So hello again. You ready for more?
In the previous video we've looked at a very simplified version of the Diamond
Cartel. We've had South Africa and Australia as
the only two players in this game. And we said okay, well they repeat to set
prices or they repeat setting prices or quantities forever.
But there's a likelihood of P that there's going to be a market in the next
period, right? So that was all we looked at in
this in previous video. But, reality of course is going to
look slightly different because there are other factors that may affect
the likelihood of cooperation. So in this video we'll look at precisely
that. So just to recap, each country in our
previous model would cooperate this year if the payoff from cooperating is higher
than the payoff from deviating. So, the payoff from cooperating, just to
recap, it's half the monopoly profit, that's what you get each period.
And if you just continue doing that until the market disappears, the overall value
of this is going to be 1 divided by 1 minus p.
The payoff from deviating is $49 million, because you only get this deviating
payoff once. And after that, it's going to be
competition forever. Of course we know that p is the
probability that the game goes on, and if p is a small number,
so if p is very close to zero, then this would mean that this number, this goes
close to one, and therefore this gets close to 25 million, 50 million divided
by 2. So it's less likely that we'll cooperate
if the likelihood that the game goes on is very small.
On the other hand, if p is close to 1, this is going to be very close to zero.
This is going to mean that this will become very large, and therefore this
overall payoff from cooperating will become very large.
So if the likelihood that the market is going to continue is high,
then we're likely to coorporate. But, this is really not the only thing
that will play a role in determing if we want to coorporate or not.
So one thing that will play a role is the number of competitors.
So we've looked at South Africa and Australia as the only two diamond
producing countries. That's not realistic, there's more
countries that produce diamonds, and that may be part of this cartel.
So, if there are more competitors, that's going to mean that the share of the
monopoly profit that I get as a participant in this cartel gets smaller.
So here, we've had the payoffs from cooperating were monopoly profits divided
by 2 i.e., South Africa got half of the profits and Australia got a half of the
profits. Now if there are ten players, ten
countries producing diamonds, then this number will be replaced by ten.
So this means that our payoff per period, from being part of the cooperative
agreement, is going to be much smaller. And I'm less likely to cooperate, okay?
I still have the same potential payoff from deviating, but my payoff from
cooperating decreases. Another aspect is the importance of the
future. So how important do I value the future?
If the future becomes more important, the interest rate, where this is usually
reflected by a decrease in the interest rate.
So let's do a quick thought experiment. if the interest rate would be very high,
then the difference that I assign to an amount of money that I get today and an
amount of money that I get tomorrow is going to be very strong, is going to be
very large. Because I could reinvest the money that I
get today for one period, and that would give me a very high return.
So that means that if I assign a high value to the future, that means it's
almost worth as much to me as the payoff today.
So this is going to mean that if I have a low interest rate, then payoffs today and
payoffs tomorrow are almost the same size.
So this is going to change our payoffs from cooperating.
And it's going to change it by adding this additional formula.
So if we just play this through for a second, if the interest rate is small
this is going to be close to 1, this here is going to be close to zero.
This is going to become very large, and therefore this whole expression is going
to become very large. So a small interest rate is going to make
it more likely that I'm going to cooperate.
Another possible influencing factor is the degree of punishment.
Remember what we played before?
The agreement was I'm going to cooperate, but the moment you don't cooperate I will
punish you very hard. So I will go down and charge low prices
forever. This is what's called the trigger
strategy the moment you pull the trigger, there's never going to be cooperation any
more. It's a very harsh punishment.
And let's see what happens if we make this degree of punishment a little bit
softer. So we change the agreement to say well as
soon as one country charges a low price in one year, the other one will charge
low prices for the next five years. So it's only a limited period of time
that we punish the other player, the player that didn't cooperate.
So the payoff from cooperating is not going to change, right?
Because that's the path where we never have to use the punishment.
On the other hand the payoff from deviating does change.
So we get the 49 million from cheating once, we get 0 profits for five periods,
but then after that we get 50 million divided by 2.
Because we revert back to monopoly prices and we share the monopoly profits.
So this means that I have to take this into account that maybe the market isn't
going to exist by then. But if it does then we start reverting
back to monopoly prices after these punishment periods of five.
Okay, so to sum up, we found what some of the factors enhancing cooperation are.
In particular we looked at the importance of future payoffs.
If I value the future highly, then it's more likely to behave cooperatively in
the present. We also identified some factors that make
cooperation more difficult. In particular if we are in the market
with a large number of competitors, it's more difficult to maintain cooperation.
Where does that come from? Comes from the fact that if we have to
split the profit from cooperation over a large number of
players, then each individual player will get
a smaller piece of the pie, and he's more inclined to deviate.
The low degree of punishment, or a forgiving punishment, also makes
cooperation more difficult. Now that's quite interesting because if
you have a very draconian punishment, like the trigger strategy,
the moment you misbehave I'm going to start punishing you
forever, okay? That draconian punishment is actually
very good for cooperation, because people are deterred by this punishment.
Okay, so to sum up, we've basically built up a a model of cooperation in a
prisoner's dilemma and we found that in a finitely
repeated game, cooperation is very unlikely.
Because we just look at the endpoint of the game, and we solve the endpoint by
moving the game forward. On the other hand, an infinitely repeated
game can give us cooperation under certain circumstances.
And these circumstances are basically affected by a number of structural
parameters in the market. Instructional parameters like the
likelihood of future payoffs, the importance of future payoffs, the number
of competitors and the degree of punishment.
So with that I hope that we brought that model, that seemingly abstract model of
cooperation a little bit closer into reality.
And look forward to seeing you in future sessions.
Thanks.