The prisoner's dilemma is a canonical example of a game, analyzed in game theory that shows why two individuals might not cooperate, even if it appears that it is in their best interest to do so. 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). A classic example of the prisoner's dilemma (PD) is presented as follows:
If it is supposed here that each player is only concerned with lessening his time in jail, the game becomes a non-zero sum game where the two players may either assist or betray the other. In the game, the sole worry of the prisoners seems to be increasing his own reward. The interesting symmetry of this problem is that the logical decision leads both to betray the other, even though their individual ‘prize’ would be greater if they cooperated.
In the regular version of this game, collaboration is dominated by betraying, and as a result, the only possible outcome of the game is for both prisoners to betray the other. Regardless of what the other prisoner chooses, one will always gain a greater payoff by betraying the other. Because betraying is always more beneficial than cooperating, all objective prisoners would seemingly betray the other.
In the extended form game, the game is played over and over, and consequently, both prisoners continuously have an opportunity to penalize the other for the previous decision. If the number of times the game will be played is known, the finite aspect of the game means that by backward induction, the two prisoners will betray each other repeatedly.
In casual usage, the label "prisoner's dilemma" may be applied to situations not strictly matching the formal criteria of the classic or iterative games, for instance, those in which two entities could gain important benefits from cooperating or suffer from the failure to do so, but find it merely difficult or expensive, not necessarily impossible, to coordinate their activities to achieve cooperation.
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The normal game is shown below:
| Prisoner B stays silent (cooperates) | Prisoner B confesses (defects) | |
|---|---|---|
| Prisoner A stays silent (cooperates) | Each serves 1 month | Prisoner A: 1 year Prisoner B: goes free |
| Prisoner A confesses (defects) | Prisoner A: goes free Prisoner B: 1 year |
Each serves 3 months |
Here, regardless of what the other decides, each prisoner gets a higher pay-off by betraying the other. For example, Prisoner A can (according to the payoffs above) state that no matter what prisoner B chooses, prisoner A is better off 'ratting him out' (defecting) than staying silent (cooperating). As a result, based on the payoffs above, prisoner A should logically betray him. The game is symmetric, so Prisoner B should act the same way, Since both rationally decide to defect, each receives a lower reward than if both were to stay quiet. Rational decision making results in both players being worse off than if each chose to lessen the sentence of his accomplice at the cost of spending more time in jail himself.
Note that the players cannot communicate or threaten retaliation, and they play the game only once. It is sometimes claimed that if the prisoners trust each other, they can both rationally choose to remain silent, lessening the penalty for both of them. That is incorrect: given the game payoffs above, the rational decision for each player is to defect. Arguments that other behavior can be "rational" must either sneak in extra payoffs (so that the game is no longer explicit) or abandon standard concepts of rationality.
We can expose the framework of the traditional Prisoners’ Dilemma by removing its original prisoner setting, presented as the following:
Based on the rules of a typical understanding of the prisoner’s dilemma, if the two players are represented by colors, red and blue, and the choices made are assigned point values it becomes clear that if the red player plays betrayal and the blue player assists the other, red gets the T prize of 5 points while blue doesn't get payoff at all. If both cooperate they get the R payoff of 3 points each, while if they both betray they get the P payoff of 1 point. The payoffs are shown below.
| Cooperate | Defect | |
|---|---|---|
| Cooperate | 3, 3 | 0, 5 |
| Defect | 5, 0 | 1, 1 |
In simple terms, the matrix looks like this:
| Cooperate | Defect | |
|---|---|---|
| Cooperate |
win-win
|
lose more-win more |
| Defect |
win more-lose more
|
lose-lose
|
It is then possible to make general the point values:
| Cooperate | Defect | |
|---|---|---|
| Cooperate | R, R | S, T |
| Defect | T, S | P, P |
Where T means the desire to betray, R for the Repayment for total unity, P for the Punishment for total betrayal and S for No reward. To be a prisoner's dilemma, the following must be true:
T > R > P > S
The above form guarantees that the balanced outcome is betrayal, but that collaboration rules the sense of middle-play. In addition to the above condition, if the game is repeated more than once, the following should be included:[1]
2 R > T + S
If the above is not true, togetherness is not always necessary, as the players are, in actuality, better off by having each player alternate between Cooperation and Betrayal.
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-player's defection is independent of one's own action, i.e. T+S = P+R.
If two players play prisoners' dilemma more than once in succession and they remember previous actions of their opponent and change their strategy accordingly, the game is called iterated prisoners' dilemma.
The iterated prisoners' dilemma game is fundamental to certain theories of human cooperation and trust. On the assumption that the game can model transactions between two people requiring trust, cooperative behaviour in populations may be modeled 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 prisoners' dilemma has also been referred to as the "Peace-War game".[2]
If the game is played exactly N times and both players know this, then it is always game theoretically optimal to defect in all rounds. The only possible Nash equilibrium is to always defect. The proof is inductive: 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. The same applies if the game length is unknown but has a known upper limit.
Unlike the standard prisoners' dilemma, in the iterated prisoners' dilemma the defection strategy is counter-intuitive and fails badly to predict the behavior of human players. Within standard economic theory, though, this is the only correct answer. The superrational strategy in the iterated prisoners' dilemma with fixed N is to cooperate against a superrational opponent, and in the limit of large N, experimental results on strategies agree with the superrational version, not the game-theoretic rational one.
For cooperation to emerge between game theoretic rational players, the total number of rounds N must be random, or at least unknown to the players. In this case always defect may no longer be a strictly dominant strategy, only a Nash equilibrium. Amongst results shown by Robert Aumann in a 1959 paper, rational players repeatedly interacting for indefinitely long games can sustain the cooperative outcome.
Interest in the iterated prisoners' dilemma (IPD) was kindled by Robert Axelrod in his book The Evolution of Cooperation (1984). In it he reports on a tournament he organized of the N step prisoners' dilemma (with N fixed) 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 or her 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.
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.
Deriving the optimal strategy is generally done in two ways:
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 [1] and consisting of Rajdeep Dash, Sarvapali Ramchurn, Alex Rogers, Perukrishnen Vytelingum) introduced a new strategy at the 20th-anniversary iterated prisoners' 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,[4] 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 prisoners' 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.
Most work on the iterated prisoners' 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 prisoners' dilemma, in which players are able to make a variable contribution to the other player. Le and Boyd[5] found that in such situations, cooperation is much harder to evolve than in the discrete iterated prisoners' dilemma. The basic intuition for this result is straightforward: in a continuous prisoners' 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 prisoners' 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 prisoners' dilemma may help explain why real-life examples of tit for tat-like cooperation are extremely rare in nature (ex. Hammerstein[6]) even though tit for tat seems robust in theoretical models.
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.[7] In his opinion, it is most prudent to give up straight-forward 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. If players treat each other as they would treat themselves, then they will cooperate.
If the two prisoners were offered a deal because the police do not have enough information for a conviction, then they will both walk free by simply remaining silent.
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. 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. Either state will benefit from military expansion regardless of what the other state does; 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 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 immediate benefit to an individual country to maintain current behavior is perceived to be greater than the eventual benefit to all countries if behavior was changed, therefore explaining the current impasse concerning climate change.[8]
In addiction research/behavioral economics, George Ainslie points out[9] that addiction can be cast as an intertemporal PD problem between the present and future selves of the addict. In this case, defecting means relapsing, and it is easy to see that not defecting both today and in the future is by far the best outcome, and that defecting both today and in the future is the worst outcome. The case where one abstains today but relapses in the future is clearly a bad outcome—in some sense the discipline and self-sacrifice involved in abstaining today have been "wasted" because the future relapse means that the addict is right back where he started and will have to start over (which is quite demoralizing, and makes starting over more difficult). The final case, where one engages in the addictive behavior today while abstaining "tomorrow" will be familiar to anyone who has struggled with an addiction. The problem here is that (as in other PDs) there is an obvious benefit to defecting "today", but tomorrow one will face the same PD, and the same obvious benefit will be present then, ultimately leading to an endless string of defections.
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.[10] This analysis is likely to be pertinent in many other business situations involving advertising.
Another example of the prisoner's dilemma in economics is competition-oriented objectives. [11] When firms are aware of the activities of their competitors, they tend to pursue policies that are designed to oust their competitors as opposed to maximizing the performance of the firm. This approach impedes the firm from functioning at its maximum capacity because it limits the scope of the strategies employed by the firms.
Without enforceable agreements, members of a cartel are also involved in a (multi-player) prisoners' dilemma.[12] 'Cooperating' typically means keeping prices at a pre-agreed minimum level. 'Defecting' means selling under this minimum level, instantly stealing business (and profits) from other cartel members. Anti-trust authorities want potential cartel members to mutually defect, ensuring the lowest possible prices for consumers.
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.
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"). 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 people 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. Subsequent research by Elinor Ostrom, winner of the 2009 Nobel Prize in Economics, proved that the tragedy of the commons is oversimplified, with the negative outcome influenced by outside influences. Without complicating pressures, groups communicate and manage the commons among themselves for their mutual benefit, enforcing social norms to preserve the resource and achieve the maximun good for the group, an example of effecting the best case outcome for PD.[13][14]
Hofstadter[15] 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, assuming there are no gains from trade, so chances of mutual cooperation, even in repeated games, are few.
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, it plays a game similar to the prisoner's dilemma to determine how the 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 the game of Chicken.
The payoff matrix is
| Cooperate | Defect | |
|---|---|---|
| Cooperate | 1, 1 | 0, 2 |
| Defect | 2, 0 | 0, 0 |
This payoff matrix was later used on the British television programmes Shafted and Golden Balls. The latter show has been analyzed by a team of economists. See: Split or Steal? Cooperative Behavior When the Stakes are Large.
It was also used earlier in the UK Channel 4 gameshow Trust Me, hosted by Nick Bateman, in 2000.