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. To do what? To improve his or her payoffs. 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. 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. 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. 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. So now, it's actually your turn. And have a look at this short game and try to find out what the Nash Equilibrium is. I'll see you in a second. 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? 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. 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. 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? 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. 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. 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. 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.