Prisoner A"},"2":{"wt":"Prisoner B"}},"i":0}}," "]}" id="mwXQ">
It is assumed that both prisoners understand the nature of the game, have no loyalty to each other, and will have no opportunity for retribution or reward outside the game. Regardless of what the other decides, each prisoner gets a higher reward by betraying the other ("defecting"). The reasoning involves analyzing both players' best responses: B will either cooperate or defect. If B cooperates, A should defect, because going free is better than serving 1 year. If B defects, A should also defect, because serving 2 years is better than serving 3. So either way, A should defect since defecting is A's best response regardless of B's strategy. Parallel reasoning will show that B should defect.
Because defection always results in a better payoff than cooperation regardless of the other player's choice, it is a strictly dominant strategy for both A and B. Mutual defection is the only strong Nash equilibrium in the game (i.e. the only outcome from which each player could only do worse by unilaterally changing strategy). The dilemma, then, is that mutual cooperation yields a better outcome than mutual defection but is not the rational outcome because the choice to cooperate, from a selfinterested perspective, is irrational. Thus, Prisoner's dilemma is a game where the Nash equilibrium is not Pareto efficient.
The structure of the traditional prisoner's dilemma can be generalized from its original prisoner setting. Suppose that the two players are represented by the colors red and blue and that each player chooses to either "cooperate" (stay silent) or "defect" (betray).
If both players cooperate, they both receive the reward R for cooperating. If both players defect, they both receive the punishment payoff P. If Blue defects while Red cooperates, then Blue receives the temptation payoff T, while Red receives the "sucker's" payoff, S. Similarly, if Blue cooperates while Red defects, then Blue receives the sucker's payoff S, while Red receives the temptation payoff T.
This can be expressed in normal form:
Red Blue  Cooperate  Defect 

Cooperate  R R  T S 
Defect  S T  P P 
and to be a prisoner's dilemma game in the strong sense, the following condition must hold for the payoffs:
The payoff relationship implies that mutual cooperation is superior to mutual defection, while the payoff relationships and imply that defection is the dominant strategy for both agents.
The "donation game"^{ [12] } is a form of prisoner's dilemma in which cooperation corresponds to offering the other player a benefit b at a personal cost c with b > c. Defection means offering nothing. The payoff matrix is thus
Red Blue  Cooperate  Defect 

Cooperate  b−c b−c  b −c 
Defect  −c b  0 0 
Note that (i.e. ) which qualifies the donation game to be an iterated game (see next section).
The donation game may be applied to markets. Suppose X grows oranges, Y grows apples. The marginal utility of an apple to the orangegrower X is b, which is higher than the marginal utility (c) of an orange, since X has a surplus of oranges and no apples. Similarly, for applegrower Y, the marginal utility of an orange is b while the marginal utility of an apple is c. If X and Y contract to exchange an apple and an orange, and each fulfills their end of the deal, then each receive a payoff of bc. If one "defects" and does not deliver as promised, the defector will receive a payoff of b, while the cooperator will lose c. If both defect, then neither one gains or loses anything.
This section needs additional citations for verification .(November 2012) 
If two players play prisoner's dilemma more than once in succession and they remember previous actions of their opponent and change their strategy accordingly, the game is called iterated prisoner's dilemma.
In addition to the general form above, the iterative version also requires that , to prevent alternating cooperation and defection giving a greater reward than mutual cooperation.
The iterated prisoner's dilemma game is fundamental to some 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 multiplayer, 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 "peacewar game".^{ [13] }
If the game is played exactly N times and both players know this, then it is 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 later retaliate. Therefore, both will defect on the last turn. Thus, the player might as well defect on the secondtolast 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 prisoner's dilemma, in the iterated prisoner's dilemma the defection strategy is counterintuitive 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 prisoner's 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 gametheoretic rational one.
For cooperation to emerge between game theoretic rational players, the total number of rounds N must be 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.
According to a 2019 experimental study in the American Economic Review which tested what strategies reallife subjects used in iterated prisoners' dilemma situations with perfect monitoring, the majority of chosen strategies were always defect, titfortat, and grim trigger. Which strategy the subjects chose depended on the parameters of the game.^{ [14] }
Interest in the iterated prisoner's 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 prisoner's 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 selfinterest. 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 winning deterministic strategy was 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 lineup of opponents.
By analysing the topscoring strategies, Axelrod stated several conditions necessary for a strategy to be successful.
The optimal (pointsmaximizing) strategy for the onetime PD game is simply defection; as explained above, this is true whatever the composition of opponents may be. However, in the iteratedPD 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 alwaysdefectors 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.
In the strategy called Pavlov, winstay, loseswitch, faced with a failure to cooperate, the player switches strategy the next turn.^{ [15] } In certain circumstances,^{[ specify ]} Pavlov beats all other strategies by giving preferential treatment to coplayers using a similar strategy.
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 introduced a new strategy at the 20thanniversary iterated prisoner's dilemma competition, which proved to be more successful than tit for tat. This strategy relied on collusion 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.^{ [18] } 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 nonSouthampton player, it would continuously defect in an attempt to minimize the score of the competing program. As a result, the 2004 Prisoners' Dilemma Tournament results show University of Southampton's strategies in the first three places, despite having fewer wins and many more losses than the GRIM strategy. (In a PD tournament, the aim of the game is not to "win" matches – that can easily be achieved by frequent defection). This strategy ended up taking the top three positions in the competition, as well as a number of positions towards the bottom.
The Southampton 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 highestscoring player (meaning that the use of selfsacrificing 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 analyzing single agent strategies as compared to Axelrod's seminal tournament. However, it provided a basis for analysing how to achieve cooperative strategies in multiagent frameworks, especially in the presence of noise. In fact, long before this newrules tournament was played, Dawkins, in his book The Selfish Gene , pointed out the possibility of such strategies winning if multiple entries were allowed, but he 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, which the Southampton programs arguably did with their preprogrammed "ten move dance" to recognize one another; this only reinforces just how valuable communication can be in shifting the balance of the game.
Even without implicit collusion between software strategies (exploited by the Southampton team) tit for tat is not always the absolute winner of any given tournament; it would be more precise to say that its long run results over a series of tournaments outperform its rivals. (In any one event a given strategy can be slightly better adjusted to the competition than tit for tat, but tit for tat is more robust). The same applies for the tit for tat with forgiveness variant, and other optimal strategies: on any given day they might not "win" against a specific mix of counterstrategies. An alternative way of putting it is using the Darwinian ESS simulation. In such a simulation, tit for tat will almost always come to dominate, though nasty strategies will drift in and out of the population because a tit for tat population is penetrable by nonretaliating nice strategies, which in turn are easy prey for the nasty strategies. Richard Dawkins showed that here, no static mix of strategies form a stable equilibrium and the system will always oscillate between bounds.
In a stochastic iterated prisoner's dilemma game, strategies are specified by in terms of "cooperation probabilities".^{ [19] } In an encounter between player X and player Y, X's strategy is specified by a set of probabilities P of cooperating with Y. P is a function of the outcomes of their previous encounters or some subset thereof. If P is a function of only their most recent n encounters, it is called a "memoryn" strategy. A memory1 strategy is then specified by four cooperation probabilities: , where is the probability that X will cooperate in the present encounter given that the previous encounter was characterized by (ab). For example, if the previous encounter was one in which X cooperated and Y defected, then is the probability that X will cooperate in the present encounter. If each of the probabilities are either 1 or 0, the strategy is called deterministic. An example of a deterministic strategy is the tit for tat strategy written as P={1,0,1,0}, in which X responds as Y did in the previous encounter. Another is the win–stay, lose–switch strategy written as P={1,0,0,1}, in which X responds as in the previous encounter, if it was a "win" (i.e. cc or dc) but changes strategy if it was a loss (i.e. cd or dd). It has been shown that for any memoryn strategy there is a corresponding memory1 strategy which gives the same statistical results, so that only memory1 strategies need be considered.^{ [19] }
If we define P as the above 4element strategy vector of X and as the 4element strategy vector of Y, a transition matrix M may be defined for X whose ij th entry is the probability that the outcome of a particular encounter between X and Y will be j given that the previous encounter was i, where i and j are one of the four outcome indices: cc, cd, dc, or dd. For example, from X's point of view, the probability that the outcome of the present encounter is cd given that the previous encounter was cd is equal to . (The indices for Q are from Y's point of view: a cd outcome for X is a dc outcome for Y.) Under these definitions, the iterated prisoner's dilemma qualifies as a stochastic process and M is a stochastic matrix, allowing all of the theory of stochastic processes to be applied.^{ [19] }
One result of stochastic theory is that there exists a stationary vector v for the matrix M such that . Without loss of generality, it may be specified that v is normalized so that the sum of its four components is unity. The ij th entry in will give the probability that the outcome of an encounter between X and Y will be j given that the encounter n steps previous is i. In the limit as n approaches infinity, M will converge to a matrix with fixed values, giving the longterm probabilities of an encounter producing j which will be independent of i. In other words, the rows of will be identical, giving the longterm equilibrium result probabilities of the iterated prisoners dilemma without the need to explicitly evaluate a large number of interactions. It can be seen that v is a stationary vector for and particularly , so that each row of will be equal to v. Thus the stationary vector specifies the equilibrium outcome probabilities for X. Defining and as the shortterm payoff vectors for the {cc,cd,dc,dd} outcomes (From X's point of view), the equilibrium payoffs for X and Y can now be specified as and , allowing the two strategies P and Q to be compared for their long term payoffs.
In 2012, William H. Press and Freeman Dyson published a new class of strategies for the stochastic iterated prisoner's dilemma called "zerodeterminant" (ZD) strategies.^{ [19] } The long term payoffs for encounters between X and Y can be expressed as the determinant of a matrix which is a function of the two strategies and the short term payoff vectors: and , which do not involve the stationary vector v. Since the determinant function is linear in f, it follows that (where U={1,1,1,1}). Any strategies for which is by definition a ZD strategy, and the long term payoffs obey the relation .
Titfortat is a ZD strategy which is "fair" in the sense of not gaining advantage over the other player. However, the ZD space also contains strategies that, in the case of two players, can allow one player to unilaterally set the other player's score or alternatively, force an evolutionary player to achieve a payoff some percentage lower than his own. The extorted player could defect but would thereby hurt himself by getting a lower payoff. Thus, extortion solutions turn the iterated prisoner's dilemma into a sort of ultimatum game. Specifically, X is able to choose a strategy for which , unilaterally setting to a specific value within a particular range of values, independent of Y's strategy, offering an opportunity for X to "extort" player Y (and vice versa). (It turns out that if X tries to set to a particular value, the range of possibilities is much smaller, only consisting of complete cooperation or complete defection.^{ [19] })
An extension of the IPD is an evolutionary stochastic IPD, in which the relative abundance of particular strategies is allowed to change, with more successful strategies relatively increasing. This process may be accomplished by having less successful players imitate the more successful strategies, or by eliminating less successful players from the game, while multiplying the more successful ones. It has been shown that unfair ZD strategies are not evolutionarily stable. The key intuition is that an evolutionarily stable strategy must not only be able to invade another population (which extortionary ZD strategies can do) but must also perform well against other players of the same type (which extortionary ZD players do poorly, because they reduce each other's surplus).^{ [20] }
Theory and simulations confirm that beyond a critical population size, ZD extortion loses out in evolutionary competition against more cooperative strategies, and as a result, the average payoff in the population increases when the population is larger. In addition, there are some cases in which extortioners may even catalyze cooperation by helping to break out of a faceoff between uniform defectors and win–stay, lose–switch agents.^{ [12] }
While extortionary ZD strategies are not stable in large populations, another ZD class called "generous" strategies is both stable and robust. In fact, when the population is not too small, these strategies can supplant any other ZD strategy and even perform well against a broad array of generic strategies for iterated prisoner's dilemma, including win–stay, lose–switch. This was proven specifically for the donation game by Alexander Stewart and Joshua Plotkin in 2013.^{ [21] } Generous strategies will cooperate with other cooperative players, and in the face of defection, the generous player loses more utility than its rival. Generous strategies are the intersection of ZD strategies and socalled "good" strategies, which were defined by Akin (2013)^{ [22] } to be those for which the player responds to past mutual cooperation with future cooperation and splits expected payoffs equally if he receives at least the cooperative expected payoff. Among good strategies, the generous (ZD) subset performs well when the population is not too small. If the population is very small, defection strategies tend to dominate.^{ [21] }
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^{ [23] } 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 noncooperative equilibrium, players who are only marginally more cooperative than noncooperators 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 noncooperative equilibrium, relative to noncooperators. 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 reallife examples of tit for tatlike cooperation are extremely rare in nature (ex. Hammerstein^{ [24] }) even though tit for tat seems robust in theoretical models.
Players cannot seem to coordinate mutual cooperation, thus often get locked into the inferior yet stable strategy of defection. In this way, iterated rounds facilitate the evolution of stable strategies.^{ [25] } Iterated rounds often produce novel strategies, which have implications to complex social interaction. One such strategy is winstay loseshift. This strategy outperforms a simple TitForTat strategy – that is, if you can get away with cheating, repeat that behavior, however if you get caught, switch.^{ [26] }
The only problem of this titfortat strategy is that they are vulnerable to signal error. The problem arises when one individual cheats in retaliation but the other interprets it as cheating. As a result of this, the second individual now cheats and then it starts a seesaw pattern of cheating in a chain reaction.
The prisoner setting 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 environmental studies, the PD is evident in crises such as global climatechange. It is argued all countries will benefit from a stable climate, but any single country is often hesitant to curb CO_{2} emissions. The immediate benefit to any one country from maintaining current behavior is wrongly perceived to be greater than the purported eventual benefit to that country if all countries' behavior was changed, therefore explaining the impasse concerning climatechange in 2007.^{ [27] }
An important difference between climatechange politics and the prisoner's dilemma is uncertainty; the extent and pace at which pollution can change climate is not known. The dilemma faced by governments is therefore different from the prisoner's dilemma in that the payoffs of cooperation are unknown. This difference suggests that states will cooperate much less than in a real iterated prisoner's dilemma, so that the probability of avoiding a possible climate catastrophe is much smaller than that suggested by a gametheoretical analysis of the situation using a real iterated prisoner's dilemma.^{ [28] }
Osang and Nandy (2003) provide a theoretical explanation with proofs for a regulationdriven winwin situation along the lines of Michael Porter's hypothesis, in which government regulation of competing firms is substantial.^{ [29] }
Cooperative behavior of many animals can be understood as an example of the prisoner's dilemma. Often animals engage in longterm partnerships, which can be more specifically modeled as iterated prisoner's dilemma. For example, guppies inspect predators cooperatively in groups, and they are thought to punish noncooperative inspectors.^{ [30] }
Vampire bats are social animals that engage in reciprocal food exchange. Applying the payoffs from the prisoner's dilemma can help explain this behavior:^{ [31] }
In addiction research / behavioral economics, George Ainslie points out^{ [32] } 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. The case where one abstains today but relapses in the future is the worst outcome – in some sense the discipline and selfsacrifice involved in abstaining today have been "wasted" because the future relapse means that the addict is right back where they started and will have to start over (which is quite demoralizing, and makes starting over more difficult). Relapsing today and tomorrow is a slightly "better" outcome, because while the addict is still addicted, they haven't put the effort in to trying to stop. 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.
John Gottman in his research described in "The Science of Trust" defines good relationships as those where partners know not to enter the (D,D) cell or at least not to get dynamically stuck there in a loop. In cognitive neuroscience, fast brain signaling associated with processing different rounds may indicate choices at the next round. Mutual cooperation outcomes entail brain activity changes predictive of how quickly a person will cooperate in kind at the next opportunity;^{ [33] } this activity may be linked to basic homeostatic and motivational processes, possibly increasing the likelihood to shortcut into the (C,C) cell of the game.
The prisoner's dilemma has been called the E. coli of social psychology, and it has been used widely to research various topics such as oligopolistic competition and collective action to produce a collective good.^{ [34] }
Advertising is sometimes cited as a realexample 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, then the advertisement from each firm negates the other's, 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, which makes it slightly different from a prisoner's dilemma. 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 making of laws banning cigarette advertising, understanding that this would reduce costs and increase profits across the industry.^{ [35] }^{ [loweralpha 2] } This analysis is likely to be pertinent in many other business situations involving advertising.^{[ citation needed ]}
Without enforceable agreements, members of a cartel are also involved in a (multiplayer) prisoner's dilemma.^{ [36] } 'Cooperating' typically means keeping prices at a preagreed minimum level. 'Defecting' means selling under this minimum level, instantly taking business (and profits) from other cartel members. Antitrust authorities want potential cartel members to mutually defect, ensuring the lowest possible prices for consumers.
Doping in sport has been cited as an example of a prisoner's dilemma.^{ [37] }
Two competing athletes have the option to use an illegal and/or dangerous drug to boost their performance. If neither athlete takes the drug, then neither gains an advantage. If only one does, then that athlete gains a significant advantage over their competitor, reduced by the legal and/or medical dangers of having taken the drug. If both athletes take the drug, however, the benefits cancel out and only the dangers remain, putting them both in a worse position than if neither had used doping.^{ [37] }
In a conversation with Ken Griffey Jr. after the 1998 MLB season, Barry Bonds expressed his frustration with other players' use of steroids. Bonds stated "I had a helluva season last year, and nobody gave a crap. Nobody. As much as I've complained about McGwire and Canseco and all of the bull with steroids, I'm tired of fighting it. I turn 35 this year. I've got three or four good seasons left, and I wanna get paid. I'm just gonna start using some hardcore stuff, and hopefully it won't hurt my body. Then I'll get out of the game and be done with it."^{ [38] } Bonds found himself in the prisoner's dilemma that is doping in baseball, the feeling that he has to use steroids so that his competitors don't have such a significant advantage over him, putting him on an even playing field, though everyone is worse off than if no one had used steroids at all.
In international political theory, the Prisoner's Dilemma is often used to demonstrate the coherence of strategic realism, which holds that in international relations, all states (regardless of their internal policies or professed ideology), will act in their rational selfinterest given international anarchy. A classic example is an arms race like the Cold War and similar conflicts.^{ [39] } During the Cold War the opposing alliances of NATO and the Warsaw Pact both had the choice to arm or disarm. From each side's point of view, disarming whilst their opponent continued to arm would have led to military inferiority and possible annihilation. Conversely, arming whilst their opponent disarmed would have led to superiority. If both sides chose to arm, neither could afford to attack the other, but both incurred the high cost of developing and maintaining a nuclear arsenal. If both sides chose to disarm, war would be avoided and there would be no costs.
Although the 'best' overall outcome is for both sides to disarm, the rational course for both sides is to arm, and this is indeed what happened. Both sides poured enormous resources into military research and armament in a war of attrition for the next thirty years until the Soviet Union could not withstand the economic cost.^{ [40] } The same logic could be applied in any similar scenario, be it economic or technological competition between sovereign states.
Many reallife dilemmas involve multiple players.^{ [41] } Although metaphorical, Hardin's tragedy of the commons may be viewed as an example of a multiplayer 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"). A commons dilemma most people can relate to is washing the dishes in a shared house. By not washing dishes an individual can gain by saving his time, but if that behavior is adopted by every resident the collective cost is no clean plates for anyone.
The commons are not always exploited: William Poundstone, in a book about the prisoner's dilemma, 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.^{ [42] } Subsequent research by Elinor Ostrom, winner of the 2009 Nobel Memorial Prize in Economic Sciences, hypothesized 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 maximum good for the group, an example of effecting the best case outcome for PD.^{ [43] }^{ [44] }
Douglas Hofstadter ^{ [45] } 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 tradeoff. One of several examples he used was "closed bag exchange":
Two people meet and exchange closed bags, with the understanding that one of them contains money, and the other contains a purchase. Either player can choose to honor the deal by putting into his or her bag what he or she agreed, or he or she can defect by handing over an empty bag.
Friend or Foe? is a game show that aired from 2002 to 2003 on the Game Show Network in the US. It is an example of the prisoner's dilemma game tested on real people, but in an artificial setting. On the game show, three pairs of people compete. When a pair is eliminated, they play 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 reward matrix is slightly different from the standard one given above, as the rewards 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 a contestant knows that their opponent is going to vote "Foe", then their own choice does not affect their own winnings. In a specific sense, Friend or Foe has a rewards model between prisoner's dilemma and the game of Chicken.
The rewards matrix is
Pair 2 Pair 1  "Friend" (cooperate)  "Foe" (defect) 

"Friend" (cooperate)  1 1  2 0 
"Foe" (defect)  0 2  0 0 
This payoff matrix has also been used on the British television programmes Trust Me, Shafted , The Bank Job and Golden Balls , and on the American game shows Take It All , as well as for the winning couple on the Reality Show shows Bachelor Pad and Love Island . Game data from the Golden Balls series has been analyzed by a team of economists, who found that cooperation was "surprisingly high" for amounts of money that would seem consequential in the real world, but were comparatively low in the context of the game.^{ [46] }
Researchers from the University of Lausanne and the University of Edinburgh have suggested that the "Iterated Snowdrift Game" may more closely reflect realworld social situations. Although this model is actually a chicken game, it will be described here. In this model, the risk of being exploited through defection is lower, and individuals always gain from taking the cooperative choice. The snowdrift game imagines two drivers who are stuck on opposite sides of a snowdrift, each of whom is given the option of shoveling snow to clear a path, or remaining in their car. A player's highest payoff comes from leaving the opponent to clear all the snow by themselves, but the opponent is still nominally rewarded for their work.
This may better reflect real world scenarios, the researchers giving the example of two scientists collaborating on a report, both of whom would benefit if the other worked harder. "But when your collaborator doesn't do any work, it's probably better for you to do all the work yourself. You'll still end up with a completed project."^{ [47] }


In coordination games, players must coordinate their strategies for a good outcome. An example is two cars that abruptly meet in a blizzard; each must choose whether to swerve left or right. If both swerve left, or both right, the cars do not collide. The local left and righthand traffic convention helps to coordinate their actions.
Symmetrical coordination games include Stag hunt and Bach or Stravinsky.
A more general set of games are asymmetric. As in the prisoner's dilemma, the best outcome is cooperation, and there are motives for defection. Unlike the symmetric prisoner's dilemma, though, one player has more to lose and/or more to gain than the other. Some such games have been described as a prisoner's dilemma in which one prisoner has an alibi, whence the term "alibi game".^{ [48] }
In experiments, players getting unequal payoffs in repeated games may seek to maximize profits, but only under the condition that both players receive equal payoffs; this may lead to a stable equilibrium strategy in which the disadvantaged player defects every X games, while the other always cooperates. Such behaviour may depend on the experiment's social norms around fairness.^{ [49] }
It is not only prisoners who face dilemmas. Guardians also confront situations in which there are only unattractive choices from which to choose. Examples can easily be found in cases where one agent must smooth tensions between its own partners: one can think of two colleagues jockeying for career advancement and the troubles this causes their company's managing director; two officials competing for promotion and the tension this causes for the head of their bureau; or in parenting when two siblings vie for attention and the anxiety this causes their parents. If the behaviour of the guardian satisfies one side, the other side feels exposed and alienated.
From an international relations perspective, Dr Spyros Katsoulas introduces the concept of the guardian's dilemma.^{ [50] } The guardian's dilemma is defined as the condition in which two states maintain their enmity towards one another despite sharing a stronger common ally. By default, a dilemma is a situation with unsatisfactory choices. The guardian's dilemma lies in the fact that the stronger state can neither stay out of a crisis between its allies nor get actively involved without affecting the fragile equilibrium. If the guardian abstains, the situation may spin out of control; if the guardian gets involved, any tilt against one side may be seen as a victory or a window of opportunity for the other. Expanding on Glenn Snyder's concept of the alliance security dilemma,^{ [51] } the outcomes of the interaction between the guardian and the two smaller partners are described as abandonment, entrapment, and emboldening.
Several software packages have been created to run prisoner's dilemma simulations and tournaments, some of which have available source code.
Hannu Rajaniemi set the opening scene of his The Quantum Thief trilogy in a "dilemma prison". The main theme of the series has been described as the "inadequacy of a binary universe" and the ultimate antagonist is a character called the AllDefector. Rajaniemi is particularly interesting as an artist treating this subject in that he is a Cambridgetrained mathematician and holds a PhD in mathematical physics – the interchangeability of matter and information is a major feature of the books, which take place in a "postsingularity" future. The first book in the series was published in 2010, with the two sequels, The Fractal Prince and The Causal Angel , published in 2012 and 2014, respectively.
A game modeled after the (iterated) prisoner's dilemma is a central focus of the 2012 video game Zero Escape: Virtue's Last Reward and a minor part in its 2016 sequel Zero Escape: Zero Time Dilemma .
In The Mysterious Benedict Society and the Prisoner's Dilemma by Trenton Lee Stewart, the main characters start by playing a version of the game and escaping from the "prison" altogether. Later they become actual prisoners and escape once again.
In The Adventure Zone: Balance during The Suffering Game subarc, the player characters are twice presented with the prisoner's dilemma during their time in two liches' domain, once cooperating and once defecting.
In the 8th novel from the author James S. A. Corey Tiamat's Wrath, Winston Duarte explains the prisoners dilemma to his 14yearold daughter, Teresa, to train her in strategic thinking. ^{[ citation needed ]}
An extreme version of the prisoner's dilemma is featured in the 2008 film The Dark Knight in which the Joker rigs two ferries, one containing prisoners and the other containing civilians, arming both groups with the means to detonate the bomb on each other's ferries. Ultimately, the two sides decide not to act.
An evolutionarily stable strategy (ESS) is a strategy that is impermeable when adopted by a population in adaptation to a specific environment, that is to say it cannot be displaced by an alternative strategy which may be novel or initially rare. Introduced by John Maynard Smith and George R. Price in 1972/3, it is an important concept in behavioural ecology, evolutionary psychology, mathematical game theory and economics, with applications in other fields such as anthropology, philosophy and political science.
The Evolution of Cooperation is a 1984 book by political scientist Robert Axelrod that expanded upon a highly influential paper of the same name written by Axelrod and evolutionary biologist W.D. Hamilton. It details a theory on the emergence of cooperation between individuals, drawing from game theory and evolutionary biology. Since 2006, reprints of the book have included a foreword by Richard Dawkins and been marketed as a revised edition.
Tit for tat is an English saying meaning "equivalent retaliation". It developed from "tip for tap", first recorded in 1558.
In economics and game theory, a participant is considered to have superrationality if they have perfect rationality but assume that all other players are superrational too and that a superrational individual will always come up with the same strategy as any other superrational thinker when facing the same problem. Applying this definition, a superrational player playing against a superrational opponent in a prisoner's dilemma will cooperate while a rationally selfinterested player would defect.
Evolutionary game theory (EGT) is the application of game theory to evolving populations in biology. It defines a framework of contests, strategies, and analytics into which Darwinian competition can be modelled. It originated in 1973 with John Maynard Smith and George R. Price's formalisation of contests, analysed as strategies, and the mathematical criteria that can be used to predict the results of competing strategies.
In game theory, the centipede game, first introduced by Robert Rosenthal in 1981, is an extensive form game in which two players take turns choosing either to take a slightly larger share of an increasing pot, or to pass the pot to the other player. The payoffs are arranged so that if one passes the pot to one's opponent and the opponent takes the pot on the next round, one receives slightly less than if one had taken the pot on this round, but after an additional switch the potential payoff will be higher. Therefore, although at each round a player has an incentive to take the pot, it would be better for them to wait. Although the traditional centipede game had a limit of 100 rounds, any game with this structure but a different number of rounds is called a centipede game.
In game theory, grim trigger is a trigger strategy for a repeated game.
In game theory, the stag hunt, sometimes referred to as the assurance game, trust dilemma or common interest game, describes a conflict between safety and social cooperation. The stag hunt problem originated with philosopher JeanJacques Rousseau in his Discourse on Inequality. In Rousseau's telling, two hunters must decide separately, and without the other knowing, whether to hunt a stag or a hare. However, both hunters know the only way to successfully hunt a stag is with the other's help. One hunter can catch a hare alone with less effort and less time, but it is worth far less than a stag and has much less meat. Rousseau therefore posits it would be much better for each hunter, acting individually, to give up total autonomy and minimal risk, which brings only the small reward of the hare. Instead, each hunter should separately choose the more ambitious and far more rewarding goal of getting the stag, thereby giving up some autonomy in exchange for the other hunter's cooperation and added might. Commentators have seen the situation as a useful analogy for many kinds of social cooperation, such as international agreements on climate change.
In game theory, a trigger strategy is any of a class of strategies employed in a repeated noncooperative game. A player using a trigger strategy initially cooperates but punishes the opponent if a certain level of defection is observed.
Regime theory is a theory within international relations derived from the liberal tradition that argues that international institutions or regimes affect the behavior of states or other international actors. It assumes that cooperation is possible in the anarchic system of states, as regimes are, by definition, instances of international cooperation.
In game theory, Deadlock is a game where the action that is mutually most beneficial is also dominant. This provides a contrast to the Prisoner's Dilemma where the mutually most beneficial action is dominated. This makes Deadlock of rather less interest, since there is no conflict between selfinterest and mutual benefit. On the other hand, deadlock game can also impact the economic behaviour and changes to equilibrium outcome in society.
In game theory, folk theorems are a class of theorems describing an abundance of Nash equilibrium payoff profiles in repeated games. The original Folk Theorem concerned the payoffs of all the Nash equilibria of an infinitely repeated game. This result was called the Folk Theorem because it was widely known among game theorists in the 1950s, even though no one had published it. Friedman's (1971) Theorem concerns the payoffs of certain subgameperfect Nash equilibria (SPE) of an infinitely repeated game, and so strengthens the original Folk Theorem by using a stronger equilibrium concept: subgameperfect Nash equilibria rather than Nash equilibria.
In game theory, a repeated game is an extensive form game that consists of a number of repetitions of some base game. The stage game is usually one of the wellstudied 2person games. Repeated games capture the idea that a player will have to take into account the impact of his or her current action on the future actions of other players; this impact is sometimes called his or her reputation. Single stage game or single shot game are names for nonrepeated games.
In game theory, a subgame perfect equilibrium is a refinement of a Nash equilibrium used in dynamic games. A strategy profile is a subgame perfect equilibrium if it represents a Nash equilibrium of every subgame of the original game. Informally, this means that at any point in the game, the players' behavior from that point onward should represent a Nash equilibrium of the continuation game, no matter what happened before. Every finite extensive game with perfect recall has a subgame perfect equilibrium. Perfect recall is a term introduced by Harold W. Kuhn in 1953 and "equivalent to the assertion that each player is allowed by the rules of the game to remember everything he knew at previous moves and all of his choices at those moves".
Peace war game is an iterated game originally played in academic groups and by computer simulation for years to study possible strategies of cooperation and aggression. As peace makers became richer over time it became clear that making war had greater costs than initially anticipated. The only strategy that acquired wealth more rapidly was a "Genghis Khan", a constant aggressor making war continually to gain resources. This led to the development of the "provokable nice guy" strategy, a peacemaker until attacked. Multiple players continue to gain wealth cooperating with each other while bleeding the constant aggressor. The Hanseatic League for trade and mutual defense appears to have originated from just such concerns about seaborne raiders.
In game theory, an epsilonequilibrium, or nearNash equilibrium, is a strategy profile that approximately satisfies the condition of Nash equilibrium. In a Nash equilibrium, no player has an incentive to change his behavior. In an approximate Nash equilibrium, this requirement is weakened to allow the possibility that a player may have a small incentive to do something different. This may still be considered an adequate solution concept, assuming for example status quo bias. This solution concept may be preferred to Nash equilibrium due to being easier to compute, or alternatively due to the possibility that in games of more than 2 players, the probabilities involved in an exact Nash equilibrium need not be rational numbers.
In game theory, the traveler's dilemma is a nonzerosum game in which each player proposes a payoff. The lower of the two proposals wins; the lowball player receives the lowball payoff plus a small bonus, and the highball player receives the same lowball payoff, minus a small penalty. Surprisingly, the Nash equilibrium is for both players to aggressively lowball. The traveler's dilemma is notable in that naive play appears to outperform the Nash equilibrium; this apparent paradox also appears in the centipede game and the finitelyiterated prisoner's dilemma.
Subjective expected relative similarity (SERS) is a normative and descriptive theory that predicts and explains cooperation levels in a family of games termed Similarity Sensitive Games (SSG), among them the wellknown Prisoner's Dilemma game (PD). SERS was originally developed in order to (i) provide a new rational solution to the PD game and (ii) to predict human behavior in singlestep PD games. It was further developed to account for: (i) repeated PD games, (ii) evolutionary perspectives and, as mentioned above, (iii) the SSG subgroup of 2×2 games. SERS predicts that individuals cooperate whenever their subjectively perceived similarity with their opponent exceeds a situational index derived from the game's payoffs, termed the similarity threshold of the game. SERS proposes a solution to the rational paradox associated with the single step PD and provides accurate behavioral predictions. The theory was developed by Prof. Ilan Fischer at the University of Haifa.
Reciprocal altruism in humans refers to an individual behavior that gives benefit conditionally upon receiving a returned benefit, which draws on the economic concept – ″gains in trade″. Human reciprocal altruism would include the following behaviors : helping patients, the wounded, and the others when they are in crisis; sharing food, implement, knowledge.
The Berge equilibrium is a game theory solution concept named after the mathematician Claude Berge. It is similar to the standard Nash equilibrium, except that it aims to capture a type of altruism rather than purely noncooperative play. Whereas a Nash equilibrium is a situation in which each player of a strategic game ensures that they personally will receive the highest payoff given other players' strategies, in a Berge equilibrium every player ensures that all other players will receive the highest payoff possible. Although Berge introduced the intuition for this equilibrium notion in 1957, it was only formally defined by Vladislav Iosifovich Zhukovskii in 1985, and it was not in widespread use until half a century after Berge originally developed it.
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