Nash Equilibrium

Futuristic control room with multiple glowing computer screens

Game Theory can help us find the ...

... where the outcome for each player depends on their decision and the decisions of other players.

The Nash Equilibrium is a point where a game "settles" because no player can improve their outcome by changing their strategy alone.

This happens in the classic "Prisoner's Dilemma".

Prisoner's Dilemma

Casey and Dana are arrested after a burglary. They are in separate rooms and cannot cooperate.

Casey has been told:

  • if you both stay quiet you both get 1 month in prison for trespass
  • if you accuse Dana: you go free, Dana gets 10 months
  • if Dana accuses you: you get 10 months, Dana goes free
  • if you both blame each other you both get 6 months

What is Casey's best move? If Dana stays quiet, Casey should "Blame" (0 is better than 1 month). If Dana blames, Casey should still "Blame" (6 is better than 10 months).

Dana
Stay Quiet
Blame Casey
Casey
Stay Quiet
−1, −1
−10, 0
Blame Dana
0, −10
−6, −6

The outcome (−6,−6) is "stable" because neither person can improve their own sentence by changing their mind. This is the Nash Equilibrium.

Nash Equilibrium

Portrait of mathematician John Nash

The Nash Equilibrium is named after John Nash (the subject of the movie "A Beautiful Mind").

It is when no player is better off by changing only their own strategy.

At (−6,−6) in our example above:

"If I change my mind, I do worse. If you change your mind, you do worse. So, we stay where we are."

So, it is a Nash Equilibrium.

Nash Equilibrium doesn't mean the "best" outcome for the group. As seen above, they would both be better off staying quiet (−1, −1), but their individual incentives pull them toward −6. Nash Equilibrium only looks at individual choices.

Let's see another example.

Example: Jade and Page travel by train to new places to earn money

  • if Jade takes a camera and Page a printer they can take people's portraits and earn $300 each
  • or they can take their own cleaning gear and clean windows for $200 total
  • but they can't carry two lots of things

The strategies look like this:

Page
Printer
Cleaning
Jade
Camera
300, 300
0, 200
Cleaning
200, 0
100, 100

There are two Nash Equilibria here! If they are both cleaning (100,100), neither wants to be the only one to switch, because they would earn $0 while the other keeps cleaning.

The players can end up stuck in a less effective strategy just because it is the current "habit."

Getting stuck in a less effective strategy (100,100) vs (300,300) can be more about habit than anything else.

No Police Needed

One way of thinking about Nash Equilibria is that (for rational players!) no police are needed to keep the rules. The players will naturally "self-police".

Example: The Intersection

Imagine two people arrive at an intersection from different sides.

  • If they both go they crash, with $9,000 worth of damage each
  • If one stops, the other goes with a benefit of $1
  • But if they both stop they will be sitting there a long time and cost them $10
Driver B
Go
Stop
Driver A
Go
−9000, −9000
−1, 0
Stop
0, −1
−10, −10

The Nash Equilibria are (Go, Stop) and (Stop, Go). As long as one person is going, the other is better off stopping to avoid a crash.

In the real world, traffic lights help us "choose" which of these two equilibria to use!

So it is better to stop and wait for the other driver rather than risk a bad time.

But an important point:

This assumes players are rational.

Have a safe day!

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