games.

Signs are encouraging that Table Tennis will work for this experiment. There are very few “in-game” factors that could affect minimax play and judging serve direction is very objective (even more so than tennis).

Dr. Wooders found that players play according to minimax in tennis, and so I will continue with my idea for a project with his help (his studies is where I got the idea from because the studies are very similar). This has never been done before though, so I have no idea how my results will turn out, but it should be very interesting and fun. (Watch Ping-Pong for work :])

All of your projects look excellent and so I wish all of you good luck!

Jayanth AKA Juanito6561