# Game Theory Introduction

Game Theory can help us find the ...

**best**decision in a competitive situation, or**fairest**decision in a cooperative situation

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

It is useful in business, military, sports, finance, personal life, games and more.

Let's have a look at an example to see how Game Theory can help us find the best decision.

## 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 do you advise Casey to do?

*... think about it for a bit ...*

So maybe both Casey and Dana should keep quiet, right? They get only 1 month each that way..

But that outcome is called **unstable.**

Because either side can **do better** by making the "I go free, you get 10 months" decision.

So what to do?

Well, sadly, Casey is better off blaming Dana.

We can see it in a table like this:

Dana | |||

Stay Quiet | Blame Casey | ||

Casey | Stay Quiet | -1, -1 | -10, 0 |

Blame Dana | 0, -10 | -6, -6 |

Casey risks getting 10 months by staying quiet!

So they will most likely get 6 months each.

**Strategy** is a player's action or series of actions to complete the game.

- Can be as simple as "Blame Dana"
- Or something like "Kick left 60% randomly"
- Or more complex like a system to play a multi-player game.

## Nash Equilibrium

The set of player's strategies where Casey and Dana both blame each other **(-6,-6)** is a **Nash Equilibrium**, 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.

In the above example: at **(-6,-6)** Casey is **not better off** by changing to "quiet", and Dana is also **not better off** by changing to "quiet", so this is a Nash Equilibrium.

(If they **both** changed to "quiet" they would both be better off, but we are only looking at individual choices here.)

Another way of viewing it: if **any** player **is** better off changing then it is **not** a Nash Equilibrium.

### 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 |

At (300,300) Jade is **not** better off changing to (200,0). And Page is **not** better of changing to (0,200). So this **is** a Nash Equilibrium.

At (0,200) Jade **is** better off changing to (100,100). So it is **not** a Nash Equilibrium, and we don't need to check any more.

At (200,0) Page **is** better off changing to (100,100). So it is **not** a Nash Equilibrium.

At (100,100) Jade is **not** better off changing to (0,200). And Page is **not** better of changing to (200,0) So this **is** a Nash Equilibrium.

So in this example there are **two** Nash Equilibria!

The previous example shows that players can end up **stuck in a less effective strategy** (100,100) vs (300,300) that 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: 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 |

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

But an important point:

This assumes that players are **rational**.

In the real world some people do stupid things and cause accidents, so we need police to help keep us safe.

## Strictly Dominant

When a player is better off switching away from a choice (no matter what the other player chooses) then we can eliminate that choice.

See Strictly Dominated Strategies to learn more.

## Thinking Clearly

By now you will be getting the idea: we set up a table listing the options for each player, then estimate the benefit (or cost) for each entry, and then use logic to work out our player's best strategy.

## Pure vs Mixed Strategies

What we have seen so far are "Pure Strategies": we end up with a clear choice.

But when chance is involved we may need a "Mixed Strategy": a combination of choices with **probabilities**.

Read Game Theory Mixed Strategies for more.

## Big Subject

This has just been an introduction, there is much more to learn about Game Theory.

Learn more to become a master strategist.