Prisoner's Dilemma

10 Key excerpts on "Prisoner's Dilemma"

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  Game Theory in Communication Networks

  Cooperative Resolution of Interactive Networking Scenarios

  • Josephina Antoniou, Andreas Pitsillides(Authors)
  • 2012(Publication Date)
  • CRC Press

    (Publisher)

  Chapter 2 Cooperation for two: Prisoner’s Dilemma type of games 2.1 Introduction This chapter investigates how to model and study the cooperative aspects between any two entities whose interaction satisfies the requirements of the well-known game theoretic interaction model of the Prisoner’s Dilemma. The Prisoner’s Dilemma is an important model because it models the interaction between two seemingly antagonistic entities luring them into cooperative be-havior in the repetitive version of the game, even though in the one-shot game this does not appear as the most profitable choice for the involved players. However, even in the one-shot game, mutual cooperation is the most socially efficient choice, since it pays more than mutual defection, which, as we will later show, is the Nash Equilibrium for the one-shot game. The Prisoner’s Dilemma and Iterated Prisoner’s Dilemma 1 have been a rich source of research material since the 1950s. However, the publication of Axelrod’s book in 1984 [2] was the main driver that brought this research to the attention of other areas outside of game theory, as a model for promoting cooperation. Prior to emerging into the details of the Prisoner’s Dilemma Game and applying it to a networking scenario, we offer an overview of similar two-player games that model interactions between two antagonistic entities that have the same two options: to cooperate or to defect. We hence justify the Prisoner’s Dilemma game as our chosen game for a situation that aims to promote cooperation between the two antagonistic players. 1 The text on Prisoner’s Dilemma is based on [5]. 15 16 Game Theory in Communication Networks 2.2 Prisoner’s Dilemma and similar two-player games The Prisoner’s Dilemma is basically a model of a game, where two players must decide whether to cooperate with their opponent or whether to defect from cooperation.

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  Game Theory and Society

  • Weiying Zhang(Author)
  • 2017(Publication Date)
  • Taylor & Francis

    (Publisher)

  The Prisoner’s Dilemma is also known as “the Cooperative Paradox” or “the Collective Action Paradox”. Even if cooperating will benefit both parties, neither parties will cooperate. The individual rational option is to not cooperate, but the collective rational option is to cooperate.

  2.2 Examples of the Prisoner’s Dilemma

  There are numerous examples of conflicts between individual rationality and collective rationality in real life. An example would be the issue of Chinese school students concerning study time. In addition to normal class from Monday to Friday, on the weekends they also have to study math and English, among other topics. This is also a Prisoner’s Dilemma. We can imagine that if all students take break on the weekends, then only the smartest children would test into prestigious secondary schools and universities. The problem is that if one student rests on the weekend, but others study more, even though the first student is smarter, he may not pass the test. The optimal choice is to study on the weekend. The result is that all students study seven days a week, so in the end, it is still only the smartest children that test into prestigious secondary schools and universities.

  Competition brings about an irrational result in which everyone is busy, but in the end not everyone is better off. China’s students today study intently, but from a social perspective this cannot be optimal.

  Competition between businesses is also a Prisoner’s Dilemma. On June 9, 2000, nine Chinese color television companies held a conference in Shenzhen. They formed a price cartel and set the minimum price on some models of color televisions. Three days later, certain companies already began to lower prices in Nanjing and other markets. The price alliance existed in name only.7 Generally speaking, this type of alliance is very difficult to maintain. Given that one enterprise does not lower prices, but another one does, the second enterprise can increase sales and hold more market share.

  Similarly, advertising is a kind of Prisoner’s Dilemma.8

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  Evolutionary Dynamics

  Exploring the Equations of Life

  • Martin A. Nowak, Martin A. Nowak(Authors)
  • 2006(Publication Date)
  • Belknap Press

    (Publisher)

  Figure 5.1 The Prisoner’s Dilemma captures the essential problem of cooperation. Assuming that I will cooperate, you can get 3 points if you cooperate but 5 points if you defect. Assuming that I will defect, you can get 0 points if you cooperate but 1 point if you defect. Hence, no matter what I will do, it is better for you to defect. Defection is the “rational” (= payoff maximizing) strategy. If, however, I analyze the game in the same way that you do, then we both choose defection and both get 1 point. We could have received 3 points each, had we both chosen cooperation. But cooperation is “irrational.” This is the dilemma. This is the dilemma: rational players who act in order to maximize their payoff defect in the Prisoner’s Dilemma (PD). Mutual cooperation leads to a higher payoff than mutual defection, but cooperation is irrational (Figure 5.1). Do not take offense at the terms “rational” and “irrational” in this context. Note that the payoff describes exactly what the players want. There is no hid-den agenda. The rewards (material or immaterial) are completely specified by the payoff matrix. Under this assumption, a rational player is defined as one who acts in a way so as to maximize her payoff. This is a simple and straight-forward concept. Experimental game theory, however, more often than not shows that hu-mans do not behave rationally. They are guided by instincts that might have evolved via different situations. In the Prisoner’s Dilemma, humans often try to cooperate. Only when they learn that it does not work will they switch to defection. Returning to our prisoners, cooperation means not to cooperate with the state attorney but to cooperate with your partner and remain silent. If both of you remain silent, nothing can be proved. Defection means confession. If both of you defect, both will get a long prison sentence. No matter what your partner does, it is better for you to defect.

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  Agents, Games, and Evolution

  Strategies at Work and Play

  • Steven Orla Kimbrough(Author)
  • 2011(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Starting simply, we focus in this chapter on two-player, two-choice social dilemmas. 3.2 Iterated Prisoner’s Dilemma The Prisoner’s Dilemma is the prototypical game representation for two-player social dilemmas. We can represent the game in strategic form; see Figure 3.1. Note we require for Prisoner’s Dilemma that T > R > P > S and 2 R > ( T + S ). Further, [N] labels an outcome of Nash equilibrium play, and 45 46 Agents, Games, and Evolution: Strategies at Work and Play C D R T C [P] [P] R S S P D [P] [N] T P C D 3 5 C [P] [P] 3 0 0 1 D [P] [N] 5 1 FIGURE 3.1 : Canonical and default (Axelrod) Prisoner’s Dilemma. [P] labels a Pareto-optimal outcome.(See § A.2.11 and § A.2.9 for information on Nash equilibria and Pareto optimal outcomes.) It is transparent from the representation that each player has a cooperating choice (labeled C) and a defecting choice (labeled D). The D strategy strictly dominates the C strategy: no matter what the counterpart player chooses, the player does better with D than with C. Both players should choose D by this reasoning, yet both would do better if both choose C rather than both choosing D. In addition, we assume that each player has full knowledge of the game. 1 There are many who describe social dilemmas, and in particular the Pris-oner’s Dilemma, as paradoxes. Those who do find it paradoxical that rational choice (ideally rational choice, in the strong sense of Rational Choice Theory 2 ) should result in a Pareto-inefficient outcome. After all, both players can do bet-ter. Is it not paradoxical that ideal rationality should lead to such an inferior outcome? Worse, careful experimental studies (Sally [269] reviews hundreds of experiments, conducted during a 35-year period) robustly find that subjects in fact often choose the cooperating strategy in Prisoner’s Dilemma and will on average achieve a higher return than merely P.

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  Iterated Prisoners' Dilemma, The: 20 Years On

  20 Years On

  • Xin Yao, Graham Kendall, Siang Yew Chong(Authors)
  • 2007(Publication Date)
  • World Scientific

    (Publisher)

  Chapter 2 Iterated Prisoner’s Dilemma and Evolutionary Game Theory Siang Yew Chong 1 , Jan Humble 2 , Graham Kendall 2 , Jiawei Li 2 , 3 , Xin Yao 1 University of Birmingham 1 , University of Nottingham 2 , Harbin Institute of Technology 3 2.1. Introduction The prisoner’s dilemma is a type of non-zero-sum game in which two players try to maximize their payoff by cooperating with, or betraying the other player. The term non-zero-sum indicates that whatever benefits accrue to one player do not necessarily imply similar penalties imposed on the other player. The Prisoner’s dilemma was originally framed by Merrill Flood and Melvin Dresher working at RAND Corporation in 1950. Albert W. Tucker formalized the game with prison sentence payoffs and gave it the “Prisoner’s Dilemma” name. The classical prisoner’s dilemma (PD) is as follows: Two suspects, A and B, are arrested by the police. The police have insufficient evidence for a conviction, and, having sepa-rated both prisoners, visit each of them to offer the same deal: if one testifies for the prosecution against the other and the other remains silent, the betrayer goes free and the silent ac-complice receives the full 10-year sentence. If both stay silent, the police can sentence both prisoners to only six months in jail for a minor charge. If each betrays the other, each will re-ceive a two-year sentence. Each prisoner must make the choice of whether to betray the other or to remain silent. However, neither prisoner knows for sure what choice the other prisoner will make. So the question this dilemma poses is: What will happen? How will the prisoners act? The general form of the PD is represented as the following matrix [Scodel et al. (1959)]: 23

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  An Introduction to Game-Theoretic Modelling

  Third Edition

  • Mike Mesterton-Gibbons(Author)
  • 2019(Publication Date)
  • American Mathematical Society

    (Publisher)

  Chapter 5 Cooperation and the Prisoner’s Dilemma To team, or not to team? That is the question. Chapters 3 and 4 have already shown us that circumstances abound in which players do better by cooperating than by competing. Indeed if ν denotes a normalized characteristic function, then Player i has an incentive to cooperate with coalition S − { i } whenever ν ( S ) > 0. But an incentive to cooperate does not imply cooperation, for we have also seen in § 3.4 that if one of two players is committed to cooperation, then it may be rational for the other to cheat and play noncooperatively. How, in such circumstances, is cooperation achieved? To reduce this question to its barest essentials, we focus on the symmetric, two-player, noncooperative game with two pure strategies C (strategy 1) and D (strategy 2) whose payoff matrix (5.1) A = R S T P satisfies (5.2) R > 1 2 max(2 P, S + T ) . The combined payoffs associated with the four possible strategy combinations CC , CD , DC , and DD are, respectively, 2 R , S + T , T + S , and 2 P . From (5.2), the best combined payoff 2 R can be achieved only if both individuals select C . We therefore say that C is a cooperative strategy and that D (for defect) is a noncooperative strategy. We will refer to this game as the cooperator’s dilemma [ 217 ]. 1 If it is also true that (5.3) T > R, P > S, 1 Not every problem of cooperation can be reduced to its barest essentials in this way. CD and DC will yield the best combined payoff if cooperation requires players to take complementary actions [ 78 ], but the cooperator’s dilemma has broad applicability. 175 176 5. Cooperation and the Prisoner’s Dilemma Table 5.1. Payoff matrix for the classic prisoner’s dilemma remain silent implicate remain silent Short Long implicate Very short Medium then the cooperator’s dilemma reduces to the prisoner’s dilemma , with which we are familiar from the exercises.

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  Collective Action

  • Russell Hardin(Author)
  • 2015(Publication Date)
  • RFF Press

    (Publisher)

  strategic calculations and should not be expected to produce collectiely rational results; (2) the most telling critique of this analysis invokes strategic calculations, but these carry no behavioral weight.

  At the first level, without strategic calculations, Prisoner’s Dilemma is merely a veridical paradox. One might at first think that individuals acting rationally to maximize their well-being should collectively make themselves so well off that they would not miss any potential gains. This is a compelling assumption in market activity: if bargaining and communication are costless, trading for mutual benefit will lead to a Pareto-optimal state of affairs in which no further gains are possible. When, however, collective goods are added to private consumption goods, market behavior may no longer be collectively rational and may result in Pareto-suboptimal outcomes. If enforceable contracts are possible and costless, 2-person collective goods and those n -person collective goods that would not yield net benefits to n — 1 persons may be provided, but goods that are more interesting and more beneficial may not. Individual behavior in welfare economics is not very strategic.

  The source of paradox in the Voter’s Paradox is structural: it follows clearly from the preferences of the individuals. The source of the first sense of paradox in Prisoner’s Dilemma is similarly structural: it depends only on the individuals’ preferences over their own outcomes, and takes no account of their strategic interactions with their adversary-partners. It follows from the fact, discussed in chapter 2 , that each player has a clear preference for the outcomes from one strategy over those from the other strategy. This can be seen in matrix la in figure 9-1 , which dis-plays the semigame played by the Row player in the Prisoner’s Dilemma game of matrix 1 in the same figure. (These matrices basically duplicate those in figure 2-1

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  Microeconomics

  Theory and Applications

  • Edgar K. Browning, Mark A. Zupan(Authors)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  The resulting equilibrium—an overall high level of government spending (and the taxes that must support such spending) across districts—is inferior to the outcome that would emerge if representatives exercised more restraint in their pursuit of spending on government projects in their individual districts. APPLICATION 14.2 2 Russell Roberts, “If You’re Paying, I’ll Have Top Sirloin,” Wall Street Journal, May 18, 1995, p. A18. 366 Chapter Fourteen • Game Theory and the Economics of Information • is that they believe the other will also. The reason for the predicted outcome is stronger than that: it is in each suspect’s interest to confess, regardless of the other’s actions. When they first encounter the prisoner’s dilemma, many people try to figure out some way in which the prisoners could realize the best all-around outcome, the two-year sentence drawn if neither one confesses. By adding some additional elements to the scenario, it is indeed pos- sible to spin a game-theoretic tale where both refuse to confess. For example, if Nancy and Sid are lovers, such that each feels as much pain if the other goes to jail as if they go to jail them- selves, they would not confess. Another possibility might be that each suspect believes that if he or she is the only one to confess, the suspect who subsequently does 15 years would be wil- ling to commit murder on release in revenge. In that case, there would probably be no confes- sions. (Note that the payoffs would be different than just the jail sentences shown in the table.) Taken on its own terms, however, the prisoner’s dilemma does show how the individual pursuit of self-interest can, in certain situations, produce results that are inferior for all players. The prisoner’s dilemma has wide-ranging applicability. It was used to model the inter- actions between the United States and the Soviet Union in the days of the Cold War.

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  Strategy and Politics

  An Introduction to Game Theory

  • Emerson Niou, Peter C. Ordeshook(Authors)
  • 2015(Publication Date)
  • Routledge

    (Publisher)

  Of course, the Prisoners’ Dilemma would be of little interest if its application were confined to prisoners. But, as we’ve already suggested in referring to the motives of political elites leading up to WWI, this game has had broad application and allows us to understand such things as the reason why people might agree to the establishment of a state that has the power to coerce them, why collectivities sometimes seem to act in ways that are irrational from the perspective of the individuals involved, why various industries might even lobby to be regulated by government, why governments regulate and license barbers and taxi cab drivers but do little in the way of regulating automobile safety aside from requiring the installation of seat belts, why our actions can depend critically on whether we anticipate interacting with the same person again in the future, why the simple basic act of voting in a democracy can be a difficult thing to explain, why labor unions and various associations often offer low cost life insurance to their members, why attempts at reforming the economy of one society prove to be a success whereas similar reforms applied elsewhere lead to massive inefficiency and corruption, and why nations sometimes war even when they might prefer other means of resolving disputes.

  The universality of Prisoners’ Dilemma-type situations is no better illustrated than the example from Chapter 1 of the school of herring that forms a swirling ball when approached by predators, only to offer those predators a ready and efficiently consumed target. The optimal strategy for the school, of course, is to scatter itself in every direction, thereby thinning out so a predator can, at best, consume a fraction of the species and perhaps even exhaust itself in the process. Each individual fish in the school has two choices: Swim away from the others or attempt to move to the center of the slowly shrinking ball so as to not be the first to be eaten and perhaps hope that predators will sate themselves before the ball is fully consumed. It seems reasonable to suppose now that staying put and/or trying to move into whatever remains of the swirling ball yields a higher chance of survival than attempting to flee does, regardless of what all others of your species do. If some or all others attempt to flee, predators will be preoccupied picking off those swimming in their direction. If few or none attempt to flee, then by being one of the few defectors, one becomes an inviting target for predators. So in either case, it seems best to stick to the swirling ball. In this context we are reminded once again of Pastor Martin Niemöller’s famous quote in reference to the crimes committed by the Nazis with which we introduce Chapter 2

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  Ergodic Theory

  Advances in Dynamical Systems

  • Idris Assani(Author)
  • 2016(Publication Date)
  • De Gruyter

    (Publisher)

  Ethan Akin The iterated Prisoner’s Dilemma: good strategies and their dynamics Abstract: For the iterated Prisoner’s Dilemma, there exist Markov strategies that solve the problem when we restrict attention to the long-term average payoff. When used by both players, these assure the cooperative payoff for each of them. Neither player can benefit by moving unilaterally any other strategy, i.e., these are Nash equilibria. In addition, if a player uses instead an alternative that decreases the opponent’s pay-off below the cooperative level, then his own payoff is decreased as well. Thus, if we limit attention to the long-term payoff, these good strategies effectively stabilize co-operative behavior. We characterize these good strategies and analyze their role in evolutionary dynamics. Keywords: Prisoner’s Dilemma, stable cooperative behavior, iterated play, Markov strategies, zero-determinant strategies, Press–Dyson equations, evolutionary game dynamics. 1 The iterated Prisoner’s Dilemma The Prisoner’s Dilemma is a two-person game that provides a simple model of a dis-turbing social phenomenon. It is a symmetric game in which each of the two play-ers, X and Y, has a choice between two strategies, c and d . Thus, there are four out-comes that we list in the order: cc , cd , dc , dd , where, for example, cd is the outcome when X plays c and Y plays d . Each then receives a payoff. The following 2 × 2 chart describes the payoff to the X player. The transpose is the Y payoff. X/Y c d c R S d T P (1.1) Alternatively, we can define the payoff vectors for each player by S X = ( R , S , T , P ) and S Y = ( R , T , S , P ). (1.2) Davis [6] and Straffin [17] provide clear introductory discussions of the elements of game theory. Either player can use a mixed strategy , randomizing by choosing c with probabil-ity p c and d with the complementary probability 1 − p c .

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