Question

Explain Game Theory and draw an example of a matrix. Please phrase things clearly/simply so I can understand, as I am still fairly new to economics. Thank you!!

Answer 1

Game Theory is an extremely wide concept which cannot be fully understood with just one example. I'll try to explain what a payoff matrix is and how to find a Pure Strategy Nash Equilibrium using some examples. I'll also explain what these terms mean.

Say, we are playing the following game:

Two cars are approaching each other from opposite directions. They both have a choice of either going left or right. Let's call the cars Car A and Car B. If Car A goes left and Car B goes right, both cars collide and receive a payoff of (-10) each. Payoff is the return from choosing a particular strategy. If car A goes left and Car B also goes left, both cars receive a payoff of (10) each. If Car B goes left and Car A goes right, both cars receive a payoff of (-10) each. If Car B goes left and Car A also goes right, they end up in wrong lanes and receive a payoff of (0) each.

We can represent this game in the form of a payoff matrix as follows:

Car A/B	Left	Right
Left	(10,10)	(-10,-10)
Right	(-10,-10)	(0,0)

Car B is the column player and Car A is the row player. We have just represented the problem in the form of a table which represents each player's payoff from playing a particular strategy. For example, if we want to know what is the players' payoff when Car B goes left and Car A goes right, we have to look in the first column and second row of the table to find the entry (-10,-10).

The solution of this game would be a strategy which would be followed by both players which would give them the highest payoff out of all the available strategies.

Let's start with Car A. If Car B plays left, Car A has a choice of playing left or right. Playing left gives Car A a payoff of (10) and playing right gives Car A a payoff of (-10). Hence, when Car B plays left, Car A will always play left. This can be done for the situation when Car B plays right. When Car B plays right, Car A will always play right to receive a payoff of (0).

We highlight Car A's best strategies:

Car A/B	Left	Right
Left	(10,10)	(-10,-10)
Right	(-10,-10)	(0,0)

Now, for Car B. If Car A plays left, Car B can play left or right. To maximize payoffs, Car B will play left when Car A plays left. If Car A plays right, Car B will play Right.

Now we highlight Car B's strategies:

Car A/B	Left	Right
Left	(10,10)	(-10,-10)
Right	(-10,-10)	(0,0)

Therefore, this game has two Nash Equilibria. This kind of Nash Equilibria is known as a mixed strategy Nash Equilibria. It requires some additional knowledge of probability to arrive at the final answer. But since these are the very basics of Game Theory, learning the process of finding Nash Equilibria is a good place to start.

Nash Equilibrium is a state where none of the players have an incentive to change their strategies and are receiving the maximum payoff possible given the responses of other players.