What happens when we have games without a dominant equilibrium,

without an equilibrium in dominant strategies?

Most games do not have dominated strategies at all.

There is no one strategy that is clearly what should be done.

Players in this case,

they cannot affect the outcome by themselves independently.

So, without a dominant strategy,

the optimal decision for what you have to do every time

depends also on what other people in this game are doing,

doesn't depend only on you.

So that is, now you are not only concerned about your actions,

you're also concerned about your actions and the reactions of the other players.

How are we going to predict the outcome of the game in this case?

Sounds very complicated.

Till the 1950s, the only thing we had in economics,

analyzing equilibrium was the dominant equilibrium.

After the 50s, mathematician John Nash came up with a new version of equilibrium.

That was brilliant and solved a lot of problems.

So that today, it affected the economic theory as much as nothing else ever will.

The Nash equilibrium is a more general but weaker equilibrium concept. How does it work?

A Nash equilibrium is a combination of strategies from which,

no player has an incentive to deviate unilaterally.

The reason that I have underlined the word unilaterally,

is because it's the most important word in this definition,

and distinguishes this equilibrium from the previous equilibrium.

First of all, word unilaterally means,

to be the only one.

So, it's a combination of strategies from which,

no player has any incentive to deviate by themself,

alone, to be the only one to deviate,

to deviate without the other player, to deviate.

A definition for the dominant equilibrium is very similar.

It's a combination of strategies from which,

no player has an incentive to deviate, period.

In the Nash equilibrium,

is a combination of strategies from which,

no player has an incentive to deviate alone.

And we'll see what the big difference this makes.

At the Nash equilibrium,

each player is doing their best it can,

given what they expect that their opponents will do.

So, your expectations, how you read others play a huge role here.

It doesn't matter what others will do,

it matters to be able to predict what the others will do,

so the Nash equilibrium,

Once you are the Nash equilibrium,

no player wants to deviate alone from there.

If you are the only one deviating from this equilibrium,

you will end up at the worse state than at the Nash equilibrium.

So, no one wants to deviate from Nash equilibrium liberal alone.

In this sense, the Nash equilibrium is a quasi-stable notion of equilibrium.

This means that, both together,

you might want to deviate from this equilibrium.

So, it's not stable if both players

deviate or all players deviate if you have from more than two.

But if you want to deviate alone,

this is not going to be a good idea.

So, Nash equilibrium will be stable in the sense

that no one wants to deviate alone from there.

How do we derive the Nash equilibrium?

Some cases, it might seem complicated.

Like consider this game,

which now is elevated to be a three by three game,

three strategies for each player.

And what you see in this game, the first thing,

you have to look for dominated strategies,

and there is none.

I know because I've made this game, so there is none.

And the second, is that we have to check if

there are deviation tendencies from each cell.

So, I will show you now a way that this will make

your job to finding the Nash equilibrium's a piece of cake.

I can be giving you tables,

and you will find the Nash equilibrium within 30 seconds.

Here's how it works.

You have to take the alternative strategies of each player,

and examine which one is the best response to each of the strategies of the other player.

Lets start. I will ask the same question nine different times,

and now we'll underline the best response.

All right, so if player two place left.

What player one wants to do?

Does he want zero, four, or three?

Four, I underline four.

If player two plays middle,

what player one wants?

Four, zero, or three?

Four, I underline four.

If player two plays right,

what player one wants?

One, one, or two? Two, of course.

So, underline two. Now, let's do the same for player two.

If player one plays up,

what player two wants?

Five, zero, or three?

Five, of course. If player one plays center,

what player two wants?

Four, five, or three?

Five. If player one plays down,

what player 2 wants to do?

Once one, one, or two?

Probably two, and we finished with this process.

Now, you look at your cells,

and the cells that they have both players underlined, a Nash equilibria.

That easy and that fast.

So first, you take the columns for the first layer,

then you take their rows of the second player,

you underline what's the result,

then you look at your matrix,

and you have the Nash equilibrium.

They only tell that no one wants to deviate is DR,

down right. Why is that?

Because if you have read and you deviate from down right.

So, player two plays the right and you want to deviate alone,

then you will play center, you will get one.

You will play up, you will get one, yourself.

You don't want to deviate by yourself.

If you have the green, same situation happens.

If you deviate to middle or left, again,

you're going to get one,

or one, you don't want to do that.

So, no one wants to deviate alone.

Now, what if they had an agreement to deviate together?

They end up at down right,

and they say, "Hey, mate.

This doesn't work very well because there's another cell that we both get four and four.

Why don't we go there?"

So, this is a better cell,

so the Nash equilibrium does not have to be the social optimum.

The social optimum in our case is a center left.

Because both players are doing better there,

and we have the higher amount of payoffs,

eight total units are distributed

while in the Nash equilibrium are distributed only four units.

But, if players decided cooperatively and they would select CL,

the social optimum in this case,

then this would not be Nash equilibrium.

Why? Because player one doesn't have a tendency to deviate from there,

he likes this cell.

But player two, she doesn't like it that much.

Because if player two deviate,

she can play middle,

given that player one played center,

and she can get one more unit.

Of course, it makes the other player to lose 40 units in order to do that,

but she has an incentive to cheat.

So, Nash equilibrium is a down right.

The social optimum, the collusion optimum as we say sometimes, is center left.

But there is any incentive for a player two,

unilateral incentive to cheat,

and she will go to the cheating point middle center.

That's why the center left is not a Nash equilibrium.

Player two would want to unilateral deviate to the cheating point C,

M. Now, let's consider another game,

that I bet you have played several times in your life.

You're walking on the sidewalk,

and it's a busy sidewalk,

and people are coming towards you.

Most of the times you pick a side and the other people pick

another side of the pavement and you actually do not cross.

So, what happens if we have more than one Nash equilibria?

The sidewalk game can be represented with this payoffs.

You can either take the inner or the outer side of the pavement,

and the other person who is coming from the other side

can also take the outer or they inner.

Now, if you take the same side,

you have zero payoff because you will bump to each other.

If you take different sides,

you have one payoff because you achieved your goal,

which is crossing with the other person without actually meeting them.

So, going to your business without having to deal with this person face to face,

bumping other so closely.

So, how do we play this game?

If we look at this game carefully,

and we do this underlining process that I described in the previous slide,

that you will see that now we have two cells,

with double underlining, so we have two Nash equilibria.

Which one is the outcome of the game?

Which one is going to be the one that will materialize?

Because you cannot have two outcomes for the same game.

Game will be played. Players will select one strategy each.

You will have one outcome,

which one is it?

It depends on two things.

First of all, it depends where we start from.

If I am on this side of the sidewalk,

and you're coming towards me from the other side of the sidewalk,

so we're not going to cross.

I will keep walking on my side,

then you will keep walking on your side,

and everything else would be weird like if you're coming from the other side.

And suddenly, I decide to step on your side.

That will be very aware.

Okay, it means that they have a different objective than going to my business.

So, another one is how initial perceptions.

Are formed like a similar game is played on the street also with cars.

But now, the payoffs are not like that.

The payoffs are like mindless and loads.

If you crashed through each other because it's not

like you come face to face and you say, okay.

You go this way, I go this way, and you solve it.

Doing the same thing with the car industry.

That would be much more dangerous.

So in this case, we have initial perceptions.

Like in some countries say would drive on their right,

and everyone drives on their right in some other counties.

We say that we drive on the left and everyone drives on the left.

All right, so we form initial perceptions in order to

solve this double equilibria problem,

and we throw it away when we label,

and we keep the other one.

However, you see that the sidewalk problem is a problem that occurs very often.

You walk on one side of your street and someone is bumping into you

because you both decide to change at the same time,

or you both decide to speak at the same course from the beginning.

So, you bumped to each other or some other times

even you change at the same time and then you change again at the same time.