So hi there, and good to have you here. As we've just seen in the last video, it's difficult or even impossible to establish and maintain cooperations in games that are only repeated a finite number of times. So in this video then, we'll shift the focus to games that are repeated without having clear end point. So do still keep in mind that we're still interested in mechanisms that will help us to enable and maintain cooperation. So just to repeat, what is finite and what's infinite repetition, it's clear in finite repetition when the end point is going to come and how often the game is repeated. With infinite repetition, there's no defined end point and we cannot predict when the game precisely is going to end. Okay? And one of these games that are infinitely repeated is the diamond cartel. Diamond cartel has basically been running for a long period and it's been very successful. So let's just take this very stylized example, where we just take two countries, and say, will these two represent the overall diamond market? And again, keep in mind, these are fictitious numbers. So we have South Africa and Australia that control the market for diamonds as a luxury good. Each of those countries has enough resources and production capacity, so they can supply the market for diamonds every year. And every January they will decide about the prices that they want to charge or the quantity that they want to produce. Okay, it comes out the same way. And the game is repeated every year and there's a probability of p that it goes on in the next year. So, how does that represent itself as a game? Well, what are the actions that this game gives us. So, if both countries charge the monopoly price then the market is shared equally. And the overall profit is going to be 50 million. Think of this as Australians and South Africa are getting together and behaving as a monopoly. A monopoly is a firm that is just, the only firm on the market and, therefore, can set any price that they want to maximize profits. Okay? So here, South Africa and Australia get together and behave as one monopoly. And make overall profits of 50 million and they then subsequently share these profits amongst themselves. If one country charges a slightly lower price and the other firm charges the monopoly price, then the country with the slightly lower price serves the entire market and makes profits of 49 million. If both countries charge lower price, it's going to be the start of a downward spiral. And, therefore, it's going to end in fierce competition. And both firms, both countries make zero profits. Okay? So if we played this game just a single time, how is it going to look like? Well we've got Australia and we've got South Africa charging monopoly price or low price. If they both charge the monopoly price they make an overall level of profits that's 50 million and they divide it equally amongst themselves, so they make half the monopoly price each. If South Africa charges a slightly lower price than Australia, they're going to make 49 million, and Australia is going to be left without any market share. And vice versa, if Australia undercuts South Africa they're going to make 49 million, and South Africa will make zero profits. And equally, if both charge a lower price. They're going to make zero profits. And if we play this game as you can figure out easily, the one stage game is going to have one Nash Equilibrium, and that Nash Equilibrium is to charge for both Australia and South Africa the lower price, and both make zero profits. So let's suppose that they agree on the following strategy. Each firm charges the monopoly price, to begin with, but as soon as one country charges a low price in one year, the other firm will charge low prices for all periods in the future. So forever, one's cooperation has broken down, there's never going to be cooperation any more. This is called a trigger strategy, because each firm continues to behave as planned until a trigger is pulled. And that trigger is pulled, in this case, if the other firm charges a price slightly below monopoly prices. So does this ensure cooperation? Is this strategy enough to ensure cooperation? Well lets have a look. For each country, we're going to have to look at the payoffs. The payoffs if we cooperate this year are going to be 50 million divided by 2. Okay, we make half monopoly profits. If we cooperate, the other firm cooperates, we continue getting 50 million divided by 2, and in the following years, this is going to continue. What do we get if we deviate? If we deviate this year we're going to make 49 million for one period. Alright? So if we deviate today we're going to make 49 million but the next year, we've pulled the trigger and, therefore, the other firm is going to charge low prices. So we're going to be left with zero profits in the next year. And in following years again we're going to make zero profits. However, the future profits are not certain, because the game only continues with a probability of p. Otherwise, it would be very easy to figure out that, of course, cooperating is a good thing to do because once you start adding these up they're going to very quickly outweigh the profits from deviating for one period. But there's a probability that's going to tell us if the game continues or if it's not going to continue. So, therefore, if we cooperate this year, if we continue cooperating, our expected profits is going to be 50 million divided by 2 times 1 divided by 1 minus p. That's basically the simplified version of today's profits, tomorrow's expected profits, the day after tomorrow's expected profits and so on and so forth. And of course, p is between 0 and 1. So, if the likelihood that there's going to be a market that is going to be profits, next year is close to zero. Then this expression is going to be close to 1. And therefore, this overall expression is going to be close to 50 million divided by 2. If, on the other hand, p is close to 1, this is going to mean that it is very likely that the market will continue tomorrow and the day after tomorrow and so on. So, this expression here, becomes almost infinity and so there for the profit step we expect from cooperating also grow almost to infinity. If we compare that to deviating one year and getting zero afterwards, this is always going to be 49 million. Okay. Because we know we get that if we behave badly once. Okay? So, therefore, each country will cooperate in this year and the same logic applies to future years as well. If the payoff from cooperating is higher than the payoff from deviating. What is the payoff from cooperating here? It's 50 million divided by 2, that's half my monopoly profits, times 1 divided by 1 minus p. So, therefore, this is going to be bigger than the payoff from deviation if p is very close to 1. Right? The closer he gets to 1, the larger this expression gets and, therefore, the larger this overall expression gets. It's going to be more difficult to cooperate if p is very small. So if I don't expect the market to be successful or to exist the next year and the year after and so on, then we might as well not cooperate at all, okay. We might as well try to deviate early on. So to summarize, if we repeat the game infinitely, then we can get cooperation. What's the difference between finitely and infinitely repeated games? Well, in finitely repeated games, we can simply use backward induction and we fold the game forward. In an infinitely repeated game, we have to compare the expected payoffs in both cases. If one country deviates it sacrifices long-term profits for short-term gains. Okay? So deviating means I get 49 million once, but after that I basically or after that I basically make zero profits. Whereas, long-term profits would be, I get 50 million divided by 2, I get half the monopoly profits forever. So what determines if I want to cooperate or not cooperate? The likelihood of future payoffs, the p, right? The likelihood that there is going to be profits next year and the period after and so on. The relative value of payoffs, so the difference between making half the monopoly profits and getting and cheating once and making higher profits. So if the benefit from cheating is very high, then it's going to be more attractive to cheat, and therefore it's less likely that we'll have cooperation. So, as we've seen in this video, cooperation can be established and maintained in games with infinite repetition. To derive the likelihood of cooperation, we made some strong assumptions, and left some important aspects aside. So in the following video we relax some of these assumptions and we'll bring some important aspects in. So that way, we'll try to move closer to reality. But I hope for now this simple version of a cooperation game caught your interest and we'll see you back in the next video. So see you very soon.