Introduction to Game Theory: a Discovery Approach

Now that we are experts at finding equilibrium pairs, what happens when a game doesn’t have any equilibrium pairs? What should our players do? As we saw in Section 2.6, if there is no equilibrium point then no matter which strategies are played, at least one player wants to switch.

Up until this point we have used the term “strategy” to mean which row or column a player chooses to play. Now we want to refer to how a player plays a repeated game as the player’s strategy. In order to avoid confusion, in repeated games we will define some specific strategies.

For example, if Player 1 always plays Row A, we say she is playing pure strategy A.

For example, if a player plays Row A 40% of the time and Row B 60% of the time, we will say he is playing a (.4, .6) strategy, as we generally use the probability rather than the percent. The probabilities of each strategy will be listed in the same order as the strategies in the matrix.

It is not enough just to determine how often to play a strategy. Suppose Player 1 just alternates rows in Activity 3.1.1. Can Player 2 “out-guess” Player 1? What might be a better way for Player 1 to play?

We’d really like to find a way to determine the best mixed strategy for each player in a repeated game. Let’s start with what we already know: games with equilibrium points. If a game has an equilibrium pair of strategies, would a player want to play a mixed strategy? Recall that a strategy pair is an equilibrium pair if neither player gains by switching strategy.

Thus, if the game has an equilibrium we know that players will play the pure strategies determined by the equilibrium pairs. So let’s get back to thinking about games without equilibrium pairs. If we play such a game once, can we predict the outcome? What about if we repeat the game several times, can we predict the outcome? Think about tossing a coin. If you toss it once, can you predict the outcome? What if you toss it 100 times, can you predict the outcome? Not exactly, but we can say what we expect: if we toss a coin 100 times we expect to have half of the coins turn up heads and half turn up tails. This may not be the actual outcome, but it is a reasonable prediction. Now is a good time to remind yourself about finding the expected value, Definition 2.3.4.

Recall the familiar game of Rock-Paper-Scissors: Rock beats Scissors, Scissors beat Paper, and Paper beats Rock. Using the payoff matrix and experimentation, we will try to determine the best strategy for this game.

Although you may have come up with a good strategy, let’s see if we can’t decide what the “best” strategy should be for Rock-Paper-Scissors.

Hopefully, you concluded that the best strategy against our perfect player would be to play Rock, Paper, Scissors 1/3 of the time each, and to play randomly. We can say that our strategy is to play each option randomly with a probability of 1/3, and call this the Random(1/3, 1/3, 1/3) strategy.

Using ideas about probability and expected value we can more precisely predict the long term payoff for each player when playing a random mixed strategy.

You may have had an idea about the best way to play Rock-Paper-Scissors before working through this section, but how can we find solutions to other games, such as the one in Activity 3.1.8? We don’t want to just use a “guess and check” method. Especially since there are infinitely many possible mixed strategies to try! The rest of the chapter will develop mathematical methods for solving repeated games with no equilibrium point.