The **Nash Equilibrium** is the most important concept in game theory. It describes a situation where no player can do better by changing their strategy, as long as the other players keep their strategies unchanged. Let's find it using the Prisoner's Dilemma!
The Prisoner's Dilemma Payoff Matrix
Sentences are shown as (Your Sentence, PC's Sentence). Lower numbers are better.
PC's Choice
Cooperate
Defect
Your Choice
Cooperate
(1 year, 1 year)
(10 years, 0 years)
Defect
(0 years, 10 years)
(5 years, 5 years)
Step 1: Analyze Your Choices (Assuming PC Cooperates)
First, let's assume the PC chooses to Cooperate. We are only looking at the first column.
If the PC Cooperates, what is your best move?
Step 2: Analyze Your Choices (Assuming PC Defects)
Now, let's assume the PC chooses to Defect. We are now only looking at the second column.
If the PC Defects, what is your best move?
Step 3: Finding the Equilibrium
Notice a pattern? No matter what the PC does, your best option is always to Defect. This is called a **Dominant Strategy**.
The PC is also a rational player and has the exact same dominant strategy. Therefore, both players will logically choose to Defect.
The Nash Equilibrium: (Defect, Defect)
The (Defect, Defect) square is the **Nash Equilibrium**. Why? Because if you are both defecting, neither of you can improve your situation by *unilaterally* (on your own) changing your choice.
If you switched to Cooperate (while the PC Defects), your sentence would go from 5 years to 10 years. That's worse.
If the PC switched to Cooperate (while you Defect), their sentence would go from 5 years to 10 years. That's also worse.
Since no one has an incentive to move, this is a stable (but not always optimal!) outcome.