# Game theory: Mixed Strategies and Nash Equilibrium

So I've recently become interested in game theory, and I've visited this site to help me understand what exactly game theory is and the applications of it.

In the lesson, they use an example of tennis and one of the players is at the net and the other is at the baseline. The player at the baseline is determining where to hit her shot: right or left, and where the volleyer will move left, or right. They called the probability that the volleyer will volley to the left is q, and to the right is 1- q.

They then state that in order to determine the nash equilibrium, the "expected payoffs" must be equal? Could someone please explain to me why? This pdf may also be of help to you. This pdf may help as well.

I also have another question. I'm thinking about using game theory in my science fair project to determine where tennis players should hit their balls (right, left, middle, short, or deep) based on the data from the two players facing one another in order for one of them to win the point. If the data was already given, would it be possible for you to conduct this experiment using game theory?

My try was to use the knowledge that I had learned up above but I didn't know how to handle multiple cases, in the example provided you only had to deal with 2 cases.

Thanks in advance to anyone who can help.

• Why the downvote? – Varun Iyer Jul 16 '14 at 18:28
• Some people give downvote without mentioning their reason(s)! :| It would be better if they tell why? – Mohammad Khosravi Jul 16 '14 at 18:30
• @MohammadKhosravi I understand. Could you help me :). – Varun Iyer Jul 16 '14 at 18:38
• Nice piece of mathematical research!!! – Bourbaki is ALIVE Sep 9 '14 at 17:22

I think you already understand why the players need to take expectations. Now why do the expectations need to be equal? The idea is, if there was one strategy which gave you strictly higher expected payoff, you would just stick to playing that strategy, instead of randomizing between 2 or more strategies, right? So, the only reason that might prompt you to play a mixed strategy is when all strategies give equal expected payoff. $\\$ Also, you can obviously extend this to randomizing over 3 or more strategies. Do you know about the previous applications of game theory in sports? I would encourage you to do some research into that before embarking into the project. For example, https://econ.duke.edu/uploads/assets/dje/2006_Symp/Wiles.pdf