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Welcome back. In the last three sessions this week,

we'll turn to the topic of game theory and cover

the first three sections of chapter 14 for those that are following along in the text.

Game theory has been applied to oligopolistic settings,

where you have to worry about what actions will my rivals

take and how should I respond in kind,

and knowing that my rivals in this market will

also be trying to figure out what actions I'm going

to take and how they should respond in kind.

Game theory can be applied to other settings besides oligopolies.

For example, looking at defense budgeting.

What should be the size of the US defense budget will depend on what's

happening in the Cold War days to the USSR's defense budget,

or nowadays to the actions China is taking regarding its military.

So there is a complex interdependency that game theory tries to take into account.

All game theory has three different characteristics.

We assume that their players,

number one, that each of the players has a set of strategies,

has a set of choices they can take and these will be their actions,

and third, that there are payoffs that result

from the different players choosing different strategies.

In this session, we will look at how

the equilibrium gets determined in a few simple cases.

We'll look at a case of individual players having a dominant strategy,

where they'll have one best choice no matter what the rivals do,

what the other players do in the setting,

and we'll see what happens to

the equilibrium in cases where each of the players have a dominant strategy,

and then we'll also cover Nash equilibrium,

which Nobel Prize winning economist mathematician John Nash came up with the concept.

Let's apply game theory to a particular setting,

the simple oligopoly game,

two firms, a duopoly,

firm A and firm B.

Both firms have two choices they can make.

Firm A can choose low output or high output,

it can choose different rows,

whereas firm B can choose different columns,

also low output and high output.

The third element of any game theoretic situation we said was the payoffs.

If both firms choose a low output,

then firm A's payoff will be 10 and firm B will earn 20.

If both firms produce a high output,

then firm A will earn 18 and firm B 25.

Now, in this particular setting,

we can show that each firm has a dominant strategy.

What do we mean by that?

Let's say you were firm A and

you knew firm B would choose a low output,

so you knew for sure that firm B was going to go low,

what's your best choice as firm A?

You effective have a choice between a payoff of 10 or 20.

You're better off going high.

What if firm B chose a high output and you

knew for sure as firm A that firm B was going to go high?

Your best choice would be between a payoff of 9 or 18.

So your best choice,

your dominant strategy, as firm A is to go high.

No matter what B does,

a dominant strategy says,

"You're better off choosing this strategy no matter what the rival does."

Does firm B have a dominant strategy?

Let me clear the screen.

Suppose you are firm B and you knew for sure that firm was going to go low,

what's your best choice?

You're choosing between 20 and 30.

Your best choice is to go high.

What if firm A chooses

high output and you knew for sure that firm B that A was going to go high?

B has a choice between the columns that give it 17 or 25.

Firm B has a dominant strategy to go high.

No matter what A does,

B always finds the best to go high.

So, both firms have a dominant strategy to produce higher output.

In a sense, this captures an oligopoly setting,

where both firms have an incentive to go high.

And in this case,

let me clear the screen again,

we'll end up if firm A has a dominant strategy

of high and firm B a dominance strategy of high,

the dominant strategy equilibrium will be high,

high, A earning $18, B, 25.

So we'll end up in this case,

and it's known as a dominant strategy equilibrium

in the lower right-hand corner or DSE for short.

Now, let's change just one thing.

In this case, and let's compare it with the previous one, in this case,

in the table 14.1,

if both firms went low,

firm A earn 10.

Now we've altered it so that the numbers are all the same,

but if both firms go low,

firm A earns 22.

Does firm A still have a dominant strategy?

If you knew for sure that firm B was going to choose low and we're firm A,

we're better off going low.

If you knew for sure firm B was going to choose high,

then we're better off going with the higher output strategy.

So, in this case,

firm A doesn't have a dominant strategy.

Its optimal strategy depends on knowing what B's going to do.

So, A doesn't have a dominant strategy.

Does B still have a dominant strategy?

And you can test yourself,

B still does, none of these payoffs have changed.

If A chooses low,

B is better off going high.

If A chooses high,

B is better off going high.

So, B still has a dominant strategy.

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And what a Nash equilibrium,

it's a broader set of equilibrium,

where once you're there,

each firm is making the best choice knowing what choice the other firm has made,

in this case, knowing what output strategy the other firm has made.

A has made its best choice of high output knowing that B has chosen high.

Similarly, B has a dominant strategy,

so no matter what a chooses, and in this case,

A ends up choosing high,

B is also better off choosing high.

That makes 25 vs 17.

If we think about Nash equilibria,

the space depicted by that blob,

Nash equilibria, dominant strategy equilibria are a smaller subset of Nash equilibrium.

So, the dominant strategy equilibria are automatically Nash equilibrium,

but not vice versa.

Now, it can be difficult to figure out a Nash equilibrium,

and if you've watched or read the book A Beautiful Mind,

the struggles John Nash had how to determine

your best choice when you're trying to figure out what

somebody else is going to do and they're trying to do likewise.

So you can see why it almost might have driven

John Nash to madness to think about these complex interactions.

We'll end up applying these concepts now to

a very important game theoretic situation

in the next session called the prisoner's dilemma.