Hello, my name is Ovanes Petrosian.

This online course is devoted to the field of Game Theory.

During this course, we will study static games,

dynamic, and differential games.

The static games are the models for processes where agents or

players make their moves

simultaneously and independently of each other.

For dynamic and differential games,

for the simplest case,

we suppose that the players they

make their moves one after another.

In each step, they observe the actions of other agents.

So, they know the information about the game.

Also, for both of this class of games,

we will study zero-sum games,

non-cooperative games and cooperative games.

Zero-sum games are the games where we

have an agent or a player who

tries to maximize his utility and there is

the other player or the agent who is acting against him.

The question is, how to define

the optimal strategies for both of the players,

especially for the player one.

Non-cooperative games are the models of

processes where we have N participants or N agents.

Each of these agent has his own goal or his own utility function,

which of course depends on the strategies

or the actions of all players.

In here, the question is of how these players would behave.

How can we forecast the behavior of these players?

The next class is cooperative games.

Cooperative games could be used in

order to model the cooperative agreements,

where the question is of how to allocate

the joint utility or the joint payoff of all players.

The next thing we can do is we can

define of how the players would cooperate,

which actions or strategies should they choose in order

to achieve cooperation or

in order to sustain cooperative agreement.

But today, we will start with zero-sum games in normal form.

Consider a classical game theoretical example

called Colonel Blotto game.

Suppose that we have a Colonel who has M regiments,

and we have his enemy who has N regiments.

The question is of how to allocate regiments of

Colonel Blotto on two battlefields?

And the same question we have for his enemy.

In this game model,

we suppose that the side that allocates the biggest number

of regiments on one battlefields wins there.

Of course, both sides try to maximize the number

of battlefields they win and most importantly,

we assume that the Colonel Blotto and

his opponent make their

move simultaneously and independently of each other.

So, we suppose that both of these players

do not have spies and cannot get

the information about the other's actions.

In order to construct a mathematical model for this process,

we need to introduce a notion zero-sum game in normal form.

The system Gamma=(X,Y,K)

is called a zero-sum game in normal form

if X and Y are the non-empty set of

strategies of player one and player two correspondingly,

and function K, which is a function defined on the set of

possible realizations of this strategy

is a payoff function of player one.

We will denote x from X as a strategy of player one,

and we will denote y from Y as a strategy of player two,

a pair (x,y) is a strategy profile in the game,

and K(x,y) is a payoff function of player

one as a function of a strategy profile.

Since we consider a zero-sum game,

then the payoff of the second player is equal to

the minus payoff of the first player because

we suppose that the first player tries

to maximize his utility or his payoff,

and the second one tries to minimize it or acts against him.