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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.

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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.

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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.

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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.

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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.

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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?

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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.

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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.

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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

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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,

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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.

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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.