[MUSIC] Now, we will talk about dynamic games. Games which happen in turns, and they keep evolving till all players have played the strategies. So let's see how these dynamic games work. First of all, we will consider a game of an entrant and an incumbent. Imagine that you have some market, and then one specific firm is in this market for a long time, it's called the incumbent. It's a monopolist that is already there. And you have another firm that looks at the profit of this incumbent and says, perhaps we should enter this business. All right, so we'll consider this game between the entrant and the incumbent. So this is the payoff matrix. We have the incumbent represented in their rows and the entrant in the columns. And then, the question is what happens if the entrant plays first? Now, this is slightly a different situation, because we have the game to not be simultaneously, that the players do not play at the same time. First, we'll go the entrant and after the entrant has committed to one strategy, then the incumbent will answer this specific strategy on their own. This, therefore, allows for commitment. Once you play a strategy, that's it, you cannot take it back, you are committed. And commitment makes a big difference in games, but not only in games, in life also. If you are able to commit into something, this will change your course for sure. Let's see how this works. So the incumbent doesn't like the possibility of someone else entering. If you look at the payoff matrix, you will see that there are two Nash equilibrium there, accommodate and enter or actually enter and accommodate, or pass and fight. But again, this would be the Nash equilabrium, the game was simultaneous. This is not a simultaneous game, because we know that the entrant plays first. So what will happen? The incumbent can threaten the entrant to not enter, to scare the entrant to not enter. Can say, if you enter, I will fight you. I may lose one, but you will lose three. Is this a good threat? Is this threat going to have result? The answer is that this is a problematic threat. It has a credibility problem. It's what we call an empty threat. Once the entrant has entered, the incumbent will not want to go ahead and punish by fighting. If the incumbent does that, we'll have a payoff of minus 1. We'll lose, let's say 1 million, but if it accommodates, it will win 1 million. So in this case, we have something that we call dynamic inconsistency. You can threaten for something, but when the time comes to deliver your threat, you may not want to do it any more. All right, so let's see how specifically this works. In dynamic games, players move sequentially. They do not move all together. Dynamic games are represented better with a succinct game tree, what we call a decision tree. This is because in the bi-matrix you see simultaneous decisions, you do not see the succession of actions. So we face such games by splitting them in smaller games, what we call sub-games. So we decompose them into smaller parts that they are easier to solve. Every sub-game then is a standalone game on its own. We face the sub-games as if they were an entire game. We start from the end. Don't forget that we have sequentiality, meaning that we take always the smallest sub-game which is the one that is at the end. And then, we find the Nash equilibrium for this sub-game, and we replace the sub-game with the Nash equilibrium, because we know that this is the only possible outcome for this sub-game. So we kind of simplify the game by exchanging, replacing the sub-games with their Nash equilibria till we get back to the initial game which will be now much more simple to solve. So we will finish the same logic by applying it to the next-to-last game, and all of the games, as I said, till we get back to the first game. The definition of the sub-game perfect Nash equilibrium is the profile of strategies that is a Nash equilibrium at every sub-game that contains this profile. So if a Nash equilibrium is a Nash equilibrium in the last sub-game, and to the next-to-last, and to the next next-to-last till the first then, this is called the sub-game perfect Nash equilibrium and is the right way to solve dynamic games. If you want to read more about that, you can go to Church and Ware, our textbook, and you can read starting from page 287. There is a very extensive discussion, and it's a very good discussion. I would advise you to go and take a glance there, because you will learn more about the concept of the sub-game perfect Nash equilibrium. What is better, to play first or to play second? In some games, you might have what we call the first-movers advantage. The first-mover advantage is when it gives you an advantage to play first. Think of some game that it will be better if you play first. Let's say the game of tic-tac-toe. If you play tic-tac-toe, and you play first, then in this particular round you can make it so you will not lose. So playing first gives you an advantage. There are other games that playing first gives you a disadvantage. Imagine a game of rock, scissors, paper in which you have to play first. So you play rock, scissors, paper, and the other player will play second, so will go okay, like that. You always lose, because once you know what the other players is playing, you can always answer with the answer that wins. So this will give you a first mover disadvantage. Now, in the game that we saw before with the entrant and the incumbent, the entrant gained two instead of one, just because it's allowed to move first. On the other hand, the incumbent gained only one. This is why this is because it is allowed to move only second. So can the follower do something to gain the advantage? Don't forget that a simple threat will not work. If you just go out and threaten people, you better have threats that they are not empty. If your threats are non-credible or empty, then this is not going to reverse the advantage. So we will examine a strategy, I will show you a strategy in which we have preemptive restriction. This is a very cool example in which you will restrict yourself in a way that will give you the advantage when you don't even have it. And this will be an example that seems crazy and weird in the beginning, but you will see how nicely it work. Stay with us, and we'll be together in a little bit. [SOUND]