Now, we will talk about the games that they have some time depth. In other words time, is a factor. You may have games that they're repeating in rounds, the same game is repeated again and again and again several times and each round you can change your actions, and you can also have dynamic games in which every round you can make different decisions. All right, so we'll see these games. Let's Start with repeated games the same game is repeated for several round the same game is played again and again and again. Imagine that you're pulling a rock scissors paper with your friends 10, 20, 50 times. All right. The same game is played again and again and again and you can make different decisions about your strategy in every round. So, if you have repeated games two important things may happen. The first is that now you develop a reputation. You have a reputation in which will give the advantage and the information to the other players about in general how you behave like for example when I play rock scissors paper I have a reputation of playing rock too many times. So, this is reputation. People start learning you. The second one is that you have a chance for retaliation. Think the case of the prisoner's dilemma. I told you before that I will refer to Prisoner's Dilemma several times and I will I already started. So, think about the prisoner's dilemma. In the prisoner's dilemma, your best, possible alternative is to convince the other person to deny and then you will confess. So, in this case you cheat the other person, you tell them let's collude, but you cheat on them. All right. So, if this happens now that this game is repeated you can do it once, but from the other time the other player that is cheated upon might start retaliating, taking a revenge to what you do. So, now that we have time depth you know that the revenge is a blade that is served cold, it can come to a different ground. All right so let's call these two factors affect games. So, players may jointly try to impose an outcome that is better than the Nash equilibrium: the collusive outcome. The collusive outcome is not a Nash equilibrium. So, if we both decide to deny, we collude to deny, this is not an Nash equilibrium, it's better than Nash, but this is not Nash. There is still incentive for cheating in this game in every round, but because there is another round that follows, this incentive may be eliminated because there is a chance for retaliation. Also if you start cheating, you develop a reputation of cheater and no one really wants to trust you and collude with you. So, if a player decides to cheat you can get away with a higher payoff only in that round. From the next round, a punishment phase will start in which players will naturally revert to the Nash equilibrium solution because there is no trust between the players anymore. So, how does this work? Retaliation is often called a tit-for-tat strategy. Tit-for-tat is an English expression that means retaliation. That is I trust you and I play the collusive strategy, but if you cheat I will be playing my Nash strategy forever. And that strategy will give both players worse results, collusive strategy will give them better but there is an opportunity for cheating, so I trust you that you will not cheat but once you cheat, that's it. We finish these and will go to the Nash equilibrium forever. Tit-for-tat strategy is also called the trigger strategy. Because everybody trusts everybody else until someone pushes the trigger. Once you push the trigger, the cheater loses from next period because you go from the collusive of outcome to the Nash equilibrium, and the fair player starts losing too but for the field prayer, there's no other alternative because doesn't trust any more the other person. So, for the fair player the Nash equilibrium is better than be cheated upon. So, if you decide to cheat, you get the money for one period, but then in the other periods you revert to a solution that these wars that you had previously agreed. What happens if the game is infinitely repeated or even randomly terminated so you have a game that is repeated either an infinite amount of times or is terminated, it stops at the random point without anyone knowing before that this is going to be the last time that the game is played. So, what happens in this case. Well, in this case, the tit-for-tat strategy it does make sense. If the player cheats then the other player will be playing the Nash strategy forever. So, they have an agreement, they have a collusive agreement, if one cheats on this agreement, the other player will be playing the Nash strategy forever by penalizing the first player who cheated, so the cheater will get high payoff for a single time because they got away with cheating once. But from the next period, the next round will revert to that lower Nash Equilibrium payoff. This is because, the threat of retaliation will be credible, if the one time pay off from cheating does not exceed the net present value from the benefits of the collusion. So, if it is good enough for you to cheat once and you get enough money, in order to run and not care about the penalty that will keep happening forever, collusion is not going to be sustainable. But if their onetime payoff from cheating is not enough to cover the benefits that you will get by colluding all the other periods which will be your penalty in case you cheat, then threat of retaliation is credible and inclusive outcome can be sustained. In this case, we say that the Turkish strategy is not a simple Nash equilibrium, but we extend this to a dynamic notion of equilibrium which is called the sub-game-perfect Nash equilibrium. In other words, we split the game into smaller sub-games, and we call these a sub-game-perfect Nash equilibrium. We will discuss about that in a little bit. For now, let's see what happens if we have finite repetition. That is what if the game is repeated for a known finite number of times. So, from the beginning you know that you are going to be playing this game for five, 10, 50 times. But you know that from the beginning. In order to solve this game, we will go in a slightly different way. We will use a philosophical device that is called The backwards induction. The backwards induction is when you start looking at the result from backwards, from the last period of things happening. Let's see if something would be different in this case if you have finite repetition. In the last period, everyone knows that this is the last period. Everyone knows that there's no possibility of retaliation. So, therefore, everyone has an incentive to cheat and cheating will get away without punishment. So, in the last period, everyone should expect that everybody will cheat. Yes, but if you cheat in the last period, then this means that there's no real real way to punish someone in the period before the last period. So, the second to last period, if everyone cheats in the last period, there's no fear of retaliation either. This logic can continue till we get back to the first period since you know that in all later periods everybody will cheat, you have no way of implementing a punishment for the cheaters. So, we can say that the threat of retaliation is not credible when you have finite repetition and therefore collusion is not sustainable. Next, we will talk about dynamic games of complete information games in which you do not cover a petition but you have players taking different actions and this will be a very interesting case. Stay with us.
