and the ball turns out and come over here then, you lose.

And this may be, you know, a -five, say, of [unknown] of cost.

Whereas, if he catches it, then you would get a reward of, say, ten.

So, that is the function in that map. The strategy that you choose to number.

Of course, it's determined, in part, by your decision, but also, in part, by

whoever is kicking the ball. So, there comes the coupling of

everybody's strategy. Your strategy and her strategy, together

determine the outcome. Now, we're going to later come back to our

power control and view it as a game. But first, let's take a look at two simple

but very famous examples. Perhaps, by far, the most famous example

of simple game theory is the so-called Prisoner's Dilemma.

It's a two-user game, modeling competition, and each user has two

possible strategies. So, we can plot this in a two-dimensional

space, in a table, on a piece of paper with two columns and two rows, okay.

Let's say, the two rows represent the two strategies for player A, and the two

columns represent the strategies for player B.

And players A and B are suspects of a crime, and the police is going to

investigate into this crime and examine both of them individually.

So, they are going to know the entries in this table later on, but they cannot

communicate, because they are being questioned separately.

And let's say, they have two choices to make.

One is you do not confess, no confess. The other is you confess.

Similarly, B can also say, I don't confess or I confess.

So, there are, altogether, four possible outcomes.

For each of these outcomes, jointly determined by A strategy and B strategy,

there will be a pair of numbers. The first number denotes the payoff to A,

and the second, the payoff to B. So, if they both decide not to confess,

then the police can only charge them with a lesser crime.

And they will each get, say, one year in prison.

Whereas, if they both confess, then, they will be sent to prison for longer terms,

say, three years. Whereas, if one confess, but the other not

confess, then the one that confess will be rewarded and walk away free for

cooperating with the investigation, and the other one will be left with, most

severe penalty of, say, five years in the prison.

And this is a symmetric game so we also have, this cell filled out exactly like

this one. Okay, B confess, then she walks away with

nothing, zero years. And A, do not confess, then she will be

getting five years, okay? So, this defines the game, the Prisoner's

Dilemma game. Because there are two players, each has a

strategy set consisting of two possible strategies.

And this table quantifies the payoff for each player, jointly determined by their

respective strategies. So, let's take a look at this famous

Prisoner's dilemma. Why is it called dilemma?

Because, imagine you are, suspect A and you say, I don't know what B will do.

But suppose, she chooses not to confess, then what should I do?

Well, if I don't confess, I guess my [unknown] is zero so I should pick,

confess. We call this the best response strategy by

player A in response to player B's picking no confession.

But B may also choose to confess. I don't know.

And, in that case, what should I do? Well, I'm basic comparing -five and -three

here, still confess is the better choice relative to -five, -three is better.

Therefore, confess is also the best response strategy if B chooses to confess.

Now, B can only choose between two choices.

So, A would say, hey, I have the same best response strategy no matter what.

That is called a dominant strategy. If it is the best response strategy for

all possible strategies by the other players, in this case, that is confess.

Now, by symmetry B is thinking in exactly the same way.

And B will also pick confess as the dominant strategy.

And therefore, the outcome of this game will be that both A and B will confess and

get three years each. What's wrong with that?

Why is it called dilemma? Because, they could have got, one year

each. If you think about a summation of the

payoffs and you want to, for each i, maximize this quantity, then -two would be

much better than -six, except, A and B cannot collaborate or communicate.

So, in their each strategization process, they end up picking a socially sub optimal

solution. And that's why it's called a dilemma.