Check out this game. Now, player two has three strategies left,

middle and right and player one can play up or down.

If you look at this game, just at once,

you would probably say I have no idea how this works.

But now, I will show you a way that you can see that there is

only one possible cell that will be materialized in the end.

Take a moment, look at the table and let me know if you can find it.

So, let's start.

The first thing that we do when we see such games is we

look for strategies that they are dominated.

If there is a strategy that cannot be played,

then this strategy should be ruled out.

So, if you look at the game up,

the red person player one doesn't have such a strategy,

because strategy up gives one,

two and zero, while down gives

better payoffs in the case that the player two plays left then middle.

But if the player two plays right,

gives two instead of zero.

So, player one doesn't have a dominated strategy,

but player two does have a dominated strategy,

and this dominated strategy is strategy right.

Strategy right is dominated by middle,

if in your set the strategies you have middle,

you will never play right because right will always give you

a smaller payoff than middle no matter what player one does.

So, I can erase this strategy from the game,

and now, I have a game with four possible outcomes.

If I look at the reduced version of the game,

I can see that now, clear one does

covers strategy that is much better than the other one.

Strategy down is not a smart strategy anymore,

it's dominated, it's strictly dominated by up.

So, rational players will never consider this strategy and I can also rule it out.

Now, the game has become very simple.

What do we want, one or zero, or two or two?

Rational players would rule out

strategy left and up combination of strategies left and up,

and we end up with only one possible outcome for this game.

So, the only combination that makes sense in this game is up and middle,

we call this equilibrium in dominant strategies.

Dominant Equilibrium or DE,

to be precise this is an IDE,

Iterated Dominant Equilibrium, equilibrium in dominant strategies

because these crossing out of strategies happened in an iterative way.

First, I have to cross out strategy right,

then strategy down and then combination up and left,

in order to remain with up and middle.

The outcome of this game does not depend on what other players are doing.

It's better for player one to play up and better for

the player two play middle and this is a known strategic outcome.

It's an outcome that you do regardless of what your competitor is doing.

So, the equilibrium in dominant strategies is a non-strategic concept of equilibrium.

The optimal strategy is determined without worrying about what your opponent is doing.

Notice that the dominant equilibrium does

not and should not stand under weak domination meaning that,

a strategy should not be weakly dominated it should be strictly dominated.

What is the difference?

That in strict domination,

all payoffs in one strategy are better than the payoffs in the other strategy.

In weak domination, only some payoffs are better and some are equally good.

So, you shouldn't have equally good payoffs in this case in order

to have dominant equilibrium in your game.

Now, the dominant equilibrium has some drawbacks,

there are three of them, very important.

The first is that it needs to assume that it is

common knowledge that all players are rational and moreover they are smart,

that they can run this iterating elimination in their brains and come up with results.

Second is that it usually doesn't work especially if you have many iterations,

we have seen in experiments in labs with real players asking them to play

real games for real money that if there is some depth of iteration,

many iterations, the process often

produces a very imprecise prediction about the outcome of the game.

And the third one is that we very rarely have results in life,

that what we get depends only on us.

Usually what you get depends on what you do,

but on what other players are doing.

And for this, we need another concept of

equilibrium that is called the Nash equilibrium and it's very interesting.

And we will talk about it in the segment after this one.