Punishment strategies game theory

Now consider the game before the last. The same logic applies for prior moves. Therefore, confess-confess is the Nash equilibrium for all rounds. The situation with an infinite number of repetitions is different, since there will be no last round, a backwards induction reasoning does not work here.

At each round, both prisoners reckon there will be another round and therefore there are always benefits arising form the cooperate lie strategy.

However, prisoners must take into account punishment strategies, in case the other player confesses in any round. If we assume the game can be played ad infinitum, we can apply it in a collusion agreement game, where two firms collude, forming a cartel. In this case, we simply need to apply the formula for calculating an infinite sequence and a discount factor to compensate for the fact that the gains to be derived are over time accounting for impatience, inflation , loss of interest, etc.

For our threats or offers to be credible, this left hand side must be greater than the right hand side, which represents the one off payoff to be gained from deviating and breaking our cartel.

It is worth reminding here that fair competition is regulated in almost all countries, with cartels being banned, so most markets that lend themselves to reduced competition and price fixing are closely monitored. Although this example is widely used in game theory and for the analysis of market structures , it can be easily seen that it does not represent a real situation. Therefore, considering a Stackelberg duopoly might seem more realistic.

We could cheat both on the quality of the fruit that I provide or the quantity of the fruit that I provide to Jake, and he can cheat on the quantity or quality of the vegetables that he provides to me. Our central intuition is: perhaps what can give us good incentives is the idea that if Jake cooperates today, then I might cooperate tomorrow, I might not cheat tomorrow. Similarly for me, if I provide Jake with lousy fruit today he can provide me with lousy vegetables tomorrow.

So what do we need? We need the difference in the value of the promise of good behavior tomorrow and the threat of bad behavior tomorrow to outweigh the temptation to cheat today. So that temptation to cheat has to be outweighed by the promise of getting good vegetables in the future from Jake and vice versa. What we need is the gain if I cheat today to be outweighed by the difference between the value of my relationship with Jake after cooperating and the value of my relationship with Jake after cheating tomorrow.

Now what we discovered last time—this was an idea I think we kind of knew, we have kind of known it since the first week—but we discovered last time, somewhat surprisingly, that life is not quite so simple. So in particular, if we think of the value of the relationship after cooperating tomorrow as being a promise, and the value of the relationship after cheating as being a threat, we need these promises and threats to be credible.

And one very simple area where we saw that ran immediately into problems was if this repeated relationship, although repeated, had a known end. Why did known ends cause problems for us? What we do has to be consistent with our incentives in the last period.

So if we look at the second to last period we might hope that we could promise to cooperate, if you cooperate today, tomorrow. We know that sub-game perfect equilibria have the property that they have Nash behavior in every sub-game, so in particular in the last period of the game and so on.

So what we want to be able to do here, is try to find scope for cooperation in relationships without contracts, without side payments, by focusing on sub-game perfect equilibria of these repeated games. We already noticed last time some things about this. Why is it not a sub-game perfect equilibrium? Because in particular, if Jake is smart and he is , Jake will look at this equilibrium and say: Ben is going to cooperate no matter what I do, so I may as well cheat, and in fact, I may as well go on cheating.

So Jake has a very good deviation there which is simply to cheat forever. And we need to focus on strategies that contain subtle behavior that generates promises of rewards and threats of punishment that induce people to actually stick to that equilibrium behavior.

We focused on what we called the grim trigger strategy. And the grim trigger strategy is what? It says in the first period cooperate and then go on playing cooperate as long as nobody has ever defected, nobody has ever cheated.

But if anybody ever plays D, anybody ever plays the defect strategy, then we just play D forever. So this is a strategy, it tells us what to do at every possible information set. And what we left ourselves last time was checking that this actually is an equilibrium, or more generally, under what conditions is this actually an equilibrium.

So we got halfway through that calculation last time. So what we need to do is we need to make sure that the temptation of cheating today is less than the value of the promise minus the value of the threat tomorrow. So the temptation today is: if I cheat today I get 3, whereas if I went on cooperating today I get 2.

So the temptation is just 1. The threat is playing D forever, so this is actually the value of D, D forever. The promise is the value of continuing cooperation, so the value of C,C for ever. So this object here is just 0. So this thing here, the value of 2 for ever is what? And then the day after, what is it? Everyone happy with that?

And I just need to take an account of the fact that the game may end between tomorrow and the next day, the game may end between the day after tomorrow and the day after the day after tomorrow and so on. So what is the value, what is thing? Except for that first 2 there, so this is just equal to 2. Now this is a calculation I can do. So we can put that in here as well. Now before I go onto a new board I want to do one other thing.

However, all of these objects on the right hand side, they start tomorrow, whereas, the temptation today is today. Temptation today happens today.

These differences in value start tomorrow. The world may end, or more importantly the relationship may end, between today and tomorrow. So how much do I have to weight them by? That is a way of sustaining cooperation. Everyone happy with that so far? Let me just turn my own page. So what have we shown so far? After all, the defection we just considered, the move away from equilibrium we just considered was what? We considered my cheating today, but thereafter, I reversed it back to doing what I was supposed to do: I went along with playing D thereafter.

So you might want to ask, why would I do that? Why would I go along? It tells me to play D. Why am I going along with that? Elsewise no man will ever bend the knee to you. This is a pretty good distillation of a what game theorists call tit-for-tat. The equilibrium stays stable as long as the Throne can dominate all of the potential competitors, as Aegon I could with his dragons.

Interestingly, this is often the case in situations where it seems desirable to get rid of a tyrant. If the presence of the tyrant is the only thing that keeps a stable equilibrium, then getting rid of the tyrant just disrupts the equilibrium and leads to all out conflict. You could draw on examples like Iraq after Saddam Hussain, or Yugoslavia after Tito, but it also applies to some situations with non-state actors like organised crime syndicates — and indeed company succession after a dominating figure retires.

But once the tyrant is out of place, a different, more fluid and more dangerous situation takes place, which is why binding the great houses together through marriage pacts would be so important. The marriage pact between the Lannister and Baratheon houses has held for 14 years by the time the series starts, but it is fragile, and many of the houses have reasons to want to get involved. Of course if one party has a weapon of mass destruction, then this equilibrium is supported.

The threat is non-credible in game theoretic terms because you would not choose the payoffs that come from playing the threatened strategy — in this case probably destroying your own territory, but probably not destroying your enemy.

But Game of Thrones also points to some of the limits of game theory. If and when they decide to do so, it will be in a more concentrated form that will not cause the same level of euphoria or mood altering effects that are often associated with prescription drugs. There are other uses for CBD that we do not currently have access to. One of the most promising uses of CBD is to help treat and possibly prevent the occurrence of seizures in those suffering from epilepsy.

Since hemp oil and CBD are in its most natural form, it is safe for use in food, cosmetics and dietary supplements. It can even be used for aromatherapy and skin care treatments as well. Game theory is the science of strategic reasoning, in such a way that it studies the behaviour of rational game players who are trying to maximise their utility , profits, gains, etc. However, it is not until the 20 th century, that game theory is broadly developed. We can differentiate several periods in the game theory evolution through the 20 th century.

However, many concepts were continuously developed through the years. During the earliest years of the century, from to , the main focus of game theory was on strictly competitive games commonly referred to as two-person zero-sum games.

This kind of games has been extremely productive, as they have set the bases for future development of game theory, becoming its first main milestones. With this book, game theory gained the status of an independent scientific discipline. They were the first to extensively apply game theory for practical reasons, especially for analysing economic behaviour.