Great, so thanks for staying on. In this video, we'll draw a slightly more complex game matrix than the one we did in the previous video, and we'll use the concept of dominated strategies to figure out the best strategy in a complex competitive situation. But for that, let me give you a concrete example. So please now follow me on the streets around the university here in Munich. So here we are just a few yards from university, and we're thinking about getting something for lunch. So, right now, there's a couple of places that we can choose from, and one of those is called Rosso Pizza. Obviously, they sell pizza, which is very popular with the students around here, but the issue is that just a few yards down the road, there's another place called Piccola Osteria. So these two are in close competition, they're tight competitors, and they're thinking about setting prices. So, for now, let me just get a pizza and I'll see you back in the office in a second. OK, so let's see what we have. Pizza, obviously. OK, that's some sort of veg. I'm going to keep healthy. Very nice. And we've got a Margherita, also very nice. Sorry about that, just a second. So, having seen these two firms, having seen these two restaurants, it all looked very harmonious, all looked very straightforward. But, the issue is actually, I mean these two firms, these two restaurants, are of course in competition with each other. It matters what the prices are. It matters what the relative prices are. If you're more expensive, you're cheaper than the other, then that's going to make a difference to your profits. And let's now try to structure and then analyze the situation in a bit more detail using game theory. So, we've got these two players, Piccola Osteria and Rosso Pizza, selling pizza, and they can charge one of three prices. They can charge a high price of 12 euros; they can charge a medium price of 10; or a low price of five euros. Let's assume there are 1,000 students around school interested in pizza and 300 of those will only ever buy at Piccola Osteria. Right? They swear on Piccola Osteria being the better pizza place. 300 of those will only ever buy at Rosso Pizza because they like that type of pizza more. And 400 of them will just buy whatever's cheaper or they choose randomly between one of those two whichever has free spaces. OK? So, to analyze this, to take this in a setting of a game, let's look at the players – they're our two pizza places. The actions they can choose are high prices, medium prices, and low prices. The rules of the game is that every morning, where in the morning, when they before they open up, they choose a price. OK? So, they choose prices simultaneously which as we saw in the previous video means we don't know what the other firm has chosen when we make our own decisions. Let's now look at payoffs. As an example. We just assume that there's no cost of producing pizzas, so the 10 euros or 12 euros are basically gross profit margins. So, if Piccolo Osteria charges a medium price and Russell Pizza charges a high price, then the payoffs are, for Piccolo Osteria, that they sell to their 300 loyal students plus the 400 students that are price-sensitive because they will go for the lower price, and they do that at a price of 10 pounds, 10 euros per pizza. OK, so that's going to be 300 plus 400 times 10 euros, that's going to be 7,000 euros. OK? What about Rosso Pizza? Rosso Pizza is going to sell only to their 300 loyal students, but they sell it at a higher profit margin. So they're going to sell 300 pizzas at 12 euros. That's going to be 3,600, OK? 3,600 euros. And from that, we can basically just construct the entire payoff matrix like we did in the previous example on toothpaste. Let's have a look here. Well, we've got Pizza Rosso, we've got Piccola Osteria, we've got the three strategies – high, medium, low – for both players. Now, the thing we just discussed, Pizza Rosso charging a medium price and Piccola Osteria charging a high price gives 7,000 euros for Pizza Rosso and it gives 3,600 for Piccola Osteria. OK? And we can construct the entire payoff matrix. And completed, these are, of course, in 1,000 euros. So, from that, we can try to find out what outcome is likely to happen. All right? And what I want to do to help this along is I want to look at two particular concepts. One is a dominant strategy. A dominant strategy is a strategy that always does better than any other strategy, regardless of what the other firms, what the other players do. So, a rational player will always choose a dominant strategy. Now this is easy for yourself, right? If something always does better, then that's what you're going to choose. What's actually more interesting is that if your rival, if the other player in the market, has a dominant strategy, you can rely on that player playing a dominant strategy. OK? So it makes you, you can anticipate that the other player will choose a dominant strategy as a consequence. There's a flip side to that. There's a what's called a dominated strategy. A dominated strategy is a strategy that never does better than another one. So we compare two different strategies, and if one always does worst than the other, this strategy is dominated. So of course, a rational player would never opt for that strategy. And again, it's easy for yourself. If you're trying to maximize your own profit, why would you choose a dominated strategy? But what's more interesting and what's more- takes more thinking is that a rational rival will never play a dominated strategy, right? So even if – a dominated strategy by your rival would give you massive profits. You should never count on the other player choosing a dominated strategy if he is interested in maximizing profits. OK? So, if we want to solve the pizza war and basically try to predict what's going to come out, what the prices are going to be, we can try to eliminate the dominated strategies in this case. OK? So, if we just have a look, we can eliminate the dominated strategies one by one. OK? Let's go back and see how we can do that. So, if Piccola Osteria would charge a high price – OK, we're here – Pizza Rosso can choose a high price and get six, a medium price, get seven, and a low price and get three and a half, so the best strategy here would be a medium price. If Piccola Osteria chooses a medium price, then the best response, the best strategy that Pizza Rosso can choose is to choose a medium price again. If the Piccolo Osteria chooses a low price, the best strategy by Pizza Rosso is actually to choose a high price. Right? 3.6 versus 3 versus 2.5. Why do they do it? They've basically given up the market for the price-sensitive consumers and they only sell to their loyal consumers at a high margin. Now, if you look at this, what's interesting is that if you compare the low price and a medium price, it's always better for Pizza Rosso to choose a medium price rather than a low price. And what that means is that there is no circumstance, no strategy by Piccola Osteria under which Pizza Rosso would want to choose a low price. Now what we can do with this strategy is basically just eliminate it. It's not going to happen. Right? There's never going to be a low price set by Pizza Rosso. We can do it exactly the other way around, and given this is a symmetric problem, we'll find that also for Piccola Osteria, charging a low price is never going to be a viable strategy. OK? So we eliminate these dominated strategies. We can eliminate them one by one, and we will continue with that until we can find a solution to the game or you just can't eliminate more. And let's go back to the game here. Once we've simplified this game, let's now have a look again. If Piccola Osteria charges a high price, we've already figured out that it's better for Pizza Rosso to charge a medium price than a high price because six is less than seven. If Piccola Osteria charges a medium price, then it's better for Pizza Rosso to charge a medium price than high, which means that here, charging a high price is dominated again. And by the same token, we can do the same thing for this simplified game and eliminate the high price for Piccola Osteria as well. And this is going to let us solve this game, make a prediction of what the outcome of this game is, namely, both Pizza Rosso and Piccola Osteria charging a medium price and both making profits of 5,000 euros. In this video, we analyzed very briefly the strategic problem of choosing the right price, and we structured this problem with the help of the matrix. It was slightly more complicated than the first one we saw. We used the proceeds of iterated elimination of dominated strategies, so we kept on cutting up the game to make it simpler and simpler to eventually find the optimal strategies for Piccola Osteria and Pizza Rosso, respectively. So, in the next video, we'll go on and demonstrate to you a more formal and a more precise way of thinking about optimal strategies, so please stay on to go on to the next video. But first, let's have a look at the exercise and figure out all the dominated strategies in these exercises. Again, any case, you'll find help in our forum. Thanks very much and see you in the next video.