Chapter 15: Game Theory
A simple introduction to game theory
The whole point of studying game theory is to give us a really good way of thinking about strategic interaction. Game theory allows us to account for the fact that decision makers respond to each other. Of course these aren’t just random responses; they’re purposeful, and intended to be optimal. This is actually a really big deal: the key building block in game theory is the idea of a best response.
Let’s back up a bit. To be able to apply game theory in the first place we need to identify whose strategic interaction we’re modeling: who are the players? We need to specify what they can do: what are the players’ strategies? We need to determine what can happen as a result of the possible strategic choices: What are the payoffs? That is, a game requires we specify the players, strategies, and payoffs. For convenience we may use a game matrix but this isn’t a strict requirement. We’ll use the matrix representation of a game only when it’s useful.
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Here’s an example to help appreciate what a best response is. First we need a game. Suppose we have two players: You and me, playing catch but with a twist. We’re standing across the room, backs to each other. There’s a table in front of each of us. My table has a basketball and a softball. Your table has a softball glove. I have two strategies: I can throw you the basketball or I can throw you the softball. You have two strategies: You can attempt to catch the ball with a glove or you can attempt to catch the ball with two hands. In game theory we like to give short labels to name each strategy. Therefore my strategies are “Basketball” and “Softball” and yours are “Glove” and “Two Hands”.
For each of my two strategies you have a best response: If I throw the basketball your best response is to catch with two hands. If I throw the softball your best response is to catch with a glove. For each of your two strategies I also have a best response: If you are catching with a glove my best response is to throw the softball. If you are catching with two hands my best response is to throw the basketball. In game theory we like to label the possible outcomes using our strategy labels. The four outcomes are therefore (Basketball, Two Hands), (Basketball, Glove), (Softball, Two Hands), and (Softball, Glove). The technical term we use to refer to how we’ve labeled these outcomes is ‘Strategy profile’ meaning the possible outcomes using one strategy from each player.
We have four possible outcomes and we can assign payoffs to each. If I throw the basketball and you catch with two hands or if I throw the softball and you try to catch with the glove, we’re both happy, and for concreteness suppose we each get a payoff of 1. If I throw the basketball and you try to catch with the glove or if I throw the softball and you try to catch with two hands, it’s awkward and doesn’t work so well and suppose we each get a payoff of 0. If we model this using a game matrix it looks like this:
My strategies correspond to the rows and your strategies correspond to the columns. I pick a row and you pick a column and jointly this determines the outcome of our interaction. The payoff numbers follow a common convention that represents the payoffs to the ‘row player’ with the first number in a cell and the payoff to the ‘column player’ with the second number in a cell.
Two of these outcomes are equilibria and two are not. An outcome is an equilibrium, specifically a Nash equilibrium, if we are both mutually best responding. So the two equilibria are (Basketball, Two Hands) and (Softball, Glove). We can verify that both are equilibria because if we’re currently playing the game in this manner neither of us has an individual incentive to switch to our other strategy. If the basketball is in the air and you’ve got two hands ready you’re not going to pick up the glove. Or if I see you with the glove and I’m winding up to throw the softball I’m not going to switch to the basketball.
However, if we were currently in one of the other two outcomes, either (Basketball, Glove) or (Softball, Two Hands), then at least one of us—indeed both of us—has an incentive to switch to a better response. If you see the basketball coming and you’re wearing the softball glove you’ll want to quickly drop the glove to the floor. If I am holding the softball and see you with two hands, I’ll want to switch to the basketball.
If we underline payoffs that correspond to best responses for each player in our game matrix it looks like this:
Continue on to the next section, Chapter 15.1: Solving Matrix Games!
Find the video walk-through here:
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