Pure vs. Mixed Strategies

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The stadium lights are blinding, and the murmuring of the crowd in the stands is amplified into a deafening roar. Yet, your senses have never been more acute. The date is January 29, 2017, and are playing to win your fifth Australian Open championship title in tennis. Millions of people are watching your every movement from across the world. You are Roger Federer. Where do you place your serves across the net?

We are often unaware of the different dimensions that everyday games are comprised of. Something seemingly as simple as a serve in tennis can be dissected into many parts, both physical and mental. In this article, we are going to explore pure and mixed strategies in game theory, using tennis as an example.

What is a pure strategy?

A pure strategy is an unconditional, defined choice that a person makes in a situation or game. For example, in the game of Rock-Paper-Scissors,if a player would choose to only play scissors for each and every independent trial, regardless of the other player’s strategy, choosing scissors would be the player’s pure strategy. The probability for choosing scissors equal to 1 and all other options (paper and rock) is chosen with the probability of 0. The set of all options (i.e. rock, paper, and scissors) available in this game is known as the strategy set.

What is a mixed strategy?

A mixed strategy is an assignment of probability to all choices in the strategy set. Using the example of Rock-Paper-Scissors, if a person’s probability of employing each pure strategy is equal, then the probability distribution of the strategy set would be 1/3 for each option, or approximately 33%. In other words, a person using a mixed strategy incorporates more than one pure strategy into a game.

The definition of a mixed strategy does not rule out the possibility for an option(s)to never be chosen (eg. pscissors= 0.5, prock = 0.5, ppaper = 0). This means that in a way, a pure strategy can also be considered a mixed strategy at its extreme, with a binary probability assignment (setting one option to 1 and all others equal to 0). For this article, we shall say that pure strategies are not mixed strategies.

In the game of tennis, each point is a zero-sum game with two players (one being the server S, and the other being the returner R). In this scenario, assume each player has two strategies (forehand F, and backhand B). Observe the following hypothetical in the payoff matrix:

The strategies FS or BS are observed for the server when the ball is served to the side of the service box closest to the returner’s forehand or backhand, respectively. For the returner, the strategies FR and BR are observed when the returner moves to the forehand or backhand side to return the serve, respectively. This gives us the payoffs when the returner receives the serve correctly (FS,FR or BS,BR), or incorrectly (FS,BR or BS,FR). The payoffs to each player for every action are given in pure strategy payoffs, as each player is only guaranteed their payoff given the opponent’s strategy is employed 100% of the time. Given these pure strategy payoffs, we can calculate the mixed strategy payoffs by figuring out the probability each strategy is chosen by each player.

So you are Roger. It is apparent to you that a pure strategy would be exploitable. If you serve to the backhand 100% of the time, it would be easy for the opponent to catch on and return from the backhand side more often than the forehand, maximizing his expected payoff. Same goes for the serve to the forehand. But how often should you mix your strategy and serve to each side to minimize your opponent’s chances of winning? Calculating these probabilities would give us our mixed strategy Nash equilibria, or the probabilities that each strategy is used which would minimize the opponent’s expected payoff. In the following article, we will look at how to find mixed strategy Nash equilibria, and how to interpret them.

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