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.