Let's see now, what happens if interaction is infinite. In other words, players start the game, the game keeps repeating every time, but we do not know when this game will finish. If our players do not know when the game will finish, we cannot use backward induction. This is because for backward induction, we have to start from the last period. And now our players they don't know which period will be the last. Maybe the last period doesn't even exist in an infinite game. Maybe the game is indefinite, they still don't know but it will have the same result. So, let's see exactly what happens in this case. Let's see, asserting now that will tell us what we should do in infinite horizon games like firms play. So suppose that firm j will follow the ij notation. Again, you can simplify that by saying i it's us, and j are the others, or i and j can stand for any two firms that they will play this simple game. So suppose that the other firm j, adopts the trigger strategy. What should we do? What is our best response? Before we answer that, we have to know what are our options. So let's take every case and see what happens with every case, and then we can make a choice and decide which one is the best for us. So suppose that the other firm adopts trigger strategy, our first option is to also follow the strategy. This means that we will have a trigger strategy too, meaning that as long as no one cheats, the game will keep playing in the same manner, everybody will collude. So everybody will have an agreement that will follow this agreement, and no one will ever defect. So this means that others are following the trigger strategy, and then under the first case a, we will follow the trigger strategy, also. Now, in this case, if we follow this option, we make half of the monopoly profit in each bill. This is the best case scenario under a. This is if we agree both of us that lets charge the best possible price for us the monopoly price, which maximize our joint profits and then let's share equally these joints profits. Now, this will give me an infinite stream of payoffs for every period because this continues to infinity, and the values of each year's, each period's payoff will not be the same to me at this period right now. So I have to discount future values. How to discount. That the formula for discount is a summation, that starts from zero and goes to infinity, and then we take the discount factor Delta and we put it in the power of the periods that these specific value is ahead, and then we take our value which is half of the monopoly price. This is simply equal to one, plus Delta plus Delta square, plus Delta in the third power, plus Delta in the fourth power, until goes to infinity times half of the monopoly profit. Now, it's easy. If you remember your high school algebra, it's easy to factor this infinite summation, and this will be equal to one over one minus Delta, and since we multiply that by half the profit, this will give us the formula that we will get here. So this will be the monopoly profit times two, times one minus Delta. This is our payoff, if we follow the strategy of imitating the other person to adopt the trigger. Now, our other choice is to defect. To be the cheaters. To take the money and run. So we will cheat for one period, get the money, and then, revert to the Nash equilibrium forever because no one will trust us again. This doesn't sound very ethical, but firms do not play ethically all the time. So let's try to see what the payoff will be here. So if firm j sets their great price of monopoly pm, what we want to do is according to Bertrand, we want to go a small increment below their price, minus Epsilon, the smallest possible value. This will be enough to make everybody come shop from us and not from them. So if this happens we'll get almost the monopoly profit, the entire monopoly profit, the other guy will get nothing if not having damages. They will not in this game, but in reality they may have. So, what do we get is the monopoly profit for this period, and then for the next period we will get zero and zero and zero and zero forever, because everybody will go to the Nash equilibrium, that in the Bertrand case the Nash equilibrium is zero profit as we have seen. So what happens here is that, we get the money and we actually run. We get the money one period, and then that's it, that's the all the money we will have out of this choice. We don't know if this is better than the previous one, but there's also a third case. A third case is that, what is going to happen after we defect. After we defect, what is the other person want to make? We have already seen that but let's see it more clearly, is that the best response for the other person is to set the prices equal to c, to follow the Bertrand paradox pricing, and this will make profits to be zero. All right. So, do we want to go for a or for b? Do we want to follow the agreement, or we want to defect? This will depend. I will take the profit, and I will compare it. So if I follow the agreement, I get pie m over two times one minus Delta. This is the value that I get, the payoff that I get if I follow the agreement and if I defect, I get just b pie m. And I have to compare those. Now the comparison, as I can see from the situation there, it's easy to understand that first of all pie m will have to cancel out because it's in both sides. So if I saw that, I see that the critical variable there is Delta. So if Delta is over one half, I want to follow the agreement. This means that if my tribute as a firm is that I care for the future and lot more than my discount factor to be just one half. So if I care for my future profits, my future stream of profits, then probably I will have a preference to not cheat. To follow the agreement. So, if Delta is over a half, trigger strategy is a sub-game perfect Nash equilibrium. We want to follow the strategy. There is no incentive for deviation from the trigger. If Delta is over one half, collusion can be sustained, if this game is infinitely repeated and firms do play trigger strategies, and they do care sufficiently for their future profits. Now, this result doesn't only hold for two firms. It can be generalized to more than two firms up to end firms, as long as the firms are finite. Let's make some very crucial points for a repetition because they are important to know. First of all, this repeated game does not have only one equilibrium. I just derived one equilibrium, the most obvious I would say, the focal point of equilibrium, but there is an infinite number of equilibria. Actually, any price between the Bertrand zero profit price and the monopoly price, can be a sub-game perfect Nash equilibrium, as long as firms agree on that specific price. For example they say, don't put the monopoly price. Do not set the monopoly price. Let's set the monopoly price minus half a dollar. If we agree on that, this can also be in the same exact manner. And Nash equilibrium, just profits will be a little less, and they may do that to avoid risk of getting caught, because as we will see a little bit, collusion is not legal. In almost every country of the world, collusion is illegal and you can get in serious trouble if you're caught colluding. So, the theory is silent as to which equilibrium will be chosen. We do not know which equilibrium will be chosen from the theory, it will be probably the one that firms prefer for their own reasons. Collusion can be sustained for indefinite repetition. Repetition in other words can be repeated for a finite amount of times, but we just don't know which one is the last period. Collusion may be sustained even when punishments do not last forever, but they do last for a specific amount of periods. Like for example if someone cheats, then for 10 periods we are going to be setting the Nash equilibrium price. This again according to the model that we showed, it will work. Collusion is sustainable in this game, but it's not necessary. So you might go as well for pricing equal to mc, and this again will be an equilibrium in this game. So it is sustainable, but it's not the necessary outcome. Now, it's interesting that in some industries, there is a verification lag. So this means that, the firms do not understand right away that they have been cheated upon. This gives the cheaters an amount of periods that they can cheat again and again till they get caught. In this case, if this exists and firms know that there exists a lag, so in this case Delta has to be higher, in order to sustain collusion. You must care for the profits much more the profits of the future, in order to be able to tolerate this kind of Iag that there is in detection. Indeed, we see from real observations in real firms, that firms almost never collude in long-run variables. Variables that you cannot understand them very quickly. Like for example, they will never collude on how much research and development they will do, this is because you cannot observe how much aren't these done exactly on time. You will observe it much later. Long term decisions they take more time to implement and more time to be detected, and therefore, we prefer if we collude, we prefer to fix the price that it's much easier to detect and much easier to enforce as an agreement. Stay with us because right away, we will see another sensitive issue, the issue of renegotiation, which is very crucial for the theory that we just set. See you soon.
