Prisoners Dilemma

The Prisoner's Dilemma constitutes a problem in game theory. It was originally framed by Merrill Flood and Melvin Dresher working at RAND in 1950. Albert W. Tucker formalized the game with prison sentence payoffs and gave it the "Prisoner's Dilemma" name (Poundstone, 1992).

If we assume that each player prefers shorter sentences to longer ones, and that each gets no utility out of lowering the other player's sentence, and that there are no reputation effects from a player's decision, then the prisoner's dilemma forms a non-zero-sum game in which two players may each "cooperate" with or "defect" from (i.e., betray) the other player. In this game, as in all game theory, the only concern of each individual player ("prisoner") is maximizing his/her own payoff, without any concern for the other player's payoff. The unique equilibrium for this game is a Pareto-suboptimal solution—that is, rational choice leads the two players to both play defectly even though each player's individual reward would be greater if they both played cooperately.

In the classic form of this game, cooperating is strictly dominated by defecting, so that the only possible equilibrium for the game is for all players to defect. In simpler terms, no matter what the other player does, one player will always gain a greater payoff by playing defect. Since in any situation playing defect is more beneficial than cooperating, all rational players will play defect, all things being equal.

In the iterated prisoner's dilemma the game is played repeatedly. Thus each player has an opportunity to "punish" the other player for previous non-cooperative play. Cooperation may then arise as an equilibrium outcome. The incentive to defect is overcome by the threat of punishment, leading to the possibility of a cooperative outcome. So if the game is infinitely repeated, cooperation may be a subgame perfect Nash equilibrium although both players defecting always remains an equilibrium and there are many other equilibrium outcomes.

Strategy for the classical prisoner's dilemma

In this game, regardless of what the opponent chooses, each player always receives a higher payoff (lesser sentence) by betraying; that is to say that betraying is the strictly dominant strategy. For instance, Prisoner A can accurately say, "No matter what Prisoner B does, I personally am better off betraying than staying silent. Therefore, for my own sake, I should betray." However, if the other player acts similarly, then they both betray and both get a lower payoff than they would get by staying silent. Rational self-interested decisions result in each prisoner's being worse off than if each chose to lessen the sentence of the accomplice at the cost of staying a little longer in jail himself. Hence a seeming dilemma. In game theory, this demonstrates very elegantly that in a non-zero sum game a Nash Equilibrium need not be a Pareto optimum.

Generalized form

We can expose the skeleton of the game by stripping it of the prisoner framing device. The generalized form of the game has been used frequently in experimental economics. The following rules give a typical realization of the game.

If player 1 (red) defects and player 2 (blue) cooperates, player 1 gets the Temptation to Defect payoff of 5 points while player 2 receives the Sucker's payoff of 0 points. If both cooperate they get the Reward for Mutual Cooperation payoff of 3 points each, while if they both defect they get the Punishment for Mutual Defection payoff of 1 point. The checker board payoff matrix showing the payoffs is given below.

These point assignments are given arbitrarily for illustration. It is possible to generalize them, as follows:

Where T stands for Temptation to defect, R for Reward for mutual cooperation, P for Punishment for mutual defection and S for Sucker's payoff. To be defined as Prisoner's dilemma, the following inequalities must hold:

This condition ensures that the equilibrium outcome is defection, but that cooperation Pareto dominates equilibrium play. In addition to the above condition, if the game is repeatedly played by two players, the following condition should be added.^[2]

If that condition does not hold, then full cooperation is not necessarily Pareto optimal, as the players are collectively better off by having each player alternate between cooperate and defect.

These rules were established by cognitive scientist Douglas Hofstadter and form the formal canonical description of a typical game of Prisoner's Dilemma.

A simple special case occurs when the advantage of defection over cooperation is independent of what the co-player does and cost of the co-players defection is independent of one's own action, i.e. T+S = P+R.

Human behavior in the Prisoner's Dilemma

One experiment based on the simple dilemma found that approximately 40% of participants played "cooperate" (i.e., stayed silent).^[3]

The iterated prisoner's dilemma

If two players play Prisoner's Dilemma more than once in succession, having memory of at least one previous game, it is called iterated Prisoner's Dilemma. Amongst results shown by Nobel Prize winner Robert Aumann in his 1959 paper, rational players repeatedly interacting for indefinitely long games can sustain the cooperative outcome. Popular interest in the iterated prisoners dilemma (IPD) was kindled by Robert Axelrod in his book The Evolution of Cooperation (1984). In this he reports on a tournament he organized in which participants have to choose their mutual strategy again and again, and have memory of their previous encounters. Axelrod invited academic colleagues all over the world to devise computer strategies to compete in an IPD tournament. The programs that were entered varied widely in algorithmic complexity, initial hostility, capacity for forgiveness, and so forth.

Axelrod discovered that when these encounters were repeated over a long period of time with many players, each with different strategies, greedy strategies tended to do very poorly in the long run while more altruistic strategies did better, as judged purely by self-interest. He used this to show a possible mechanism for the evolution of altruistic behaviour from mechanisms that are initially purely selfish, by natural selection.

The best deterministic strategy was found to be "Tit for Tat," which Anatol Rapoport developed and entered into the tournament. It was the simplest of any program entered, containing only four lines of BASIC, and won the contest. The strategy is simply to cooperate on the first iteration of the game; after that, the player does what his opponent did on the previous move. Depending on the situation, a slightly better strategy can be "Tit for Tat with forgiveness." When the opponent defects, on the next move, the player sometimes cooperates anyway, with a small probability (around 1%-5%). This allows for occasional recovery from getting trapped in a cycle of defections. The exact probability depends on the line-up of opponents.

By analysing the top-scoring strategies, Axelrod stated several conditions necessary for a strategy to be successful.

Therefore, Axelrod reached the Utopian-sounding conclusion that selfish individuals for their own selfish good will tend to be nice and forgiving and non-envious.

The optimal (points-maximizing) strategy for the one-time PD game is simply defection; as explained above, this is true whatever the composition of opponents may be. However, in the iterated-PD game the optimal strategy depends upon the strategies of likely opponents, and how they will react to defections and cooperations. For example, consider a population where everyone defects every time, except for a single individual following the Tit-for-Tat strategy. That individual is at a slight disadvantage because of the loss on the first turn. In such a population, the optimal strategy for that individual is to defect every time. In a population with a certain percentage of always-defectors and the rest being Tit-for-Tat players, the optimal strategy for an individual depends on the percentage, and on the length of the game.

A strategy called Pavlov (an example of Win-Stay, Lose-Switch) cooperates at the first iteration and whenever the player and co-player did the same thing at the previous iteration; Pavlov defects when the player and co-player did different things at the previous iteration. For a certain range of parameters, Pavlov beats all other strategies by giving preferential treatment to co-players which resemble Pavlov.

Although Tit-for-Tat is considered to be the most robust basic strategy, a team from Southampton University in England (led by Professor Nicholas Jennings [2] and consisting of Rajdeep Dash, Sarvapali Ramchurn, Alex Rogers, Perukrishnen Vytelingum) introduced a new strategy at the 20th-anniversary Iterated Prisoner's Dilemma competition, which proved to be more successful than Tit-for-Tat. This strategy relied on cooperation between programs to achieve the highest number of points for a single program. The University submitted 60 programs to the competition, which were designed to recognize each other through a series of five to ten moves at the start. Once this recognition was made, one program would always cooperate and the other would always defect, assuring the maximum number of points for the defector. If the program realized that it was playing a non-Southampton player, it would continuously defect in an attempt to minimize the score of the competing program. As a result,^[5] this strategy ended up taking the top three positions in the competition, as well as a number of positions towards the bottom.

This strategy takes advantage of the fact that multiple entries were allowed in this particular competition, and that the performance of a team was measured by that of the highest-scoring player (meaning that the use of self-sacrificing players was a form of minmaxing). In a competition where one has control of only a single player, Tit-for-Tat is certainly a better strategy. Because of this new rule, this competition also has little theoretical significance when analysing single agent strategies as compared to Axelrod's seminal tournament. However, it provided the framework for analysing how to achieve cooperative strategies in multi-agent frameworks, especially in the presence of noise. In fact, long before this new-rules tournament was played, Richard Dawkins in his book The Selfish Gene pointed out the possibility of such strategies winning if multiple entries were allowed, but remarked that most probably Axelrod would not have allowed them if they had been submitted. It also relies on circumventing rules about the prisoner's dilemma in that there is no communication allowed between the two players. When the Southampton programs engage in an opening "ten move dance" to recognize one another, this only reinforces just how valuable communication can be in shifting the balance of the game.

If an iterated PD is going to be iterated exactly N times, for some known constant N, then it is always game theoretically optimal to defect in all rounds. The only possible Nash equilibrium is to always defect. The proof goes like this: one might as well defect on the last turn, since the opponent will not have a chance to punish the player. Therefore, both will defect on the last turn. Thus, the player might as well defect on the second-to-last turn, since the opponent will defect on the last no matter what is done, and so on. For cooperation to emerge between game theoretic rational players, the total number of rounds must be random, or at least unknown to the players. However, even in this case always defect is no longer a strictly dominant strategy, only a Nash equilibrium. The superrational strategy in this case is to cooperate against a superrational opponent, and in the limit of large fixed N, experimental results on strategies agree with the superrational version, not the game-theoretic rational one.

Another odd case is "play forever" prisoner's dilemma. The game is repeated infinitely many times, and the player's score is the average (suitably computed).

The prisoner's dilemma game is fundamental to certain theories of human cooperation and trust. On the assumption that the PD can model transactions between two people requiring trust, cooperative behaviour in populations may be modelled by a multi-player, iterated, version of the game. It has, consequently, fascinated many scholars over the years. In 1975, Grofman and Pool estimated the count of scholarly articles devoted to it at over 2,000. The iterated prisoner's dilemma has also been referred to as the "Peace-War game".^[6]

Continuous Iterated Prisoner's Dilemma

Most work on the iterated prisoner's dilemma has focused on the discrete case, in which players either cooperate or defect, because this model is relatively simple to analyze. However, some researchers have looked at models of the continuous iterated prisoner's dilemma, in which players are able to make a variable contribution to the other player. Le and Boyd^[7] found that in such situations, cooperation is much harder to evolve than in the discrete iterated prisoner's dilemma. The basic intuition for this result is straightforward: in a continuous prisoner's dilemma, if a population starts off in a non-cooperative equilibrium, players who are only marginally more cooperative than non-cooperators get little benefit from assorting with one another. By contrast, in a discrete prisoner's dilemma, Tit-for-Tat cooperators get a big payoff boost from assorting with one another in a non-cooperative equilibrium, relative to non-cooperators. Since Nature arguably offers more opportunities for variable cooperation rather than a strict dichotomy of cooperation or defection, the continuous prisoner's dilemma may help explain why real-life examples of Tit-for-Tat-like cooperation are extremely rare in Nature (ex. Hammerstein^[8]) even though Tit-for-Tat seems robust in theoretical models.

Learning psychology and game theory

Where game players can learn to estimate the likelihood of other players defecting, their own behaviour is influenced by their experience of the others' behaviour. Simple statistics show that inexperienced players are more likely to have had, overall, atypically good or bad interactions with other players. If they act on the basis of these experiences (by defecting or cooperating more than they would otherwise) they are likely to suffer in future transactions. As more experience is accrued a truer impression of the likelihood of defection is gained and game playing becomes more successful. The early transactions experienced by immature players are likely to have a greater effect on their future playing than would such transactions affect mature players. This principle goes part way towards explaining why the formative experiences of young people are so influential and why, for example, those who are particularly vulnerable to bullying sometimes become bullies themselves.

The likelihood of defection in a population may be reduced by the experience of cooperation in earlier games allowing trust to build up.^[9] Hence self-sacrificing behaviour may, in some instances, strengthen the moral fibre of a group. If the group is small the positive behaviour is more likely to feed back in a mutually affirming way, encouraging individuals within that group to continue to cooperate. This is allied to the twin dilemma of encouraging those people whom one would aid to indulge in behaviour that might put them at risk. Such processes are major concerns within the study of reciprocal altruism, group selection, kin selection and moral philosophy.

Douglas Hofstadter's Superrationality

Douglas Hofstadter in his Metamagical Themas proposed that the definition of "rational" that led "rational" players to defect is faulty. He proposed that there is another type of rational behavior, which he called "superrational", where players take into account that the other person is presumably superrational, like them. Superrational players behave identically, and know that they will behave identically. They take that into account before they maximize their payoffs, and they therefore cooperate.

Superrationality is not in contradiction to ordinary rationality. If a superrational player plays against all game-theoretic rational opponents, the answer will be the same as game theoretic rationality. The concept of superrationality allows groups of superrational players to form, where members of the groups expect one-another to behave superrationally. One-shot cooperation is observed in human culture, wherever religious and ethical codes exist.

Although superrationality is not in contradiction with alternate forms of rationality, it is not studied by academic economists, who define rationality to exclude any superrational behavior.

Morality

While it is sometimes thought that morality must involve the constraint of self-interest, David Gauthier famously argues that co-operating in the prisoners dilemma on moral principles is consistent with self-interest and the axioms of game theory.^{citation needed} In his opinion, it is most prudent to give up straightforward maximizing and instead adopt a disposition of constrained maximization, according to which one resolves to cooperate in the belief that the opponent will respond with the same choice, while in the classical PD it is explicitly stipulated that the response of the opponent does not depend on the player's choice. This form of contractarianism claims that good moral thinking is just an elevated and subtly strategic version of basic means-end reasoning.

Douglas Hofstadter expresses a strong personal belief that the mathematical symmetry is reinforced by a moral symmetry, along the lines of the Kantian categorical imperative: defecting in the hope that the other player cooperates is morally indefensible.^{citation needed} If players treat each other as they would treat themselves, then they will cooperate.

Real-life examples

These particular examples, involving prisoners and bag switching and so forth, may seem contrived, but there are in fact many examples in human interaction as well as interactions in nature that have the same payoff matrix. The prisoner's dilemma is therefore of interest to the social sciences such as economics, politics and sociology, as well as to the biological sciences such as ethology and evolutionary biology. Many natural processes have been abstracted into models in which living beings are engaged in endless games of Prisoner's Dilemma (PD). This wide applicability of the PD gives the game its substantial importance.

In political science, for instance, the PD scenario is often used to illustrate the problem of two states engaged in an arms race. Both will reason that they have two options, either to increase military expenditure or to make an agreement to reduce weapons. Neither state can be certain that the other one will keep to such an agreement; therefore, they both incline towards military expansion. The paradox is that both states are acting rationally, but producing an apparently irrational result. This could be considered a corollary to deterrence theory.

In sociology or criminology, the PD may be applied to an actual dilemma facing two inmates. The game theorist Marek Kaminski, a former political prisoner, analysed the factors contributing to payoffs in the game set up by a prosecutor for arrested defendants (cf. References). He concluded that while the PD is the ideal game of a prosecutor, numerous factors may strongly affect the payoffs and potentially change the properties of the game.

In environmental studies, the PD is evident in crises such as global climate change. All countries will benefit from a stable climate, but any single country is often hesitant to curb CO₂ emissions. The benefit to an individual country to maintain current behavior is greater than the benefit to all countries if behavior was changed, therefore explaining the current impasse concerning climate change. ^[10]

In program management and technology development, the PD applies to the relationship between the customer and the developer. Capt Dan Ward, an officer in the US Air Force, examined The Program Manager's Dilemma in an article published in Defense AT&L, a defense technology journal.^[11]

Another example concerns a well-known concept in cycling races, for instance in the Tour de France. Consider two cyclists halfway in a race, with the peloton (larger group) at great distance behind them. The two cyclists often work together (mutual cooperation) by sharing the tough load of the front position, where there is no shelter from the wind. If neither of the cyclists makes an effort to stay ahead, the peloton will soon catch up (mutual defection). An often-seen scenario is one cyclist doing the hard work alone (cooperating), keeping the two ahead of the peloton. In the end, this will likely lead to a victory for the second cyclist (defecting) who has an easy ride in the first cyclist's slipstream.

Also in athletics, there is a widespread practice in high school wrestling where the participants intentionally lose unnaturally large amounts of weight so as to compete against lighter opponents. In doing so, the participants are clearly not at their top level of physical and athletic fitness and yet often end up competing against the same opponents anyway, who have also followed this practice (mutual defection). The result is a reduction in the level of competition. Yet if a participant maintains their natural weight (cooperating), they will most likely compete against a stronger opponent who has lost considerable weight.

Advertising is sometimes cited as a real life example of the prisoner’s dilemma. When cigarette advertising was legal in the United States, competing cigarette manufacturers had to decide how much money to spend on advertising. The effectiveness of Firm A’s advertising was partially determined by the advertising conducted by Firm B. Likewise, the profit derived from advertising for Firm B is affected by the advertising conducted by Firm A. If both Firm A and Firm B chose to advertise during a given period the advertising cancels out, receipts remain constant, and expenses increase due to the cost of advertising. Both firms would benefit from a reduction in advertising. However, should Firm B choose not to advertise, Firm A could benefit greatly by advertising. Nevertheless, the optimal amount of advertising by one firm depends on how much advertising the other undertakes. As the best strategy is dependent on what the other firm chooses there is no dominant strategy and this is not a prisoner's dilemma but rather is an example of a stag hunt. The outcome is similar, though, in that both firms would be better off were they to advertise less than in the equilibrium. Sometimes cooperative behaviors do emerge in business situations. For instance, cigarette manufacturers endorsed the creation of laws banning cigarette advertising, understanding that this would reduce costs and increase profits across the industry.^[9] This analysis is likely to be pertinent in many other business situations involving advertising.

Large software projects under the GPL (such as Linux) can force cooperation in an otherwise standard PD situation. Given a piece of Free Software, you can study the (modifiable) source code and make improvements. Then you can keep secret the improved version, i.e. keep the modified source code to yourself and distribute it in an unmodifiable binary form (defect). Alternatively, you could share the improved version in a modifiable source code form (cooperate). If everyone defects, then many are probably making exactly the same improvements. For any software that is under the GPL, it is illegal to distribute only the unmodifiable form, including any changes made, thus forcing cooperation. Hence, rival parties can all work on it and know that none will defect, and all share in the improvements made by the others.

Many real-life dilemmas involve multiple players. Although metaphorical, Hardin's tragedy of the commons may be viewed as an example of a multi-player generalization of the PD: Each villager makes a choice for personal gain or restraint. The collective reward for unanimous (or even frequent) defection is very low payoffs (representing the destruction of the "commons"). Such multi-player PDs are not formal as they can always be decomposed into a set of classical two-player games. The commons are not always exploited: William Poundstone, in a book about the Prisoner's Dilemma (see References below), describes a situation in New Zealand where newspaper boxes are left unlocked. It is possible for someone to take a paper without paying (defecting) but very few do, feeling that if they do not pay then neither will others, destroying the system.

Because there is no mechanism for personal choice to influence others' decisions, this type of thinking relies on correlations between behavior, not on causation. Because of this property, those who do not understand superrationality often mistake it for magical thinking. Without superrationality, not only petty theft, but voluntary voting requires widespread magical thinking, since a non-voter is a free rider on a democratic system.

Potentially, magic would be required to explain Wikipedia contributions: since text is added under the assumption that others will do the same. If the explanation depends on expected future actions it would not require magic. To those who distinguish between superrationality and magical thinking, there is no conflict.

The theoretical conclusion of PD is one reason why, in many countries, plea bargaining is forbidden. Often, precisely the PD scenario applies: it is in the interest of both suspects to confess and testify against the other prisoner/suspect, even if each is innocent of the alleged crime. Arguably, the worst case is when only one party is guilty — here, the innocent one is unlikely to confess, while the guilty one is likely to confess and testify against the innocent.

Related games

Closed-bag exchange

Hofstadter^[12] once suggested that people often find problems such as the PD problem easier to understand when it is illustrated in the form of a simple game, or trade-off. One of several examples he used was "closed bag exchange":

In this game, defection is always the best course, implying that rational agents will never play. However, in this case both players cooperating and both players defecting actually give the same result, so chances of mutual cooperation, even in repeated games, are few.

Friend or Foe?

Friend or Foe? is a game show that aired from 2002 to 2005 on the Game Show Network in the United States. It is an example of the prisoner's dilemma game tested by real people, but in an artificial setting. On the game show, three pairs of people compete. As each pair is eliminated, they play a game of Prisoner's Dilemma to determine how their winnings are split. If they both cooperate (Friend), they share the winnings 50-50. If one cooperates and the other defects (Foe), the defector gets all the winnings and the cooperator gets nothing. If both defect, both leave with nothing. Notice that the payoff matrix is slightly different from the standard one given above, as the payouts for the "both defect" and the "cooperate while the opponent defects" cases are identical. This makes the "both defect" case a weak equilibrium, compared with being a strict equilibrium in the standard prisoner's dilemma. If you know your opponent is going to vote Foe, then your choice does not affect your winnings. In a certain sense, Friend or Foe has a payoff model between "Prisoner's Dilemma" and "Chicken".

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