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Welcome back. In the last three sessions this week,

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

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

Â Game theory has been applied to oligopolistic settings,

Â where you have to worry about what actions will my rivals

Â take and how should I respond in kind,

Â and knowing that my rivals in this market will

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

Â to take and how they should respond in kind.

Â Game theory can be applied to other settings besides oligopolies.

Â For example, looking at defense budgeting.

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

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

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

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

Â All game theory has three different characteristics.

Â We assume that their players,

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

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

Â and third, that there are payoffs that result

Â from the different players choosing different strategies.

Â In this session, we will look at how

Â the equilibrium gets determined in a few simple cases.

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

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

Â what the other players do in the setting,

Â and we'll see what happens to

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

Â and then we'll also cover Nash equilibrium,

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

Â Let's apply game theory to a particular setting,

Â the simple oligopoly game,

Â two firms, a duopoly,

Â firm A and firm B.

Â Both firms have two choices they can make.

Â Firm A can choose low output or high output,

Â it can choose different rows,

Â whereas firm B can choose different columns,

Â also low output and high output.

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

Â If both firms choose a low output,

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

Â If both firms produce a high output,

Â then firm A will earn 18 and firm B 25.

Â Now, in this particular setting,

Â we can show that each firm has a dominant strategy.

Â What do we mean by that?

Â Let's say you were firm A and

Â you knew firm B would choose a low output,

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

Â what's your best choice as firm A?

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

Â You're better off going high.

Â What if firm B chose a high output and you

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

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

Â So your best choice,

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

Â No matter what B does,

Â a dominant strategy says,

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

Â Does firm B have a dominant strategy?

Â Let me clear the screen.

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

Â what's your best choice?

Â You're choosing between 20 and 30.

Â Your best choice is to go high.

Â What if firm A chooses

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

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

Â Firm B has a dominant strategy to go high.

Â No matter what A does,

Â B always finds the best to go high.

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

Â In a sense, this captures an oligopoly setting,

Â where both firms have an incentive to go high.

Â And in this case,

Â let me clear the screen again,

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

Â of high and firm B a dominance strategy of high,

Â the dominant strategy equilibrium will be high,

Â high, A earning $18, B, 25.

Â So we'll end up in this case,

Â and it's known as a dominant strategy equilibrium

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

Â Now, let's change just one thing.

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

Â in the table 14.1,

Â if both firms went low,

Â firm A earn 10.

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

Â but if both firms go low,

Â firm A earns 22.

Â Does firm A still have a dominant strategy?

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

Â we're better off going low.

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

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

Â So, in this case,

Â firm A doesn't have a dominant strategy.

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

Â So, A doesn't have a dominant strategy.

Â Does B still have a dominant strategy?

Â And you can test yourself,

Â B still does, none of these payoffs have changed.

Â If A chooses low,

Â B is better off going high.

Â If A chooses high,

Â B is better off going high.

Â So, B still has a dominant strategy.

Â 8:40

And what a Nash equilibrium,

Â it's a broader set of equilibrium,

Â where once you're there,

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

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

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

Â Similarly, B has a dominant strategy,

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

Â A ends up choosing high,

Â B is also better off choosing high.

Â That makes 25 vs 17.

Â If we think about Nash equilibria,

Â the space depicted by that blob,

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

Â So, the dominant strategy equilibria are automatically Nash equilibrium,

Â but not vice versa.

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

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

Â the struggles John Nash had how to determine

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

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

Â So you can see why it almost might have driven

Â John Nash to madness to think about these complex interactions.

Â We'll end up applying these concepts now to

Â a very important game theoretic situation

Â in the next session called the prisoner's dilemma.

Â