Introduction to Game Theory: a Discovery Approach

Two candidates, Arnold and Bainbridge, are facing each other in a state election. They have three choices regarding the issue of the speed limit on I-5: they can support raising the speed limit to 70 MPH, they can support keeping the current speed limit, or they can dodge the issue entirely. The next three examples present three different payoff matrices for Arnold and Bainbridge.

Chances are, in each of the exercises above, you were able to determine what each player should do. In particular, if both players play your suggested strategies, there is no reason for either player to change to a different strategy.

For example, consider the game matrix from Example 1.2.4, Table 1.2.5.

You determined that Player 2 should choose to play B, and thus, Player 1 should play B (i.e., we have the strategy pair [B, B]). Why is this an equilibrium pair? If Player 2 plays B, does Player 1 have any reason to change to strategy A? No, she would lose 10 instead of 1. If Player 1 plays B, does Player 2 have any reason to change to strategy A? No, she would gain 0 instead of 1. Thus neither player benefits from changing strategy, and so we say [B, B] is an equilibrium pair.

For now, we can use a “guess and check” method for finding equilibrium pairs. Take each outcome and decide whether either player would prefer to switch. Remember, Player 1 can only choose a different row, and Player 2 can only choose a different column. In our above example there are four outcomes to check: [A, A], [A, B], [B, A], and [B, B]. We already know [B, B] is an equilibrium pair, but let’s check the rest. Suppose the players play [A, A]. Does Player 1 want to switch to B? No, she’d rather get 100 than 0. Does Player 2 want to switch to B? Yes! She’d rather get 10 than \(-100\text{.}\) So [A, A] is NOT an equilibrium pair since a player wants to switch. Now check that for [A, B] Player 1 would want to switch, and for [B, A] both players would want to switch. Thus [A, B] and [B, A] are NOT equilibrium pairs. Now you can try to find equilibrium pairs in any matrix game by just checking each payoff vector to see if one of the players would have wanted to switch to a different strategy.

Generally, when we define a game matrix for a game, our rows and columns will be named with the strategy choices for the players. However, mathematically, we can just think of the matrix of payoff vectors without the row and column labels. In this case, we often identify the payoff vector itself as an equilibrium or equilibrium point. In Table 1.2.5, for example, we would say the payoff vector \((-1, 1)\) is an equilibrium point.

After trying the above examples, do you think every game has an equilibrium pair? Can games have multiple equilibrium pairs?

The next three activities give you a few more games to practice finding equilibrium pairs.

We’ve seen how to describe a zero-sum game and how to find equilibrium pairs. We’ve tried to decide what each player’s strategy should be. Each player may need to consider the strategy of the other player in order to determine his or her best strategy. But we need to be careful, although our intuition can be useful in deciding the best strategy, we’d like to be able to be more precise about finding strategies for each player. We’ll learn some of these tools in the next section.