Dilemma of the prisoner

The dilemma of the prisoner is a famous example of the Game theory. In this play, as in good of others, it is made the assumption that each player called “prisoner here” tries to maximize his benefit without trying to make some as much with those of the other player.

Principle

The usual form of this dilemma is that of two Prison niers (accessory to an offense) retained in separate cells and which cannot communicate.

so only one of both acknowledges, this one is certain to obtain a handing-over of sorrow whereas the second obtains the maximum sentence (10 years);
if both acknowledge, they will be condemned to a lighter sorrow (5 years);
if none acknowledges, the sorrow will be minimal (6 months), for lack of elements to the file.

This problem models the questions of tariff policy well: the competitor which lowers its Prix gains market shares and can thus increase its sales and to if required increase its Bénéfice… but if its principal competitor does as much of it, both can lose there.

The balance of Nash for this type of play does not lead to a Pareto's optimum (i.e. a state in which one cannot improve the wellbeing of an individual without deteriorating that of another). With balance, each prisoner chooses to be lacking even if they gained to cooperate. Unfortunately for the prisoners, each one is encouraged to cheat after having made the promise to cooperate. It is the heart of the dilemma.

This is however appreciably different from a situation of open market where two economic agents " prisonniers" can cooperate to reach a Pareto's optimum (because the optimum seen by the producers does not go not always in the interest of the consumer, the laws Antitrust of all the countries prohibit, officially at least, the agreements between competitor producers).

In a play of the dilemma of the prisoner repeated , each player with opportunity “of punishing” the other player for his preceding not-co-operation. The co-operation can thus occur in this configuration. The incentive to cheat is lower than the threat of punishment, which introduces the possibility of cooperating.

The dilemma of the prisoner is used by the economists, the mathematicians, the Psychologues, the biologists and the specialists in Political science. The Paradigme correspondent is also mentioned in Philosophie and in the field of the cognitive Sciences.

Dilemma of the traditional prisoner

Formulation

The first experiment of the dilemma of the prisoner was carried out in 1950 by Melvin Dresher and Merill Flood, which worked then for the RAND Corporation. Thereafter, Albert W. Tucker presented it in the form of a history:

Two suspects are stopped by the police force. But the agents do not have enough evidence to accuse them, therefore they separately question them in their making the same offer. “ If you denounce your accomplice and that he does not denounce himself, you will be given in freedom and the other will bail out 10 years of prison. If you denounce it and he also, you will bail out both 5 years of prison. If nobody denounces himself, you will have both 6 months of prison. ”

One often summarizes the Utilité S of each one in this table:

Each prisoner reflects on his side by considering the two possible cases of reaction of its accomplice.

“ If he would denounce me:

If I keep silent myself, I will make 10 years of prison;
But if I denounce it, I will make only 5 years. ”

“ If he would not denounce me:

If I keep silent myself, I will make 6 months of prison;
But if I denounce it, I will be free. ”

“ Whatever his choice, I thus may find it beneficial to denounce it. ”

If each accomplice makes this reasoning, both probably will choose to denounce mutually, this choice being more impresses Rationalité. In accordance with the statement, they will consequently bail out 5 years of prison each one. However, if they were both remained quiet, they would have bailed out only 6 months each one. Thus, when each one continues its individual interest, the result obtained is not optimal within the meaning of Vilfredo Pareto.

This play is with nonnull nap, i.e. the sum of the profits for the participants is not always the same one: it raises a question of co-operation.

So that there is dilemma, temptation T (I denounce it, it keep silent myself) must pay more than the co-operation R (both are keep silent), which must report more than the punishment for selfishness P (I denounce it, he denounces me), which must be more developing that deception S (I keep silent myself, it denounces me). This is formalized by:

T > R > P > S (here: 0 > -0,5 > -5 > -10)

So that a collaboration can be born in a repeated dilemma (or iterative) (see low), 2 blows of co-operation R must be more developing that the rotation Tentation/Dupe. What makes the 2R condition > T+S: 2*-0,5 > 0 + (- 10).

Examples of real situations

The dilemma of the prisoner provides a general framework to think the situations where two or several actors may find it beneficial to cooperate, but an interest even more extremely not to make it if the other does it, and no means of forcing the other. Certain fields like ecology or the economy completely use the mathematical nature of the problem by giving a concrete direction to the parameters, for example that of payments. Others do not retain that the conceptual framework to categorize situations where the payments are not very quantifiable.

The following examples will make it possible to better determine the diversity of the possible applications and the great general information of the framework of the dilemma of the prisoner.

Economy

A canonical example is the case of two companies which do not have the right to get along on a common marketing policy (what the antitrust law states-unienne and the French law prohibit) and which wonder whether it is necessary for them to carry out or not price drops to conquer market shares at the expense of their competitor. It goes without saying that if both lower their price, they will be both losers compared to the status quo , except of course if the market presents the exceptional elasticity which can characterize the sector Informatique, for example.

The dilemma of the prisoner is given, wrongly, like example that competition leads to socially deplorable situations. Indeed, if one utilizes a third player (the company), it is possible to imagine that she prefers the absence of co-operation of the criminals between them, or the absence of collusion between the companies. Especially, the price drops resulting from competition, that the game theory approaches with realism, benefits the unit from the consumers while enabling them to obtain same satisfaction at less price, and thus to use the difference to consume elsewhere, and to support the general dynamism. The general wellbeing increased even if from the subjective and incomplete point of view of the only producer, one had been able to believe that competition brought to a fall of the satisfaction of the actors concerned. In other words, the result of the play can be judged like socially preferable. In fact, the structure of the dilemma of the prisoner isolates a mode from interaction between players, makes it possible to draw from it from the direct consequences for them without damage of other consequences to specify on a case-by-case basis.

Ecology

The game theory, and the dilemma of the prisoner in particular, are frequently used in ecology to model the evolution behaviors between individuals of same a Espèce worms of the strategies évolutivement stable. The appearance and the maintenance of the Behavior S of Co-operation for example, lend themselves to this type of analysis. Richard Dawkins did of it one of the central points of its theory of the egoistic Gène, since the optimization of survival can pass by an apparently altruistic behavior.

International policy

That is to say two countries have and B. the countries has and B can choose to maintain or not a Armée. If both have an army, the war is impossible, because too expensive. The military expenditure is then a net loss for the two countries. If only one has an army, it can obviously conquer without blow to férir the other, which is worse. Lastly, if none has of army, peace reigns and the countries do not have a military expenditure. The situation of co-operation making it possible each one not to have of army is obviously preferable with the situation where the two countries maintain an army, but it is unstable: each of the two countries has a strong incentive unilaterally to obtain an army to invade the other.

Sociology and anthropology

The Social sciences also adopted the conceptual framework of the dilemma of the prisoner to speak about situations blocked in an unfavourable state by the difficulty of coordinating the actions of the various agents or of checking (and possibly of punishing) the egoistic deviances. This framework was thus used by the sociologists to explain the slowness of the adoption of the high-output seeds in the French agricultural world in the Années 1960. The whole of the farmers would have had interest to adopt these varieties, but the social pressure being exerted on those which tried the experiment dissuaded from many farmers.

Psychology

The dilemma of the prisoner frequently meets in the relations of couple. Thus let us take the example of a married couple in a state of conflict, each partner having had a extra-marital adventure without the knowledge of the other. Each of both would like to be able to acknowledge its fault and to be reconciled (co-operation). However, each of both fears the contempt of the other if it is the only one to have fault, and prefers the state of conflict.

Case of resolution of the dilemma

In a very great number of real situations, the alive species which of it are able develop of the process of socialization allowing to solve the dilemma of the prisoner, i.e. to make stable the co-operative strategy, which is optimal. There are two ways of obtaining this result: The first possibility is the installation of a memorizing of the protagonists and their behaviors, in order to be in the case of the repeated dilemma (which is not any more one dilemma). The second solution is to establish a code of conduct within the company, associated with sanctions for the contraveners. In this way, treason, if it is more heavily sanctioned than than it pays, becomes less advantageous than the co-operation. The mankind developed a faculty very important to implement these two systems of resolution of the dilemma of the prisoner, by his sympathy and storage capacities, and by a legal system and repressive very sophisticated. Thus, the Démocratie constitutes one of the most advanced means which the dilemma of the prisoner with large scales developed the mankind to solve.

Limits of the applications

the protagonists of the dilemma in his traditional form have the choice only between two attitudes: to cooperate or not to cooperate. This dichotomy is obviously very artificial. Actually, all the degrees exist between these two terms, returning the application of this dilemma much more delicate. However this form has a teaching goal and the structure lights more complex or more realistic versions (competition in Cournot for example).
It acts of a remaining model with two protagonists isolated until the advertisement from the results from the transaction. In the real life, three individuals or more can interact what notably complexes the choice between the co-operation and the not-co-operation. However phenomena of stowaway (the free-riding of the english-speaking) whose existence is not a doubt show forces it to them-also example.

The repeated dilemma

In its book The Evolution off Co-operation (evolution of the co-operation, 1984), Robert Axelrod studies a traditional extension of this dilemma: the play is repeated, and the participants keep in memory the preceding meetings. This version of the play is also called iterative dilemma of the prisoner . It gives another illustration starting from an equivalent situation: two people exchange bags, supposed to contain money respectively and a good. Each one has an immediate interest to pass an empty bag, but it is more advantageous for both than the transaction takes place.

When one durably repeats this play in a population, the players who adopt a there interested Stratégie lose with the long run, whereas the players apparently not involved see their finally rewarded “altruism”: the dilemma of the prisoner is not thus strictly speaking any more a dilemma . Axelrod saw there an explanation of the appearance of an altruistic behavior in a context of evolution darwinienne by Natural selection.

The best strategy in a deterministic context “eye for eye” ( “Tit for Tat” , another current translation being “Lump of a woman”) was conceived by Anatol Rapoport for a computerized contest. Its exceptional simplicity was right of the other proposals. It consists in cooperating with the first blow, then each time to reproduce the behavior of the adversary of the preceding blow. An alternative, “eye for eye with forgiveness”, appeared a little more effective: in the event of defection of the adversary, one cooperates sometimes (of 1 with 5 %) with the following blow. That makes it possible to avoid remaining blocked in a negative cycle. The best adjustment depends on the other participants. In particular, “eye for eye with forgiveness” is more effective if the communication is scrambled, i.e. if it happens that another participant interprets a blow wrongly.

For the dilemma of the prisoner, there does not exist always optimal strategy. If, for example, all the population is systematically lacking except an individual who respects “eye for eye”, then this last has a disadvantage with the first blow. Vis-a-vis a unanimity of defect, the best strategy is always to betray too. If there are a share of systematic traitors and d'" eye for œil" , the optimal strategy depends on the proportion and the duration of the play. While making disappear the individuals who do not obtain good totals and while making duplicate those which carry out, one can study the dynamic interesting ones. The final distribution depends on the initial population.

If the number NR of iterations is finished and known, the balance of Nash is systematically to be lacking, as for NR =1. That is shown simply by recurrence:

with the last blow, without possible sanction on behalf of the adversary, one may find it beneficial to betray;
by doing this, with the last but one blow, as it is anticipated that the adversary will betray though it arrives at the following blow, it is better to betray too;
one continues the reasoning until refusing to cooperate with all the blows.

So that the co-operation remains interesting, the future must thus remain dubious for all the participants -- a possible solution is to draw a NR random.

The situation is also astonishing if one plays dilemma of the prisoner indefinitely, the score being the Moyenne scores obtained (calculated in a suitable way).

The dilemma of the prisoner is the base of certain theories of the human co-operation and confidence. If one assimilates the situations of transactions which claim confidence with a dilemma of the prisoner, a behavior of co-operation in a population can be modelled like a play between several players, repeated - from where the fascination of many academics for a long time: in 1975, Grofman and Pool estimated already at more than 2000 the scientific articles on the subject.

This work provides a base modélisable, quantitative, for the scientific study of the laws Morale S.

Axelrod gives in its work How to make a success of in an egoistic world an example of strategy eye for eye within the framework of the dilemma of the iterative prisoner: during the war of the trenches, combatants of the two camps, and this, against the opinion of the command, applied the principle of living and leave-food . The protagonists thus never started in first the hostilities but strongly retorted with any aggression.

Alternatives

There exist alternatives of this play which, by modifying the profits slightly arrive at very different conclusions:

Play of Hen-wet

Hen-wet is another play with not-null sum, where the co-operation is rewarded. This play is similar to the dilemma of the prisoner in what it is advantageous to betray when the other cooperates. But it remotely in what it is advantageous to cooperate even if the other betrays: the double defection is worst solutions -- thus an unstable balance -- whereas in the dilemma of the prisoner it is always advantageous to betray, which returned the balance of double stable defection. The double co-operation is in the two plays an unstable balance.

A Matrice of the profits resembles:

if both cooperate, they receive +5;
if one cooperates whereas the other is trued, then the first obtains +1 and the other +10;
if both are lacking, they touch -20.

“Hen-wet” name is drawn from the automobile “play”:

Two cars launch out one towards the other, ready to return itself inside. Each player can deviate and avoid the catastrophe (co-operation) or keep the course with the collision risk (defection).

He is advantageous to seem a " dur" who will not give up and to intimidate the adversary… as long as one manages to remain concerned.

One finds examples concrete in many daily situations: the maintenance of the house common to a couple, for example, or the maintenance of a system of irrigation between two farmers. Each one can only maintain it, but they benefit from it both as much. If one of them does not ensure its share of maintenance, the other always may find it beneficial to do it in its place, to continue to sprinkle. Consequently, if one manages to establish a reputation of indelicate dominating -- i.e. if the practice is taken that it is always the different one which deals with maintenance -- it will be likely to maintain this situation.

This example can also apply in international policy, in the situation where two States maintain a disagreement which is likely to lead to a war. To pass for a wet hen is a guarantee to be later on again confronted with the same situation (like the France and the Great Britain noted it before 1939), but to maintain a reputation supposes an expenditure (maintenance military) and risks (always possible war).

Play of the insurance

A play of the insurance to the same structure as a dilemma of the prisoner, with this close the rewards in the event of mutual co-operation are more important than if the test is lacking. A matrix of profit will resemble:

if both cooperate, each one receives +10;
if one cooperates and that the other been lacking, the first has only +1 and the other +5;
if both are lacking, they draw +3 from them each one.

The play of the insurance can lead to a very stable balance, since the most advantageous balance is stable within the meaning of Nash: none may find it beneficial to deviate. It is nevertheless possible that a player deviates by error, confidence or the practice is then called into question.

Friend or enemy

" Friend or ennemi" ( “Friend gold Foe” ) is a play currently diffused on a chain cabled in the United States (Game Show Network). It is an example of dilemma of the prisoner tested on private individuals within an artificial framework. On the plate, three pairs of participants clash. When a pair is eliminated, its two members are distributed their profits according to a dilemma of the prisoner. If both cooperate (“Friend”), they equitably share the sum accumulated during the play. If none cooperates (“Foe”), they are left without anything. If he one cooperates and that the other been lacking, the first leaves the empty handeds and the other gains the whole. The situation is a little different from the canonical matrix higher: the profit is the same one for which sees its betrayed confidence or which carries the other in its loss. If a player knows that the other will betray it, its answer is indifferent for him. Nonco-operative balance is thus neutral, whereas it is stable in the usual case. The matrix to be considered is thus:

if both cooperate, each one obtains 50%;
if both are lacking, they draw 0% from them;
if one cooperates and that the other betrays it, the first receives 0% and the other 100%.

This play would be very useful for which would wish to analyze the dilemma of the prisoner on a real case. As one can take part only once in the play, all the considerations concerning the repeated plays are not applicable, the strategy “Eye for Eye” does not intervene… however, one does not know what occurs after the emission.

See too

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