In this lesson, what you'll learn exactly 
is what it is that makes up a game.

However, before we do that, we need to 
establish the intuitive notion of what is

known as a set.

For our purposes, when we hear the word 
set, all we need to think of is a bucket.

What does the bucket do?

We'll call this bucket “A.”

Buckets hold things; they are containers.

So for example, the set “A” might have 
inside of it: the numbers 1 and 2.

It may have inside of it the symbol
star.

But most importantly, a bucket can also 
contain other buckets.

So we can say that the set “B” is inside
of the set “A.”

If I want to refer to the items inside 
of “A,” I call them elements.

So for example, I say, Star is an
element of “A” or the bucket

“B” -- the set “B” -- is an 
element of “A.”

Again for our purposes, when we hear the 
word set, all we need to think about

is a container that can possibly have 
other containers. Okay? Good.

Now, we can define a game.

So we define a game,“G,” as it's set -- 
I use curly braces -- to denote the set.

With three elements, “P,” “A,” and “U.”

“P” stands for players. “A” for actions.
And “U” for utility.

Which we also, sometimes, refers to pay 
offs.

I will now discuss each of the term.

So players is very simple.

“P” is also a set that denotes the 
decision makers in a game.

So for example, if two players or two 
people are playing chess, it can be Jose

and Maria. They are playing chess.

However, decision makers don't have to be
human.

For example, robots make decisions.

You can think of an automatic 
self-driving car, making a decision.

So the game can be between robot 1,
robot 2 and robot 3.

More generically, I always or typically 
refer to players as just a set, where the

players are P1, player 1, player 2,
up into player n.

So a game that has “n”players 
is -- surprisingly enough -- just called

an n player game. Okay?

So players, “P” is a set that list the 
decision makers in a game.

Okay?

How about actions?

So “A” is a set with elements, we'll say, 
a1, a2, an.

Where each little a denotes what each
player can do.

So for example, we can think of a 2 player
game.

So players is P1 and P2.

And the action set, or just be, a1 and
a2.

Where a1 says that 
Player 1 -- his move-- can be either “Up”

or “Down”. That's his action.

And player 2 can either go “Left” or 
“Right”.

So the set, “A” of actions tells us what
each player can do in the game.

Finally, Utility.

So “U” the Utility is also a set with 
elements.

U1, U2, Un where each element is actually
a function.

The function takes an 
input -- the Actions -- of all players

and outputs -- the reward -- for 
a player.

So for example, U1 is a 
function.

It may take, it says that if player 1 
plays “Up” and player 2 plays “Left,”

then -- I'm using this example 
here with “Up” and “Left” -- then player

1 earns a reward of two.

However, we can also say that if player 1
plays “Up” and player 2 plays “Right,”

then player 1 earns a reward of: -7.

Each “u” and the set “U” gives the 
Utility function for each player.

So then, we have it.

A game that is set, with Players, 
Actions, and Utilities.

It says, who is making a decision?

What decisions can they make?

And what reward do they get as a result
of the decision of all players?

In the next lesson, we'll learn how to 
represent a game in a simple matrix form,

known as the normal form game.