# What is the prisoner's dilemma?

### Introduction

In this report is to introduce what is the Prisoner's Dilemma and the meaning about Cartel Union. First is to analyze the correlation of manufacturers and the prisoners' dilemma due to the correlation. The effectiveness of Cartel coalition is investigated and the principle of output dispense is given. Explain the factors affecting the stability of Cartel. From the game theory it is a rational behavior for interactive decision problems. In a game, several agents strive to maximize their (expected) utility index by choosing particular courses of action, and each agent's final utility payoffs depend on the profile of courses of action chosen by all agents. The interactive situation, specified by the set of participants, the possible courses of action of each agent, and the set of all possible utility payoffs, is called a game; the agents 'playing' a game are called the players. From this precondition to understand about the prisoner's dilemma from the Dominant Strategies, Extensions of the Prisoner's Dilemma and the Issues with Respect to the Prisoners' Dilemma these three parts to analysis the problem of Prisoner's Dilemma. And used the Oligopoly Problem to explain the reason why Prisoner's Dilemma can solutions the problem of oligopoly pricing and the situation in the market. Combine whit the Cartel and the Nash equilibrium

### The Prisoner's Dilemma

The Prisoner's Dilemma is one of the best-known models in game theory. In the picture, figure 1, the natural world in a ridiculous role to prove that two suspicious people help each other, or opposing each other.

In this assumptive situation, two confederates have been locked up in prison, and they tried to fake evidence of a crime and not sell out the other side do not recognize the crime. And the next step is give the serious of the punishment that each receives is determined not only by his behavior, but also by the behavior of his or her accomplice. The two prisoners are separated and cannot communicate with each other. And the result should be have four different possible.

- If one confesses to the crime and turns in the accomplice his sentence will be reduced.
- If one confesses while the accomplice does not, the first can make a deal with the police, and will be set free. But the information he provides will be used to incriminate his accomplice, who will receive the maximum sentence.
- If both prisoners confess to the crime, then each receives a reduced sentence, but neither is set free.
- If neither confesses to the crime, then each receives the minimum sentence because of the lack of evidence. This option may not be as attractive to either individual as the option of striking a deal with the police and being set free at the expense of one's partner. Since the prisoners cannot communicate with each other, the question of whether to "trust" the other not to confess is the critical aspect of this game.

Although this is a simple model, its lessons can be used to examine more complicated strategic interactions, such as arms races. If two antagonistic countries uncontrollably build up their armaments, they increase the potential for mutual loss and destruction. For each country, the value of arming itself is decreased because the costs of doing, for example-- financial costs, heightened security tensions, greater mutual destructive capabilities and so on -- provide few advantages over the opponent, resulting in an no profit outcome. Each country has a choice: cooperate to control arms development, with the goal of achieving mutual benefits, or defect from the pact, and develop armaments.

The dilemma stems from the realization that if one side arms itself and the other does not cooperates, the participant who develops armaments will be considered stronger and will win the game. If both cooperate, the best possible outcome is a tie. This is better than the payoff from mutual defection and an arms race, but it is not as attractive as winning, and so the temptation to out-arm one's opponent is always present. The fear that one's opponent will give in to such temptations often drives both players to arm; not doing so risks total loss, and the benefits of not arming can only be realized if one's opponent overcomes his or her temptation to win. Such trust is often lacking in the international environment.

The U.S.-Soviet relationship was a good example of this dynamic. For a long time, the two countries did not trust each other at all. Each armed itself to the hilt, fearing that the other one was doing so, and not wanting to risk being vulnerable. Yet the cost of the arms race was so high that it eventually bankrupted the Soviet Union. Had the Soviets been willing to trust the U.S. more, and vice versa, much of the arms race could have been prevented, at tremendous financial and security savings forboth nations, and indeed, the rest of the world.

The lessons initially drawn from the Prisoner's Dilemma can be discouraging. The game illustrates a zero-sum situation, in which one person must lose in order for the other to win. To keep from losing, each player is motivated to pursue a "winning" strategy. The collective result is unproductive, at best, and destructive, at worst.

### Dominant Strategies

What has happened here is that the two prisoners have fallen into something called” dominant strategy equilibrium." The first is Dominant Strategy: Let an individual player in a game evaluate separately each of the strategy combinations he may face, and, for each combination, choose from his own strategies the one that gives the best payoff. If the same strategy is chosen for each of the different combinations of strategies the player might face, that strategy is called a "dominant strategy" for that player in that game. The second is Dominant Strategy Equilibrium: If in a game, each player has a dominant strategy, and each player plays the dominant strategy, then that combination of strategies and the corresponding payoffs are said to constitute the dominant strategy equilibrium for that game. In the Prisoners' Dilemma game to confess is a dominant strategy, and when both prisoners confess, that is dominant strategy equilibrium.

This remarkable result -- that individually rational action results in both persons being made worse off in terms of their own self-interested purposes -- is what has made the wide impact in modern social science. For there are many interactions in the modern world that seem very much like that, from arms races through road congestion and pollution to the depletion of fisheries and the overexploitation of some subsurface water resources. These are all quite different interactions in detail, but are interactions in which individually rational action leads to inferior results for each person, and the Prisoners' Dilemma suggests something of what is going on in each of them. That is the source of its power.

### Extensions of the Prisoner's Dilemma

Few social situations can be modeled accurately by a single interaction. Rather, most situations result from a series of interactions over a long period of time. An extended version of the Prisoner's Dilemma scenario includes repeated interaction, which increases the probability of cooperative behavior.

The logic of this version of Prisoner's Dilemma suggests that a player's strategy depends on his or her experience in previous interactions, and that that strategy will also affect the future behavior of one's opponent. The result is a relationship of mutual reciprocity; a player is likely to cooperate if his or her opponent previously demonstrated willingness to cooperate, and is unlikely to cooperate if the opponent previously did not. The knowledge that the game will be played again leads players to consider the consequences of their actions; one's opponent may retaliate or be unwilling to cooperate in the future, if one's strategy always seeks maximum payoffs at the expense of the other player.

### Issues With Respect to the Prisoners' Dilemma

This remarkable result -- that individually rational action results in both persons being made worse off in terms of their own self-interested purposes -- is what has made the wide impact in modern social science. For there are many interactions in the modern world that seem very much like that, from arms races through road congestion and pollution to the depletion of fisheries and the overexploitation of some subsurface water resources. These are all quite different interactions in detail, but are interactions in which individually rational action leads to inferior results for each person, and the Prisoners' Dilemma suggests something of what is going on in each of them. That is the source of its power.

A number of critical issues can be raised with the Prisoners' Dilemma. That is a two-person game, but many of the applications of the idea are really many-person interactions. We have assumed that there is no communication between the two prisoners. If they could communicate and commit themselves to coordinated strategies, we would expect a quite different outcome. In the Prisoners' Dilemma, the two prisoners interact only once. Repetition of the interactions might lead to quite different results. Compelling as the reasoning is that leads to the dominant strategy equilibrium may be, it is not the only way this problem might be reasoned out. Perhaps it is not really the most rational answer after all. We will consider some of these points in what follows. Oligopoly prices and "Solutions" to Pricing Games there is a example to

Table 1

Perrier |
|||

price = $1 |
price = $2 |
||

Apollinaris |
price = $1 |
0,0 |
5000,-5000 |

price = $2 |
-5000,5000 |
0,0 |

In the Prisoners' Dilemma, each company has a strong rationale to choose one strategy -- and in this case it is a price cut. For example, Appolinaris might reason "Either Perrier will cut to $1 or it will not. If it does, it has a better to decrease-- otherwise it will lose all of the customers and lose $5000. On the other hand, if Perrier doesn't cut, I'm still better off to cut, since it will take the customers away and get a profit of $5000." Like this the price cut is a dominant strategy. But this is a very simplified -- unreasonable -- conception of price competition.

The Prisoners' Dilemma has been influential throughout the social sciences, because the rational and self-interested decision-makers, choosing their strategies in isolation from one another, find that the strategies interact so that they both have bad outcomes.

In application to the problem of oligopoly pricing, the examples given so far seem to give strong support to the second hypothesis of oligopoly pricing, the hypothesis that oligopoly prices will be the same as those in a P-competitive market: zero profits. But that's not really so clear. And antitrust laws are designed to make such a price-fixing agreement illegal. But we haven't always had antitrust laws -- they were enacted because many people believed that businessmen were collaborating to fix high prices. And even now, there may be ways to get around the law.

When the decision-makers in a "game" get together, agree on a common strategy, and share out the gains from it among themselves, the agreement they come to is called a "cooperative solution" to the game. The examples we have looked at so far are "noncooperative solutions."

It appears that we cannot rule out the possibility of a cooperative solution to the oligopoly pricing game, so we need to look a bit at the cooperative alternative in game theory.

### The Oligopoly Problem

It seems that game theory doesn't solve the oligopoly problem after all. There are at least two kinds of solutions to the problem of oligopoly pricing -- cooperative and noncooperative. Actually, it's a bit worse than that. In each of the two categories, there is actually more than one sort of solution, depending on how we approach the problem!

That had become pretty clear to economists by the 1960's, and many economists lost interest in game theory. But despite its failure on this specific point, game theory has proved to be a powerful tool of economic thinking, so that it has become more influential since the 1960's, culminating in the Nobel Prize for three game theorists (including John Nash, who invented the Nash-Equilibrium) in 1994.

And it is not just simply a failure in the analysis of oligopoly prices. Sometimes it's important to be confused at a higher level. We know that oligopoly pricing is a hard problem, but the reason why it is a hard problem. The pricing examples we have seen here give some insight about the reason why price competition -- when it does occur -- is so powerful in bringing prices down to the lowest stable level. And we can apply the same methods to a range of other problems, both related to imperfect competition and in other fields of economics.

### Cartel

Cartel union is an important form of business cooperation. So as to used price fixing and set limit to pursue the trade profits, then used the basis of equitable distribution of the profit-sharing mechanism. on the basis of the Cartel expansion of the alliance model, and analysis the repeated game mechanism; the result is that the static game under the conditions of the enterprises do not have the stability of the Cartel Union, in repeated games under the conditions of enterprises to cooperate balance in the balance between competition and choice depends on the size of the discount factor.

Forgiveness policy is an effective cartel enforcement policy, it will help undermine the stability of the cartel, and the cartel will help improve the efficiency of law enforcement. Design the reasonable rule and effective policy of forgiveness play a role in the foundation of concrete and determined that forgiveness is transparent the basic requirements of the policy. Forgiveness policy and the effect of the cartel legislation, law enforcement situation is closely related to severe legal sanctions, the firm's attitude toward law enforcement and strong enforcement measures to promote the effective implementation of the policy of forgiveness. Cartel is in order to strengthen the law enforcement.

### Nash equilibrium

Assume N that a player involved in the game, given other people the strategy under the condition of each player to choose their own optimal strategy (personal best strategy may or may not depend on others to rely on the strategy), so to maximize their effectiveness. All in games player will make a strategic combination（Strategy Profile）. Nash equilibrium refers to a combination of strategy, this combination of strategies by all participants the best strategy component. Even given the strategy of others, no one has sufficient reason to break this balance.

It has the case with the Prisoner's Dilemma. Assuming there are two A and B of the Joint thief was convicted, into private homes seized by the police. Police were placed in two different rooms within the two to trial, each of the suspects, the police are given the policy is: If you suspect a crime has been, frankly, to hand over the stolen goods, the evidence, both were convicted. If the other suspects also made frank, the two men were each sentenced 8 years; if another crime suspects but frankly did not deny, however, prevent the crime of official duties (as a result of evidence of their guilt has been) plus 2 penalty , To be honest and active 8-year sentence was immediately released. If they both deny, for lack of evidence the police can not be sentenced two of theft, but the accusal they can into the private home of the charges would be liable to imprisonment for 1 year.

——————————————————————————

┃ B ┃ B ┃

————————┃————————┃————————┃

┃ confess ┃ disavow ┃

————————┃————————┃————————┃

A confess ┃ -8, -8 ┃ 0, -10 ┃

————————┃————————┃————————┃

A disavow ┃ -10, 0 ┃ -1, -1 ┃

————————┃————————┃————————┃

On the case, clearly the best strategy is to deny both of them; the result is only sentenced to 1 year. However, due to two in isolation, should be the first from a psychological point of view, the parties will suspect the other party will sell out in order to protect themselves, followed by Adam Smith's theory, hypothesis that everyone to be " Rational economic man " and they will proceed from the purpose of self-selection. The two men will have a calculation process: If one frankly, one deny, would take a 10-year prison, a maximum of only 8 years frank; one would deny, one person will be able to be released, and he would take 10 years in prison. Taking all these circumstances into account, whether or not someone is honest, for another one, is frankly a cost-effective. The two would move such a brain, in the end, both chose to be honest; the results were sentenced to 8 years imprisonment.

Rational agent-based on economics premise the assumption that the two prisoners in line with the interests of their own choice is to confess frankly, was beneficial to both sides of the strategy is not to confess and thus would not have been released. Both of this option, frankly, as well as the strategy are therefore sentenced to 8 years of the end. So "Nash equilibrium" in the lead to the "invisible hand" of the principle of a paradox: from self-serving purpose, the results Dog in the manger, not altruism or self.

### The relationship between the ‘Prisoners Dilemma' and the theory of cartels

Monopolize market is the reality of the existence that there is an important market structure, which refers to products in the market. With a product that only a few companies to provide production and characterized by any decision-making will Manufacturers of other vendors have an impact on production. In the monopolistic cartel Union, Cartel's output is allocated to form a key cartel. Such as the OPEC. Since the international oil price in the 1970's sharp rise year, OPEC cartel does have some of the traditional organization of the basic features, but with other traditional organizations cartel, OPEC also has some obvious differences. most of the studies can not prove that OPEC is a cartel; OPEC can only prove that there is a similarity between the members. In addition, the OPEC cartel's interests have been significantly reduced. As a result, the conclusion can be drawn: For now, OPEC is not an effective operation of the cartel's organization; or it could be argued that the international oil market, OPEC's is not entirely influence the power of the cartel.

Cartel's output has a bearing on the distribution of the vital interests of manufacturers, as the productivity of firms, manufacturers. After forming cartels the ideal of production allocation should be based on the manufacturers' effect to determine the productivity of production quotas and more efficient production, Inefficient and less productive. Currently, to make the cartel of several companies as a monopoly vendor various Production plant. So the formula is:

Ci = C = R (I = 1, 2... n,)

“Ci” for the first “i” makers of the marginal cost

“C” for the industry as a whole marginal cost

“R” for the industry as whole marginal gains

Actually, cartel will follow the general principles to distribute the sales quota.

According to the manufacturer's production capacity to the level of sales quota allocation, the greater the scale of production, the higher ability of manufacturers will get the greater quota.

According to the manufacturer's sales in the past to the level of sales quota allocation, sales in the past, the higher the level of the companies get bigger quotas.

Base on location to distribute quotas, if the companies get a certain region or country market, while other manufacturers have been other regional or national markets.

Through the Manufacturers together to formation of a cartel agreement and is often unstable, this instability in the cartel but also as the increased manufacturers, from the chart of analysis can identify the manufactory will leave the cartel in order to Lead to the dissolution of the cartel.

From this chart, when the Manufacturers of production is mi, Ci=/=Ri, the mean is mi is not the manufacture's good output, in fact, if the manufactures reduce the price to P1, the Manufacturers amount of the request will become mtto mt', and in mt', Ci=Ri, the mean is the Manufacturers optimal output. That is to explain the manufactures Deviated from the limitative price Po, it will be increase the profit. At the same time, if there is another reality deviated from the makers of possible price-fixing. Such as the price or prices in one way to cut the price with impunity, the result is bound to deviate from the manufacturers limit price, which eventually led to the disintegration of the cartel.

### Conclusion

From this report we know about the Prisoner's Dilemma in game theory, what is the meaning of the Prisoner's Dilemma and the impact of economics. Different possibilities will bring different results. According to the first part of the chart there have four different the result of crime. But the both choices of strategy and frankly was sentenced to 5 years, therefore the result is called "Nash equilibrium", also called non-cooperative equilibrium. Each party in the choice of strategy when there is no "conspiracy", they just choose the most favorable to their strategy, without thinking about the benefit between each other. On other point of view through the prisoner's dilemma develop the resulting is the formation of monopolistic market and the price war. Lead the different companies have different price competition, under the Nash equilibrium and the influence of game theory will consider adopting a normal pricing strategy or the formation of high-price strategy of monopoly prices and make every effort to obtain the monopoly profits. If the monopoly can be formed, then will getting the largest mutual profit. In fact, perfectly competitive equilibrium is the "Nash equilibrium" or "non-cooperative game balanced." In this state, each manufacturers or consumers are all the others the price has been set for the decision-making.

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