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So hello back. Great to have you here, and hope you

enjoyed the pizza. In this video, we'll introduce the idea

of the Nash Equilibrium, which is a more formal and precise way of thinking about

optimal strategies. And to do that, let's go back to the

example of the Piccola Osteria and Pizza Rosso, which both sell pizzas to students

around the university, here in Munich. And we'll look at the simplified version

of this game, where we already eliminated the dominated strategies of charging a

low price for both of these players. Okay, so in the simplified version in the

last video we saw that really somehow, it seems to be the best thing, the best

strategy for both, to charge a medium price.

But what is it that we actually did when we looked for the best price, the optimum

price? Well, we looked for a Nash

Equilibrium. But what is a Nash Equilibrium?

A Nash Equilibrium is a combination of strategies such that no player can

deviate unilaterally from his or her strategy.

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And so indeed, once we found a Nash Equilibrium, we found a sort of a natural

resting point of the game. Because no player wants to do something

else, assuming that the other player sticks to his or her strategy.

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And to do that a little bit more practically, to try to find out what this

actually means in practice, let's go back to the game that we just looked at.

Okay, so we've got the two players. We've got Pizza Rosso and Piccola Osteria

and both of them can choose between a high and a medium price.

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So, if for example, Piccola Osteria charges a high price, what's Pizza

Rosso's best strategy? Their best strategy is to charge a medium

price because seven thousand is more than six thousand.

On the other hand, if Piccola Osteria charges a medium price, then it's best

for Pizza Rosso, again, to charge a medium price.

And we can do the same thing for Piccola Osteria, so if Pizza Rosso charges a high

price, then Osteria will charge medium. And if Pizza Rosso charges a medium

price, then it's best for the Osteria to charge a medium price again.

So what we have is a situation where both players, Osteria and Rosso charge a

medium price and that's their optimal strategy.

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So lets look at this from the perspective of a Nash Equilibrium.

Does any of the players have an incentive to deviate from the current strategy?

Okay, so let's assume that Piccola Osteria play their strategies, so they

play medium price. What's best for Pizza Rosso?

Well Pizza Rosso won't want to change from a medium price to a high price

because it would lower their profits. And by the same token Osteria would not

want to change price if Pizza Rosso charges a medium price.

Okay? So for them, it's best again to stick to

the price that they had in the Nash Equilibrium.

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Great, so now you've had a bit of practice. Let's try and put the Nash Equilibrium

into perspective with another couple of concepts that we have already learned in

the course. So let's just ask a couple of questions.

So, first, is a Nash Equilibrium the same as a dominant strategy?

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Second, can a Nash Equilibrium contain dominated strategies?

And, third, will every Nash Equilibrium contain dominant strategies?

Okay, so these are the concepts that so far

we've studied in the course. Let's now put these things into

perspective. Is a Nash Equilibrium the same as a

dominant strategy? It's not.

Why? Because a dominant strategy refers to a

single player, a Nash Equilibrium, as we know from the definition, is the

combination of strategies. It's a strategy for each player in the

game. Can a Nash Equilibrium contain dominated

strategies? Well, let's think back to what a

dominated strategy actually is. It's a strategy for which you have

another one that always does better. Okay?

So if we want to maximize our profits, if we want to maximize our payoffs,

we're going to choose something that will give us the highest payoff.

So a dominated strategy will never be part of a Nash Equilibrium.

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And finally, will every Nash Equilibrium contain dominant strategies?

Well, no, as we'll see in a second. So let's go back to the full version of

the of the pizza war and let's have a look.

Well, what we know is that here, the Piccola Osteria will never

charge low prices. That's a dominated strategy, and of course

it can't be part of the Nash Equilibrium. By the same token, Pizza Rosso cannot do

that. However, what we found is that neither

Pizza Rosso nor the Osteria has a dominant strategy.

And let's have a look at how we find this, okay?

What is a dominant strategy? A dominant strategy is one that does

better than any other strategy regardless of what the other player does.

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Let's go through this. If Piccola Osteria charges a high price

the best strategy for Pizza Rosso is to charge medium.

If Piccola Osteria charges a medium price, the best strategy is to charge medium.

But, if Piccola Osteria charges a low price, then the best strategy for Pizza

Rosso is actually, to charge a high price.

So, by that definition, Pizza Rosso does not have a dominant strategy because

there's no strategy that always does better than the other one, okay?

And we can do the same for Piccola Osteria, and we find that here the best

responses, again, are not necessarily just one strategy.

So we can have a Nash Equilibrium that's just one without any player having a

dominant strategy. Okay?

So that's just throwing these concepts around and trying to put them into

perspective. Let's go a bit further into the concept

of a Nash Equilibrium and see what the nature of a Nash Equilibrium is.

So will every game have a Nash Equilibrium in pure strategies.

In other words, will there always be a combination of one behavior of one player

and one behavior of the other player that is a natural resting point of the game?

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And for that, let's take a look at one of my favorite sports football, or soccer.

Okay, and of course one of the most exciting points in a game is when

there's a penalty, okay? So we've got a penalty taker, and we've

got a goalie, and of course each one wants to win.

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So to simplify height of the shot and so on lets just assume there's left

and right. Okay, the goalie can either dive to the

left or to the right and the penalty taker can either shoot to the left or to

the right. So if they both end up in the left side

of the goal, meaning that the goalie jumps to the left and the penalty taker

shoots to the left. Then the goalie wins, he catches the

ball. If the goalie goes to the left, the

penalty taker goes to the right, then the penalty taker wins, because he scores the

goal. And by the same token, whenever you have

both goalie and penalty taker jumping or shooting to the same side of the goal the

goalie wins. If its opposite sides the shooter wins.

Now what's interesting about this game is that there is never a situation where

price. Okay?

It's either the goalie winning or it's the penalty taker winning.

So that means that we never have a stable situation.

We don't have a natural resting point of the game.

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So the second question is, can a game have more than one Nash Equilibrium.

In every game that we've looked at so far, we had either no equilibrium, in the

case of penalty taking or we had one equilibrium, as in the game of

Pizza. Okay.

So can there be a situation where we have multiple Nash Equilibria?

So as an example here we can think of the battle for competing technical standards.

So we have firm A and firm B, both of them have developed a standard, Standard A or

Standard B. And they both want to choose one of these

two technologies. If you look at the payoffs here, what

you'll find is that there are two Nash Equilibria.

Okay. So, if Firm B chooses Standard A, Firm A,

of course, wants to choose Standard A as well.

And if firm A chooses standard A, then it's best for firm B to also choose

standard A. So we have one Nash Equilibrium here.

At the same time, if firm A chooses standard B, then of course firm B will

want to choose standard B as well. Because that gives them a higher payoff

and by the same token if B chooses their own standards, standard B,

then Firm A will want to follow and also choose standard B.

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So here we are faced with a situation of two Nash Equilibria.

Okay? So, for example one of the

real life cases that was looking at precisely such a situation was Sony and

Phillips. When they started developing the compact

disc in the late 1970s and early 80s, both of them knew they wanted to

standardize, but it was difficult to see which of the two standards was going to

be chosen. Sony, of course, wanted to get as much as

possible of their own standard into the system.

Phillips wanted to build as much of their own system into the standard.

In the end they ended with a sort of combination of both and were very

successful. So the CD was one of the most successful

introductions in consumer electronics. So we've just learned about the Nash

Equilibrium as the natural resting point of the game.

So at this point, in a Nash Equilibrium, no player can gain higher payoffs by

playing another strategy. So, in a competitive situation, we would

expect two rational rivals to choose their strategies such that they will end

up in a Nash Equilibrium. So, of course, in some situations, there

may be no Nash Equilibrium at all, as we've seen in the game of football and

soccer. In other settings, we have two or even

more Nash Equilibria. So, what's interesting there is that, how

do we select? How do we solve these sorts of games?

So in the next video, we'll have a closer look at these kinds of games.

But first, do have a look at the following game, and put down how many

Nash Equilibria there are.

Okay? Thanks very much, and see you very soon.