Introduction to Game Theory: a Discovery Approach

This section requires you to be able to solve “large” systems of equations. You will be using the matrix techniques from Section 3.6. You are encouraged to use technology such as a Desmos or Sage.

As we saw in Section 3.5, an important part of game theory is the process of translating a game to a form that we can analyze.

After playing Undercut with an opponent, try to devise a good strategy.

As we’ve seen before, a payoff matrix can help with analyzing a game.

Let’s assume we are going to play Undercut repeatedly. By the time you and your opponent are done playing, what should it mean to win the game?

Most likely, you said that someone will win the game if they have the most points. In fact, we probably don’t care if the final score is 10-12 or 110-112. In either case, Player 2 wins. Since we will play this game several times, we do care about the point difference. For example, a score of 5-1 would be better for Player 1 than 5-3. So let’s think about the game in terms of the point difference between the players in a given game. This is called the net gain. For example, with score of 5-1, Player 1 would have a net gain of 4.

The method of using net gain to describe the payoffs to each player should be familiar from some of the really early examples where we turned constant-sum payoff vectors into zero-sum vectors. But note that the original form of this game wasn’t even a constant-sum game! What we are really doing here is thinking about our payoffs not as points, but a win or loss relative to our opponent. Now that we have reframed Undercut as a zero-sum game, we can apply our methods for solving the game that we have seen in this chapter.

This game has one additional property that will help simplify our analysis. This game is symmetric, meaning the game looks the same to Players 1 and 2.

Hopefully, you determined that there is not a pure strategy equilibrium for Undercut. Thus, we would like to find a mixed strategy equilibrium. Since this is a \(5 \times 5\) game, we cannot use our graphical method. We will need to rely on our expected value method. We want to decide with what probability we should play each number. Let \(a, b, c, d, e\) be the probabilities with which Player 2 plays 1-5, respectively. For example, if Player 1 plays a pure strategy of 2, then the expected value for Player 1, \(E_1(2)\text{,}\) is \(-3a+0b+5c-2d-3e\text{.}\)

If we use that the game is symmetric, and hence the expected value of the game for each player must be 0 since neither player can have an advantage over the other, we do not need to set the equations equal to each other. We could not use this method earlier since we had no way of knowing the expected value of a general game.

We now have five equations and five unknowns. There is a sixth equation: we know that the probabilities must add up to 1. We can now solve for the equilibrium strategy.

You have now solved a rather complex two-person game. Try playing it with your friends and family. It may be difficult (or even impossible) to play randomly with the exact probabilties. It is also unlikely that your opponent will also be playing the equilibrium strategy, but can you use the solution to assure an advantage, or at least assure that your opponent doensn’t have an advantage? Can you see the difference between an exact theoretical solution to a game, and a practical strategy for playing the game? In the next chapter we will see even more differences between theoretical and practical solutions to a game.