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Christian Hilbe

Professor of Game theory and evolutionary dynamics

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Publications

TitleYear
The evolution of strategic ignorance in strategic interaction
2025
Laura Schmid, Christian Hilbe
Deliberate Ignorance: Choosing Not To Know 29
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Authors:Laura Schmid, Christian Hilbe
Publication date:05/05/2025
Journal:Deliberate Ignorance: Choosing Not To Know
Volume:29
Description:Optimal decision making requires individuals to know their available options and to anticipate correctly what consequences these options have. In many social interactions, however, we refrain from gathering all relevant information, even if this information would help us make better decisions and is costless to obtain. This chapter examines several examples of “deliberate ignorance.” Two simple models are proposed to illustrate how ignorance can evolve among self-interested and payoff-maximizing individuals, and open problems are highlighted that lie ahead for future research to explore.
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The zoo of models of deliberate ignorance
2025
Pete C Trimmer, Richard McElreath, Sarah Auster, Gordon DA Brown, Jason Dana, Gerd Gige…
Deliberate ignorance: Choosing not to know, 155-183
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Authors:Pete C Trimmer, Richard McElreath, Sarah Auster, Gordon DA Brown, Jason Dana, Gerd Gigerenzer, Russell Golman, Christian Hilbe, Anne Kandler, Yaakov Kareev, Lael J Schooler, Nora Szech
Publication date:05/05/2025
Journal:Deliberate ignorance: Choosing not to know
Pages:155-183
Publisher:MIT Press
Description:This chapter looks at deliberate ignorance from a modeling perspective. Standard economic models cannot produce deliberate ignorance in a meaningful way; if there were no cost for acquisition and processing, data could be looked at privately and processed perfectly. Here the focus is on cases where the standard assumptions are violated in some way. Cases are considered from an individual’s perspective, without gametheoretic (strategic) aspects. Different classes of “not wanting to know” something are identified: aside from the boring case of the cost of information acquisition being too high, an individual may prefer to not know some information (eg, when knowledge would reduce the enjoyment of other experiences) or may want to not use some infor mation (eg, relating to a lack of self-control). In addition, strategic cases of deliberate ignorance are reviewed, where obtaining information would also signal to others that information acquisition has occurred, and thus it may be better to remain ignorant. Finally, the possibility of deliberate ignorance emerging in population-level models is discussed, where there seems to be a relative dearth of models of the phenomenon at present. Throughout, the authors make use of examples to summarize different classes of models, ideas for how deliberate ignorance can make sense, and gaps in the literature for future modeling.
Cited by:5
Resources:[PDF] from mpg.de
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The Evolution of Extortion in Iterated Prisoner’s Dilemma Games
2025
Christian Hilbe, Martin A Nowak, Karl Sigmund
Proceedings of the National Academy of Sciences 110, 6913-6918
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Authors:Christian Hilbe, Martin A Nowak, Karl Sigmund
Publication date:05/05/2025
Journal:Proceedings of the National Academy of Sciences
Volume:110
Pages:6913-6918
Description:Iterated games are a fundamental component of economic and evolutionary game theory. They describe situations where two players interact repeatedly and have the ability to use conditional strategies that depend on the outcome of previous interactions, thus allowing for reciprocation. Recently, a new class of strategies has been proposed, so-called “zero-determinant” strategies. These strategies enforce a fixed linear relationship between one’s own payoff and that of the other player. A subset of those strategies allows “extortioners” to ensure that any increase in one player’s own payoff exceeds that of the other player by a fixed percentage. Here, we analyze the evolutionary performance of this new class of strategies. We show that in reasonably large populations, they can act as catalysts for the evolution of cooperation, similar to tit-for-tat, but that they are not the stable outcome of natural selection. In very small …
Cited by:334
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Cooperative Dilemmas in Rational Debate
2025
Toby Handfield, Julián Garcia, Christian Hilbe, Shang Long Yeo
arXiv preprint arXiv:2504.06485
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Authors:Toby Handfield, Julián Garcia, Christian Hilbe, Shang Long Yeo
Publication date:04/08/2025
Journal:arXiv preprint arXiv:2504.06485
Description:As an epistemic activity, rational debate and discussion requires cooperation, yet involves a tension between collective and individual interests. While all participants benefit from collective outcomes like reaching consensus on true beliefs, individuals face personal costs when changing their minds. This creates an incentive for each debater to let others bear the cognitive burden of exploring alternative perspectives. We present a model to examine the strategic dynamics between debaters motivated by two competing goals: discovering truth and minimizing belief revisions. Our model demonstrates that this tension creates social dilemmas where strategies that are optimal for individuals systematically undermine the collective pursuit of truth. Paradoxically, our analysis reveals that increasing debaters' motivation to seek truth can sometimes produce equilibria with worse outcomes for collective truth discovery. These findings illuminate why rational debate can fail to achieve optimal epistemic outcomes, even when participants genuinely value truth.
Resources:[PDF] from arxiv.org
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Can I afford to remember less than you? Best responses in repeated additive games
2025
Franziska Lesigang, Christian Hilbe, Nikoleta E Glynatsi
Economics Letters 250, 112300
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Authors:Franziska Lesigang, Christian Hilbe, Nikoleta E Glynatsi
Publication date:04/01/2025
Journal:Economics Letters
Volume:250
Pages:112300
Publisher:North-Holland
Description:In this paper, we study best responses in repeated additive games among two players. A stage game is additive if each player’s payoff is the sum of two components, and each component only depends on the action of a single player. We suppose one player’s strategy depends on the co-player’s last n actions. Then we prove that the other player has a best response that only depends on their own n− 1 actions. That is, for an important sub-class of games and strategies, players can achieve maximum payoffs even with less memory than their opponent.
Resources:[HTML] from sciencedirect.com
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Dynamics of cooperation in concurrent games
2025
Charlotte SL Rossetti, Oliver P Hauser, Christian Hilbe
Nature Communications 16 (1), 1524
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Authors:Charlotte SL Rossetti, Oliver P Hauser, Christian Hilbe
Publication date:02/11/2025
Journal:Nature Communications
Volume:16
Issue:1
Pages:1524
Publisher:Nature Publishing Group UK
Description:People frequently encounter situations where individually optimal decisions conflict with group interests. To navigate such social dilemmas, they often employ simple heuristics based on direct reciprocity: cooperate when others do and cease cooperation when partners defect. However, prior research typically assumes that individuals only interact in one game at a time. In reality, people engage in multiple games concurrently, and the outcome of one interaction can influence behavior in another. Here, we introduce a theoretical framework to study the resulting cross-over and spill-over effects. Participants repeatedly engage in two independent stage games, either with the same or different partners, adapting their strategies over time through an evolutionary learning process. Our findings indicate that individuals often link their behavior across games, particularly under cognitive constraints like imperfect recall. A …
Cited by:1
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Time-Dependent Strategies in Repeated Asymmetric Public Goods Games
2025
Valentin Hübner, Christian Hilbe, Manuel Staab, Maria Kleshnina, Krishnendu Chatterjee
Dynamic Games and Applications, 1-29
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Authors:Valentin Hübner, Christian Hilbe, Manuel Staab, Maria Kleshnina, Krishnendu Chatterjee
Publication date:02/07/2025
Journal:Dynamic Games and Applications
Pages:1-29
Publisher:Springer US
Description:The public goods game is among the most studied metaphors of cooperation in groups. In this game, individuals can use their endowments to make contributions towards a good that benefits everyone. Each individual, however, is tempted to free-ride on the contributions of others. Herein, we study repeated public goods games among asymmetric players. Previous work has explored to which extent asymmetry allows for full cooperation, such that players contribute their full endowment each round. However, by design that work focusses on equilibria where individuals make the same contribution each round. Instead, here we consider players whose contributions along the equilibrium path can change from one round to the next. We do so for three different models–one without any budget constraints, one with endowment constraints, and one in which individuals can save their current endowment to be used in …
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Conditional cooperation with longer memory
2024
Nikoleta E Glynatsi, Ethan Akin, Martin A Nowak, Christian Hilbe
Proceedings of the National Academy of Sciences 121 (50), e2420125121
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Authors:Nikoleta E Glynatsi, Ethan Akin, Martin A Nowak, Christian Hilbe
Publication date:12/10/2024
Journal:Proceedings of the National Academy of Sciences
Volume:121
Issue:50
Pages:e2420125121
Publisher:National Academy of Sciences
Description:Direct reciprocity is a wide-spread mechanism for the evolution of cooperation. In repeated interactions, players can condition their behavior on previous outcomes. A well-known approach is given by reactive strategies, which respond to the coplayer’s previous move. Here, we extend reactive strategies to longer memories. A reactive-n strategy takes into account the sequence of the last n moves of the coplayer. A reactive-n counting strategy responds to how often the coplayer cooperated during the last n rounds. We derive an algorithm to identify the partner strategies within these strategy sets. Partner strategies are those that ensure mutual cooperation without exploitation. We give explicit conditions for all partner strategies among reactive-2, reactive-3 strategies, and reactive-n counting strategies. To further explore the role of memory, we perform evolutionary simulations. We vary several key parameters, such as …
Cited by:3
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Indirect reciprocity under opinion synchronization
2024
Yohsuke Murase, Christian Hilbe
Proceedings of the National Academy of Sciences 121 (48), e2418364121
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Authors:Yohsuke Murase, Christian Hilbe
Publication date:11/26/2024
Journal:Proceedings of the National Academy of Sciences
Volume:121
Issue:48
Pages:e2418364121
Publisher:National Academy of Sciences
Description:Indirect reciprocity is a key explanation for the exceptional magnitude of cooperation among humans. This literature suggests that a large proportion of human cooperation is driven by social norms and individuals’ incentives to maintain a good reputation. This intuition has been formalized with two types of models. In public assessment models, all community members are assumed to agree on each others’ reputations; in private assessment models, people may have disagreements. Both types of models aim to understand the interplay of social norms and cooperation. Yet their results can be vastly different. Public assessment models argue that cooperation can evolve easily and that the most effective norms tend to be stern. Private assessment models often find cooperation to be unstable, and successful norms show some leniency. Here, we propose a model that can organize these differing results within a single …
Cited by:4
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Adaptive dynamics of direct reciprocity with N rounds of memory
2024
Nataliya A Balabanova, Manh Hong Duong, Christian Hilbe
arXiv preprint arXiv:2411.09535
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Authors:Nataliya A Balabanova, Manh Hong Duong, Christian Hilbe
Publication date:11/14/2024
Journal:arXiv preprint arXiv:2411.09535
Description:The theory of direct reciprocity explores how individuals cooperate when they interact repeatedly. In repeated interactions, individuals can condition their behaviour on what happened earlier. One prominent example of a conditional strategy is Tit-for-Tat, which prescribes to cooperate if and only if the co-player did so in the previous round. The evolutionary dynamics among such memory-1 strategies have been explored in quite some detail. However, obtaining analytical results on the dynamics of higher memory strategies becomes increasingly difficult, due to the rapidly growing size of the strategy space. Here, we derive such results for the adaptive dynamics in the donation game. In particular, we prove that for every orbit forward in time, there is an associated orbit backward in time that also solves the differential equation. Moreover, we analyse the dynamics by separating payoffs into a symmetric and an anti-symmetric part and demonstrate some properties of the anti-symmetric part. These results highlight some interesting symmetries that arise when interchanging player one with player two, and cooperation with defection.
Resources:[PDF] from arxiv.org
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The co-evolution of direct, indirect and generalized reciprocity
2024
Saptarshi Pal, Christian Hilbe, Nikoleta E Glynatsi
arXiv preprint arXiv:2411.03488
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Authors:Saptarshi Pal, Christian Hilbe, Nikoleta E Glynatsi
Publication date:11/05/2024
Journal:arXiv preprint arXiv:2411.03488
Description:People often engage in costly cooperation, especially in repeated interactions. When deciding whether to cooperate, individuals typically take into account how others have acted in the past. For instance, when one person is deciding whether to cooperate with another, they may consider how they were treated by the other party (direct reciprocity), how the other party treated others (indirect reciprocity), or how they themselves were treated by others in general (generalized reciprocity). Given these different approaches, it is unclear which strategy, or more specifically which mode of reciprocity, individuals will prefer. This study introduces a model where individuals decide how much weight to give each type of information when choosing to cooperate. Through equilibrium analysis, we find that all three modes of reciprocity can be sustained when individuals have sufficiently frequent interactions. However, the existence of such equilibria does not guarantee that individuals will learn to use them. Simulations show that when individuals mainly imitate others, generalized reciprocity often hinders cooperation, leading to defection even under conditions favorable to cooperation. In contrast, when individuals explore new strategies during learning, stable cooperation emerges through direct reciprocity. This study highlights the importance of studying all forms of reciprocity in unison.
Resources:[PDF] from arxiv.org
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Computational evolution of social norms in well-mixed and group-structured populations
2024
Yohsuke Murase, Christian Hilbe
Proceedings of the National Academy of Sciences 121 (33), e2406885121
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Authors:Yohsuke Murase, Christian Hilbe
Publication date:08/13/2024
Journal:Proceedings of the National Academy of Sciences
Volume:121
Issue:33
Pages:e2406885121
Publisher:National Academy of Sciences
Description:Models of indirect reciprocity study how social norms promote cooperation. In these models, cooperative individuals build up a positive reputation, which in turn helps them in their future interactions. The exact reputational benefits of cooperation depend on the norm in place, which may change over time. Previous research focused on the stability of social norms. Much less is known about how social norms initially evolve when competing with many others. A comprehensive evolutionary analysis, however, has been difficult. Even among the comparably simple space of so-called third-order norms, there are thousands of possibilities, each one inducing its own reputation dynamics. To address this challenge, we use large-scale computer simulations. We study the reputation dynamics of each third-order norm and all evolutionary transitions between them. In contrast to established work with only a handful of norms, we …
Cited by:6
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Evolution of reciprocity with limited payoff memory
2024
Nikoleta E Glynatsi, Alex McAvoy, Christian Hilbe
Proceedings of the Royal Society B 291 (2025), 20232493
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Authors:Nikoleta E Glynatsi, Alex McAvoy, Christian Hilbe
Publication date:06/19/2024
Journal:Proceedings of the Royal Society B
Volume:291
Issue:2025
Pages:20232493
Publisher:The Royal Society
Description:Direct reciprocity is a mechanism for the evolution of cooperation in repeated social interactions. According to the literature, individuals naturally learn to adopt conditionally cooperative strategies if they have multiple encounters with their partner. Corresponding models have greatly facilitated our understanding of cooperation, yet they often make strong assumptions on how individuals remember and process payoff information. For example, when strategies are updated through social learning, it is commonly assumed that individuals compare their average payoffs. This would require them to compute (or remember) their payoffs against everyone else in the population. To understand how more realistic constraints influence direct reciprocity, we consider the evolution of conditional behaviours when individuals learn based on more recent experiences. Even in the most extreme case that they only take into account …
Cited by:2
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Direct reciprocity among humans
2024
Charlotte SL Rossetti, Christian Hilbe
Ethology 130 (4), e13407
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Authors:Charlotte SL Rossetti, Christian Hilbe
Publication date:04/01/2024
Journal:Ethology
Volume:130
Issue:4
Pages:e13407
Description:Direct reciprocity is the tendency to repay others' cooperation. This tendency can be crucial to maintain cooperation in evolving populations. Once direct reciprocity evolves, individuals have a long‐run interest to cooperate, even if it is costly in the short run. The major theoretical framework to describe reciprocal behavior is the repeated prisoner's dilemma. Over the past decades, this game has been the major workhorse to predict when reciprocal cooperation ought to evolve, and which strategies individuals are supposed to adopt. Herein, we compare these predictions with the empirical evidence from experiments with human subjects. From a theory‐driven perspective, humans represent an ideal test case, because they give researchers the most flexibility to tailor the experimental design to the assumptions of a model. Overall, we find that theoretical models describe well in which situations people cooperate …
Cited by:15
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Efficiency and resilience of cooperation in asymmetric social dilemmas
2024
Valentin Hübner, Manuel Staab, Christian Hilbe, Krishnendu Chatterjee, Maria Kleshnina
Proceedings of the National Academy of Sciences 121 (10), e2315558121
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Authors:Valentin Hübner, Manuel Staab, Christian Hilbe, Krishnendu Chatterjee, Maria Kleshnina
Publication date:03/05/2024
Journal:Proceedings of the National Academy of Sciences
Volume:121
Issue:10
Pages:e2315558121
Publisher:National Academy of Sciences
Description:Direct reciprocity is a powerful mechanism for cooperation in social dilemmas. The very logic of reciprocity, however, seems to require that individuals are symmetric, and that everyone has the same means to influence each others’ payoffs. Yet in many applications, individuals are asymmetric. Herein, we study the effect of asymmetry in linear public good games. Individuals may differ in their endowments (their ability to contribute to a public good) and in their productivities (how effective their contributions are). Given the individuals’ productivities, we ask which allocation of endowments is optimal for cooperation. To this end, we consider two notions of optimality. The first notion focuses on the resilience of cooperation. The respective endowment distribution ensures that full cooperation is feasible even under the most adverse conditions. The second notion focuses on efficiency. The corresponding endowment …
Cited by:8
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Evolutionary games and applications: fifty years of ‘the logic of animal conflict’
2023
Christian Hilbe, Maria Kleshnina, Kateřina Staňková
Dynamic Games and Applications 13 (4), 1035-1048
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Authors:Christian Hilbe, Maria Kleshnina, Kateřina Staňková
Publication date:12/01/2023
Journal:Dynamic Games and Applications
Volume:13
Issue:4
Pages:1035-1048
Publisher:Springer US
Description:Darwin’s theory of evolution by natural selection has become one of the most influential theories in biology and adjacent fields [34]. It provided answers to many existing puzzles, but it also inspired many new questions. Some of these questions are still being addressed today, with a variety of methods. When it comes to the evolution of behavior, perhaps one of the most well-established methods is evolutionary game theory, which was introduced in ‘The logic of animal conflict’by John Maynard Smith and George Price in 1973 [95]. This method proved to be both versatile and insightful, with many applications in biology, anthropology, political science, economics, and other domains. In this special issue, we want to celebrate the fiftieth anniversary of the field by discussing some of the advances, and by highlighting potential challenges and future directions that lie ahead of the field. In their seminal article, Maynard …
Cited by:4
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Vacancies in growing habitats promote the evolution of cooperation
2023
Hye Jin Park, Christian Hilbe, Martin A Nowak, Beom Jun Kim, Hyeong-Chai Jeong
Journal of Theoretical Biology 575, 111629
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Authors:Hye Jin Park, Christian Hilbe, Martin A Nowak, Beom Jun Kim, Hyeong-Chai Jeong
Publication date:11/07/2023
Journal:Journal of Theoretical Biology
Volume:575
Pages:111629
Publisher:Academic Press
Description:We study evolutionary game dynamics in a growing habitat with vacancies. Fitness is determined by the global effect of the environment and a local prisoner’s dilemma among neighbors. We study population growth on a one-dimensional lattice and analyze how the environment affects evolutionary competition. As the environment becomes harsh, an absorbing phase transition from growing populations to extinction occurs. The transition point depends on which strategies are present in the population. In particular, we find a ‘cooperative window’ in parameter space, where only cooperators can survive. A mutant defector in a cooperative community might briefly proliferate, but over time naturally occurring vacancies separate cooperators from defectors, thereby driving defectors to extinction. Our model reveals that vacancies provide a strong boost for cooperation by spatial selection.
Cited by:1
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Indirect reciprocity with stochastic and dual reputation updates
2023
Yohsuke Murase, Christian Hilbe
PLOS Computational Biology 19 (7), e1011271
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Authors:Yohsuke Murase, Christian Hilbe
Publication date:07/20/2023
Journal:PLOS Computational Biology
Volume:19
Issue:7
Pages:e1011271
Publisher:Public Library of Science
Description:Cooperation is a crucial aspect of social life, yet understanding the nature of cooperation and how it can be promoted is an ongoing challenge. One mechanism for cooperation is indirect reciprocity. According to this mechanism, individuals cooperate to maintain a good reputation. This idea is embodied in a set of social norms called the “leading eight”. When all information is publicly available, these norms have two major properties. Populations that employ these norms are fully cooperative, and they are stable against invasion by alternative norms. In this paper, we extend the framework of the leading eight in two directions. First, we include norms with ‘dual’ reputation updates. These norms do not only assign new reputations to an acting donor; they also allow to update the reputation of the passive recipient. Second, we allow social norms to be stochastic. Such norms allow individuals to evaluate others with certain probabilities. Using this framework, we characterize all evolutionarily stable norms that lead to full cooperation in the public information regime. When only the donor’s reputation is updated, and all updates are deterministic, we recover the conventional model. In that case, we find two classes of stable norms: the leading eight and the ‘secondary sixteen’. Stochasticity can further help to stabilize cooperation when the benefit of cooperation is comparably small. Moreover, updating the recipients’ reputations can help populations to recover more quickly from errors. Overall, our study highlights a remarkable trade-off between the evolutionary stability of a norm and its robustness with respect to errors. Norms that correct errors quickly …
Cited by:11
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The effect of environmental information on evolution of cooperation in stochastic games
2023
Maria Kleshnina, Christian Hilbe, Štěpán Šimsa, Krishnendu Chatterjee, Martin A Nowak
Nature Communications 14 (1), 4153
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Authors:Maria Kleshnina, Christian Hilbe, Štěpán Šimsa, Krishnendu Chatterjee, Martin A Nowak
Publication date:07/12/2023
Journal:Nature Communications
Volume:14
Issue:1
Pages:4153
Publisher:Nature Publishing Group UK
Description:Many human interactions feature the characteristics of social dilemmas where individual actions have consequences for the group and the environment. The feedback between behavior and environment can be studied with the framework of stochastic games. In stochastic games, the state of the environment can change, depending on the choices made by group members. Past work suggests that such feedback can reinforce cooperative behaviors. In particular, cooperation can evolve in stochastic games even if it is infeasible in each separate repeated game. In stochastic games, participants have an interest in conditioning their strategies on the state of the environment. Yet in many applications, precise information about the state could be scarce. Here, we study how the availability of information (or lack thereof) shapes evolution of cooperation. Already for simple examples of two state games we find surprising …
Cited by:31
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Adaptive dynamics of memory-one strategies in the repeated donation game
2023
Philip LaPorte, Christian Hilbe, Martin A Nowak
PLoS Computational Biology 19 (6), e1010987
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Authors:Philip LaPorte, Christian Hilbe, Martin A Nowak
Publication date:06/29/2023
Journal:PLoS Computational Biology
Volume:19
Issue:6
Pages:e1010987
Publisher:Public Library of Science
Description:Human interactions can take the form of social dilemmas: collectively, people fare best if all cooperate but each individual is tempted to free ride. Social dilemmas can be resolved when individuals interact repeatedly. Repetition allows them to adopt reciprocal strategies which incentivize cooperation. The most basic model for direct reciprocity is the repeated donation game, a variant of the prisoner’s dilemma. Two players interact over many rounds; in each round they decide whether to cooperate or to defect. Strategies take into account the history of the play. Memory-one strategies depend only on the previous round. Even though they are among the most elementary strategies of direct reciprocity, their evolutionary dynamics has been difficult to study analytically. As a result, much previous work has relied on simulations. Here, we derive and analyze their adaptive dynamics. We show that the four-dimensional space of memory-one strategies has an invariant three-dimensional subspace, generated by the memory-one counting strategies. Counting strategies record how many players cooperated in the previous round, without considering who cooperated. We give a partial characterization of adaptive dynamics for memory-one strategies and a full characterization for memory-one counting strategies.
Cited by:4
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Mutation enhances cooperation in direct reciprocity
2023
Josef Tkadlec, Christian Hilbe, Martin A Nowak
Proceedings of the National Academy of Sciences 120 (20), e2221080120
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Authors:Josef Tkadlec, Christian Hilbe, Martin A Nowak
Publication date:05/16/2023
Journal:Proceedings of the National Academy of Sciences
Volume:120
Issue:20
Pages:e2221080120
Publisher:National Academy of Sciences
Description:Direct reciprocity is a powerful mechanism for the evolution of cooperation based on repeated interactions between the same individuals. But high levels of cooperation evolve only if the benefit-to-cost ratio exceeds a certain threshold that depends on memory length. For the best-explored case of one-round memory, that threshold is two. Here, we report that intermediate mutation rates lead to high levels of cooperation, even if the benefit-to-cost ratio is only marginally above one, and even if individuals only use a minimum of past information. This surprising observation is caused by two effects. First, mutation generates diversity which undermines the evolutionary stability of defectors. Second, mutation leads to diverse communities of cooperators that are more resilient than homogeneous ones. This finding is relevant because many real-world opportunities for cooperation have small benefit-to-cost ratios, which are …
Cited by:16
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Cooperation and coordination in heterogeneous populations
2023
Xiaomin Wang, Marta C Couto, Nianyi Wang, Xinmiao An, Bin Chen, Yali Dong, Christian Hi…
Philosophical Transactions of the Royal Society B 378 (1876), 20210504
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Authors:Xiaomin Wang, Marta C Couto, Nianyi Wang, Xinmiao An, Bin Chen, Yali Dong, Christian Hilbe, Boyu Zhang
Publication date:05/08/2023
Journal:Philosophical Transactions of the Royal Society B
Volume:378
Issue:1876
Pages:20210504
Publisher:The Royal Society
Description:One landmark application of evolutionary game theory is the study of social dilemmas. This literature explores why people cooperate even when there are strong incentives to defect. Much of this literature, however, assumes that interactions are symmetric. Individuals are assumed to have the same strategic options and the same potential pay-offs. Yet many interesting questions arise once individuals are allowed to differ. Here, we study asymmetry in simple coordination games. In our set-up, human participants need to decide how much of their endowment to contribute to a public good. If a group’s collective contribution reaches a pre-defined threshold, all group members receive a reward. To account for possible asymmetries, individuals either differ in their endowments or their productivities. According to a theoretical equilibrium analysis, such games tend to have many possible solutions. In equilibrium, group …
Cited by:20
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Quantitative assessment can stabilize indirect reciprocity under imperfect information
2023
Laura Schmid, Farbod Ekbatani, Christian Hilbe, Krishnendu Chatterjee
Nature Communications 14 (1), 2086
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Authors:Laura Schmid, Farbod Ekbatani, Christian Hilbe, Krishnendu Chatterjee
Publication date:04/12/2023
Journal:Nature Communications
Volume:14
Issue:1
Pages:2086
Publisher:Nature Publishing Group UK
Description:The field of indirect reciprocity investigates how social norms can foster cooperation when individuals continuously monitor and assess each other’s social interactions. By adhering to certain social norms, cooperating individuals can improve their reputation and, in turn, receive benefits from others. Eight social norms, known as the “leading eight," have been shown to effectively promote the evolution of cooperation as long as information is public and reliable. These norms categorize group members as either ’good’ or ’bad’. In this study, we examine a scenario where individuals instead assign nuanced reputation scores to each other, and only cooperate with those whose reputation exceeds a certain threshold. We find both analytically and through simulations that such quantitative assessments are error-correcting, thus facilitating cooperation in situations where information is private and unreliable. Moreover, our …
Cited by:22
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Evolution of direct reciprocity in group-structured populations
2022
Yohsuke Murase, Christian Hilbe, Seung Ki Baek
Scientific Reports 12 (1), 18645
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Authors:Yohsuke Murase, Christian Hilbe, Seung Ki Baek
Publication date:11/04/2022
Journal:Scientific Reports
Volume:12
Issue:1
Pages:18645
Publisher:Nature Publishing Group UK
Description:People tend to have their social interactions with members of their own community. Such group-structured interactions can have a profound impact on the behaviors that evolve. Group structure affects the way people cooperate, and how they reciprocate each other’s cooperative actions. Past work has shown that population structure and reciprocity can both promote the evolution of cooperation. Yet the impact of these mechanisms has been typically studied in isolation. In this work, we study how the two mechanisms interact. Using a game-theoretic model, we explore how people engage in reciprocal cooperation in group-structured populations, compared to well-mixed populations of equal size. In this model, the population is subdivided into groups. Individuals engage in pairwise interactions within groups while they also have chances to imitate strategies outside the groups. To derive analytical results, we focus on …
Cited by:14
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Cumulative reciprocity can sustain cooperation in repeated social interactions
2022
Haoxiang Xia, Christian Hilbe
NATURE COMPUTATIONAL SCIENCE 2 (11), 709-710
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Authors:Haoxiang Xia, Christian Hilbe
Publication date:11/01/2022
Journal:NATURE COMPUTATIONAL SCIENCE
Volume:2
Issue:11
Pages:709-710
Publisher:SPRINGERNATURE
Description:In social interactions, individuals are often tempted to 'free ride' (benefit without paying) on other group members' contributions. A new mathematical framework suggests a strategy based on cumulative reciprocity that can help to sustain mutual cooperation.
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Reputation effects drive the joint evolution of cooperation and social rewarding
2022
Saptarshi Pal, Christian Hilbe
Nature Communications 13 (1), 5928
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Authors:Saptarshi Pal, Christian Hilbe
Publication date:10/07/2022
Journal:Nature Communications
Volume:13
Issue:1
Pages:5928
Publisher:Nature Publishing Group UK
Description:People routinely cooperate with each other, even when cooperation is costly. To further encourage such pro-social behaviors, recipients often respond by providing additional incentives, for example by offering rewards. Although such incentives facilitate cooperation, the question remains how these incentivizing behaviors themselves evolve, and whether they would always be used responsibly. Herein, we consider a simple model to systematically study the co-evolution of cooperation and different rewarding policies. In our model, both social and antisocial behaviors can be rewarded, but individuals gain a reputation for how they reward others. By characterizing the game’s equilibria and by simulating evolutionary learning processes, we find that reputation effects systematically favor cooperation and social rewarding. While our baseline model applies to pairwise interactions in well-mixed populations, we obtain …
Cited by:29
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Evolution of cooperation through cumulative reciprocity
2022
Juan Li, Xiaowei Zhao, Bing Li, Charlotte SL Rossetti, Christian Hilbe, Haoxiang Xia
Nature Computational Science 2 (10), 677-686
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Authors:Juan Li, Xiaowei Zhao, Bing Li, Charlotte SL Rossetti, Christian Hilbe, Haoxiang Xia
Publication date:10/01/2022
Journal:Nature Computational Science
Volume:2
Issue:10
Pages:677-686
Publisher:Nature Publishing Group US
Description:Reciprocity is a simple principle for cooperation that explains many of the patterns of how humans seek and receive help from each other. To capture reciprocity, traditional models often assume that individuals use simple strategies with restricted memory. These memory-1 strategies are mathematically convenient, but they miss important aspects of human reciprocity, where defections can have lasting effects. Here we instead propose a strategy of cumulative reciprocity. Cumulative reciprocators count the imbalance of cooperation across their previous interactions with their opponent. They cooperate as long as this imbalance is sufficiently small. Using analytical and computational methods, we show that this strategy can sustain cooperation in the presence of errors, that it enforces fair outcomes and that it evolves in hostile environments. Using an economic experiment, we confirm that cumulative reciprocity is more …
Cited by:31
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Evolutionary instability of selfish learning in repeated games
2022
Alex McAvoy, Julian Kates-Harbeck, Krishnendu Chatterjee, Christian Hilbe
PNAS nexus 1 (4), pgac141
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Authors:Alex McAvoy, Julian Kates-Harbeck, Krishnendu Chatterjee, Christian Hilbe
Publication date:09/01/2022
Journal:PNAS nexus
Volume:1
Issue:4
Pages:pgac141
Publisher:Oxford University Press
Description:Across many domains of interaction, both natural and artificial, individuals use past experience to shape future behaviors. The results of such learning processes depend on what individuals wish to maximize. A natural objective is one’s own success. However, when two such “selfish” learners interact with each other, the outcome can be detrimental to both, especially when there are conflicts of interest. Here, we explore how a learner can align incentives with a selfish opponent. Moreover, we consider the dynamics that arise when learning rules themselves are subject to evolutionary pressure. By combining extensive simulations and analytical techniques, we demonstrate that selfish learning is unstable in most classical two-player repeated games. If evolution operates on the level of long-run payoffs, selection instead favors learning rules that incorporate social (other-regarding) preferences. To further …
Cited by:19
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Direct reciprocity between individuals that use different strategy spaces
2022
Laura Schmid, Christian Hilbe, Krishnendu Chatterjee, Martin A Nowak
PLoS Computational Biology 18 (6), e1010149
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Authors:Laura Schmid, Christian Hilbe, Krishnendu Chatterjee, Martin A Nowak
Publication date:06/14/2022
Journal:PLoS Computational Biology
Volume:18
Issue:6
Pages:e1010149
Publisher:Public Library of Science
Description:In repeated interactions, players can use strategies that respond to the outcome of previous rounds. Much of the existing literature on direct reciprocity assumes that all competing individuals use the same strategy space. Here, we study both learning and evolutionary dynamics of players that differ in the strategy space they explore. We focus on the infinitely repeated donation game and compare three natural strategy spaces: memory-1 strategies, which consider the last moves of both players, reactive strategies, which respond to the last move of the co-player, and unconditional strategies. These three strategy spaces differ in the memory capacity that is needed. We compute the long term average payoff that is achieved in a pairwise learning process. We find that smaller strategy spaces can dominate larger ones. For weak selection, unconditional players dominate both reactive and memory-1 players. For intermediate selection, reactive players dominate memory-1 players. Only for strong selection and low cost-to-benefit ratio, memory-1 players dominate the others. We observe that the supergame between strategy spaces can be a social dilemma: maximum payoff is achieved if both players explore a larger strategy space, but smaller strategy spaces dominate.
Cited by:25
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Introspection dynamics: a simple model of counterfactual learning in asymmetric games
2022
Marta C Couto, Stefano Giaimo, Christian Hilbe
New Journal of Physics 24 (6), 063010
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Authors:Marta C Couto, Stefano Giaimo, Christian Hilbe
Publication date:06/14/2022
Journal:New Journal of Physics
Volume:24
Issue:6
Pages:063010
Publisher:IOP Publishing
Description:Social behavior in human and animal populations can be studied as an evolutionary process. Individuals often make decisions between different strategies, and those strategies that yield a fitness advantage tend to spread. Traditionally, much work in evolutionary game theory considers symmetric games: individuals are assumed to have access to the same set of strategies, and they experience the same payoff consequences. As a result, they can learn more profitable strategies by imitation. However, interactions are oftentimes asymmetric. In that case, imitation may be infeasible (because individuals differ in the strategies they are able to use), or it may be undesirable (because individuals differ in their incentives to use a strategy). Here, we consider an alternative learning process which applies to arbitrary asymmetric games, introspection dynamics. According to this dynamics, individuals regularly compare their …
Cited by:21
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Cooperation in alternating interactions with memory constraints
2022
Peter S Park, Martin A Nowak, Christian Hilbe
Nature Communications 13 (1), 737
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Authors:Peter S Park, Martin A Nowak, Christian Hilbe
Publication date:02/08/2022
Journal:Nature Communications
Volume:13
Issue:1
Pages:737
Publisher:Nature Publishing Group UK
Description:In repeated social interactions, individuals often employ reciprocal strategies to maintain cooperation. To explore the emergence of reciprocity, many theoretical models assume synchronized decision making. In each round, individuals decide simultaneously whether to cooperate or not. Yet many manifestations of reciprocity in nature are asynchronous. Individuals provide help at one time and receive help at another. Here, we explore such alternating games in which players take turns. We mathematically characterize all Nash equilibria among memory-one strategies. Moreover, we use evolutionary simulations to explore various model extensions, exploring the effect of discounted games, irregular alternation patterns, and higher memory. In all cases, we observe that mutual cooperation still evolves for a wide range of parameter values. However, compared to simultaneous games, alternating games require …
Cited by:40
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(Mis) perceiving cooperativeness
2022
Charlotte SL Rossetti, Christian Hilbe, Oliver P Hauser
Current Opinion in Psychology 43, 151-155
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Authors:Charlotte SL Rossetti, Christian Hilbe, Oliver P Hauser
Publication date:02/01/2022
Journal:Current Opinion in Psychology
Volume:43
Pages:151-155
Publisher:Elsevier
Cited by:12
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A unified framework of direct and indirect reciprocity
2021
Laura Schmid, Krishnendu Chatterjee, Christian Hilbe, Martin A Nowak
Nature Human Behaviour 5 (10), 1292-1302
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Authors:Laura Schmid, Krishnendu Chatterjee, Christian Hilbe, Martin A Nowak
Publication date:10/01/2021
Journal:Nature Human Behaviour
Volume:5
Issue:10
Pages:1292-1302
Publisher:Nature Publishing Group UK
Description:Direct and indirect reciprocity are key mechanisms for the evolution of cooperation. Direct reciprocity means that individuals use their own experience to decide whether to cooperate with another person. Indirect reciprocity means that they also consider the experiences of others. Although these two mechanisms are intertwined, they are typically studied in isolation. Here, we introduce a mathematical framework that allows us to explore both kinds of reciprocity simultaneously. We show that the well-known ‘generous tit-for-tat’ strategy of direct reciprocity has a natural analogue in indirect reciprocity, which we call ‘generous scoring’. Using an equilibrium analysis, we characterize under which conditions either of the two strategies can maintain cooperation. With simulations, we additionally explore which kind of reciprocity evolves when members of a population engage in social learning to adapt to their environment …
Cited by:122
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The evolution of indirect reciprocity under action and assessment generosity
2021
Laura Schmid, Pouya Shati, Christian Hilbe, Krishnendu Chatterjee
Scientific Reports 11 (1), 17443
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Authors:Laura Schmid, Pouya Shati, Christian Hilbe, Krishnendu Chatterjee
Publication date:08/31/2021
Journal:Scientific Reports
Volume:11
Issue:1
Pages:17443
Publisher:Nature Publishing Group UK
Description:Indirect reciprocity is a mechanism for the evolution of cooperation based on social norms. This mechanism requires that individuals in a population observe and judge each other’s behaviors. Individuals with a good reputation are more likely to receive help from others. Previous work suggests that indirect reciprocity is only effective when all relevant information is reliable and publicly available. Otherwise, individuals may disagree on how to assess others, even if they all apply the same social norm. Such disagreements can lead to a breakdown of cooperation. Here we explore whether the predominantly studied ‘leading eight’ social norms of indirect reciprocity can be made more robust by equipping them with an element of generosity. To this end, we distinguish between two kinds of generosity. According to assessment generosity, individuals occasionally assign a good reputation to group members who would …
Cited by:33
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Evolving cooperation in multichannel games
2020
Kate Donahue, Oliver P Hauser, Martin A Nowak, Christian Hilbe
Nature communications 11 (1), 3885
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Authors:Kate Donahue, Oliver P Hauser, Martin A Nowak, Christian Hilbe
Publication date:08/04/2020
Journal:Nature communications
Volume:11
Issue:1
Pages:3885
Publisher:Nature Publishing Group UK
Description:Humans routinely engage in many distinct interactions in parallel. Team members collaborate on several concurrent projects, and even whole nations interact with each other across a variety of issues, including trade, climate change and security. Yet the existing theory of direct reciprocity studies isolated repeated games. Such models cannot account for strategic attempts to use the vested interests in one game as a leverage to enforce cooperation in another. Here we introduce a general framework of multichannel games. Individuals interact with each other over multiple channels; each channel is a repeated game. Strategic choices in one channel can affect decisions in another. With analytical equilibrium calculations for the donation game and evolutionary simulations for several other games we show that such linkage facilitates cooperation. Our results suggest that previous studies tend to underestimate the …
Cited by:58
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Social immunity modulates competition between coinfecting pathogens
2020
Barbara Milutinović, Miriam Stock, Anna V Grasse, Elisabeth Naderlinger, Christian Hilb…
Ecology letters 23 (3), 565-574
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Authors:Barbara Milutinović, Miriam Stock, Anna V Grasse, Elisabeth Naderlinger, Christian Hilbe, Sylvia Cremer
Publication date:03/01/2020
Journal:Ecology letters
Volume:23
Issue:3
Pages:565-574
Description:Coinfections with multiple pathogens can result in complex within‐host dynamics affecting virulence and transmission. While multiple infections are intensively studied in solitary hosts, it is so far unresolved how social host interactions interfere with pathogen competition, and if this depends on coinfection diversity. We studied how the collective disease defences of ants – their social immunity – influence pathogen competition in coinfections of same or different fungal pathogen species. Social immunity reduced virulence for all pathogen combinations, but interfered with spore production only in different‐species coinfections. Here, it decreased overall pathogen sporulation success while increasing co‐sporulation on individual cadavers and maintaining a higher pathogen diversity at the community level. Mathematical modelling revealed that host sanitary care alone can modulate competitive outcomes between …
Cited by:11
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Social dilemmas among unequals
2019
Oliver P Hauser, Christian Hilbe, Krishnendu Chatterjee, Martin A Nowak
Nature 572 (7770), 524-527
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Authors:Oliver P Hauser, Christian Hilbe, Krishnendu Chatterjee, Martin A Nowak
Publication date:08/22/2019
Journal:Nature
Volume:572
Issue:7770
Pages:524-527
Publisher:Nature Publishing Group UK
Description:Direct reciprocity is a powerful mechanism for the evolution of cooperation on the basis of repeated interactions, , –. It requires that interacting individuals are sufficiently equal, such that everyone faces similar consequences when they cooperate or defect. Yet inequality is ubiquitous among humans, and is generally considered to undermine cooperation and welfare, , –. Most previous models of reciprocity do not include inequality, , , –. These models assume that individuals are the same in all relevant aspects. Here we introduce a general framework to study direct reciprocity among unequal individuals. Our model allows for multiple sources of inequality. Subjects can differ in their endowments, their productivities and in how much they benefit from public goods. We find that extreme inequality prevents cooperation. But if subjects differ in productivity, some endowment inequality can be necessary for cooperation to …
Cited by:184
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Indirect reciprocity with private, noisy, and incomplete information
2018
Christian Hilbe, Laura Schmid, Josef Tkadlec, Krishnendu Chatterjee, Martin A Nowak
Proceedings of the national academy of sciences 115 (48), 12241-12246
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Authors:Christian Hilbe, Laura Schmid, Josef Tkadlec, Krishnendu Chatterjee, Martin A Nowak
Publication date:11/27/2018
Journal:Proceedings of the national academy of sciences
Volume:115
Issue:48
Pages:12241-12246
Publisher:National Academy of Sciences
Description:Indirect reciprocity is a mechanism for cooperation based on shared moral systems and individual reputations. It assumes that members of a community routinely observe and assess each other and that they use this information to decide who is good or bad, and who deserves cooperation. When information is transmitted publicly, such that all community members agree on each other’s reputation, previous research has highlighted eight crucial moral systems. These “leading-eight” strategies can maintain cooperation and resist invasion by defectors. However, in real populations individuals often hold their own private views of others. Once two individuals disagree about their opinion of some third party, they may also see its subsequent actions in a different light. Their opinions may further diverge over time. Herein, we explore indirect reciprocity when information transmission is private and noisy. We find that in the …
Cited by:164
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Evolution of cooperation in stochastic games
2018
Christian Hilbe, Štěpán Šimsa, Krishnendu Chatterjee, Martin A Nowak
Nature 559 (7713), 246-249
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Authors:Christian Hilbe, Štěpán Šimsa, Krishnendu Chatterjee, Martin A Nowak
Publication date:07/12/2018
Journal:Nature
Volume:559
Issue:7713
Pages:246-249
Publisher:Nature Publishing Group UK
Description:Social dilemmas occur when incentives for individuals are misaligned with group interests, , , , , –. According to the ‘tragedy of the commons’, these misalignments can lead to overexploitation and collapse of public resources. The resulting behaviours can be analysed with the tools of game theory. The theory of direct reciprocity, , , , , – suggests that repeated interactions can alleviate such dilemmas, but previous work has assumed that the public resource remains constant over time. Here we introduce the idea that the public resource is instead changeable and depends on the strategic choices of individuals. An intuitive scenario is that cooperation increases the public resource, whereas defection decreases it. Thus, cooperation allows the possibility of playing a more valuable game with higher payoffs, whereas defection leads to a less valuable game. We analyse this idea using the theory of stochastic games …
Cited by:240
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Partners and rivals in direct reciprocity
2018
Christian Hilbe, Krishnendu Chatterjee, Martin A Nowak
Nature human behaviour 2 (7), 469-477
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Authors:Christian Hilbe, Krishnendu Chatterjee, Martin A Nowak
Publication date:07/01/2018
Journal:Nature human behaviour
Volume:2
Issue:7
Pages:469-477
Publisher:Nature Publishing Group UK
Description:Reciprocity is a major factor in human social life and accounts for a large part of cooperation in our communities. Direct reciprocity arises when repeated interactions occur between the same individuals. The framework of iterated games formalizes this phenomenon. Despite being introduced more than five decades ago, the concept keeps offering beautiful surprises. Recent theoretical research driven by new mathematical tools has proposed a remarkable dichotomy among the crucial strategies: successful individuals either act as partners or as rivals. Rivals strive for unilateral advantages by applying selfish or extortionate strategies. Partners aim to share the payoff for mutual cooperation, but are ready to fight back when being exploited. Which of these behaviours evolves depends on the environment. Whereas small population sizes and a limited number of rounds favour rivalry, partner strategies are selected …
Cited by:161
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The signal-burying game can explain why we obscure positive traits and good deeds
2018
Moshe Hoffman, Christian Hilbe, Martin A Nowak
Nature Human Behaviour 2 (6), 397-404
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Authors:Moshe Hoffman, Christian Hilbe, Martin A Nowak
Publication date:06/01/2018
Journal:Nature Human Behaviour
Volume:2
Issue:6
Pages:397-404
Publisher:Nature Publishing Group UK
Description:People sometimes make their admirable deeds and accomplishments hard to spot, such as by giving anonymously or avoiding bragging. Such ‘buried’ signals are hard to reconcile with standard models of signalling or indirect reciprocity, which motivate costly pro-social behaviour by reputational gains. To explain these phenomena, we design a simple game theory model, which we call the signal-burying game. This game has the feature that senders can bury their signal by deliberately reducing the probability of the signal being observed. If the signal is observed, however, it is identified as having been buried. We show under which conditions buried signals can be maintained, using static equilibrium concepts and calculations of the evolutionary dynamics. We apply our analysis to shed light on a number of otherwise puzzling social phenomena, including modesty, anonymous donations, subtlety in art and fashion …
Cited by:39
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Crosstalk in concurrent repeated games impedes direct reciprocity and requires stronger levels of forgiveness
2018
Johannes G Reiter, Christian Hilbe, David G Rand, Krishnendu Chatterjee, Martin A Nowak
Nature Communications 9 (1), 555
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Authors:Johannes G Reiter, Christian Hilbe, David G Rand, Krishnendu Chatterjee, Martin A Nowak
Publication date:02/07/2018
Journal:Nature Communications
Volume:9
Issue:1
Pages:555
Publisher:Nature Publishing Group UK
Description:Direct reciprocity is a mechanism for cooperation among humans. Many of our daily interactions are repeated. We interact repeatedly with our family, friends, colleagues, members of the local and even global community. In the theory of repeated games, it is a tacit assumption that the various games that a person plays simultaneously have no effect on each other. Here we introduce a general framework that allows us to analyze “crosstalk” between a player’s concurrent games. In the presence of crosstalk, the action a person experiences in one game can alter the person’s decision in another. We find that crosstalk impedes the maintenance of cooperation and requires stronger levels of forgiveness. The magnitude of the effect depends on the population structure. In more densely connected social groups, crosstalk has a stronger effect. A harsh retaliator, such as Tit-for-Tat, is unable to counteract crosstalk. The …
Cited by:47
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The Red Queen and King in finite populations
2017
Carl Veller, Laura K Hayward, Christian Hilbe, Martin A Nowak
Proceedings of the National Academy of Sciences 114 (27), E5396-E5405
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Authors:Carl Veller, Laura K Hayward, Christian Hilbe, Martin A Nowak
Publication date:07/03/2017
Journal:Proceedings of the National Academy of Sciences
Volume:114
Issue:27
Pages:E5396-E5405
Publisher:National Academy of Sciences
Description:In antagonistic symbioses, such as host–parasite interactions, one population’s success is the other’s loss. In mutualistic symbioses, such as division of labor, both parties can gain, but they might have different preferences over the possible mutualistic arrangements. The rates of evolution of the two populations in a symbiosis are important determinants of which population will be more successful: Faster evolution is thought to be favored in antagonistic symbioses (the “Red Queen effect”), but disfavored in certain mutualistic symbioses (the “Red King effect”). However, it remains unclear which biological parameters drive these effects. Here, we analyze the effects of the various determinants of evolutionary rate: generation time, mutation rate, population size, and the intensity of natural selection. Our main results hold for the case where mutation is infrequent. Slower evolution causes a long-term advantage in an …
Cited by:30
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Memory-n strategies of direct reciprocity
2017
Christian Hilbe, Luis A Martinez-Vaquero, Krishnendu Chatterjee, Martin A Nowak
Proceedings of the National Academy of Sciences 114 (18), 4715-4720
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Authors:Christian Hilbe, Luis A Martinez-Vaquero, Krishnendu Chatterjee, Martin A Nowak
Publication date:05/02/2017
Journal:Proceedings of the National Academy of Sciences
Volume:114
Issue:18
Pages:4715-4720
Publisher:National Academy of Sciences
Description:Humans routinely use conditionally cooperative strategies when interacting in repeated social dilemmas. They are more likely to cooperate if others cooperated before, and are ready to retaliate if others defected. To capture the emergence of reciprocity, most previous models consider subjects who can only choose from a restricted set of representative strategies, or who react to the outcome of the very last round only. As players memorize more rounds, the dimension of the strategy space increases exponentially. This increasing computational complexity renders simulations for individuals with higher cognitive abilities infeasible, especially if multiplayer interactions are taken into account. Here, we take an axiomatic approach instead. We propose several properties that a robust cooperative strategy for a repeated multiplayer dilemma should have. These properties naturally lead to a unique class of cooperative …
Cited by:152
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Only the combination of mathematics and agent-based simulations can leverage the full potential of evolutionary modeling: Comment on” Evolutionary game theory using agent-based …
2016
Christian Hilbe, Arne Traulsen
Physics of life reviews 19, 29-31
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Authors:Christian Hilbe, Arne Traulsen
Publication date:12/01/2016
Journal:Physics of life reviews
Volume:19
Pages:29-31
Description:Only the combination of mathematics and agent-based simulations can leverage the full potential of evolutionary modeling: Comment on "Evolutionary game theory using agent-based methods" by C. Adami, J. Schossau and A. Hintze Only the combination of mathematics and agent-based simulations can leverage the full potential of evolutionary modeling: Comment on "Evolutionary game theory using agent-based methods" by C. Adami, J. Schossau and A. Hintze Phys Life Rev. 2016 Dec:19:29-31. doi: 10.1016/j.plrev.2016.10.004. Epub 2016 Oct 25. Authors Christian Hilbe 1 , Arne Traulsen 2 Affiliations 1 IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria. Electronic address: . 2 Department of Evolutionary Theory, Max Planck Institute for Evolutionary Biology, August-Thienemann-Straße 2, 24306 Plön, Germany. Electronic address: . PMID: 27810387 DOI: …
Cited by:6
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Asymmetric power boosts extortion in an economic experiment
2016
Christian Hilbe, Kristin Hagel, Manfred Milinski
PLoS One 11 (10), e0163867
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Authors:Christian Hilbe, Kristin Hagel, Manfred Milinski
Publication date:10/04/2016
Journal:PLoS One
Volume:11
Issue:10
Pages:e0163867
Publisher:Public Library of Science
Description:Direct reciprocity is a major mechanism for the evolution of cooperation. Several classical studies have suggested that humans should quickly learn to adopt reciprocal strategies to establish mutual cooperation in repeated interactions. On the other hand, the recently discovered theory of ZD strategies has found that subjects who use extortionate strategies are able to exploit and subdue cooperators. Although such extortioners have been predicted to succeed in any population of adaptive opponents, theoretical follow-up studies questioned whether extortion can evolve in reality. However, most of these studies presumed that individuals have similar strategic possibilities and comparable outside options, whereas asymmetries are ubiquitous in real world applications. Here we show with a model and an economic experiment that extortionate strategies readily emerge once subjects differ in their strategic power. Our experiment combines a repeated social dilemma with asymmetric partner choice. In our main treatment there is one randomly chosen group member who is unilaterally allowed to exchange one of the other group members after every ten rounds of the social dilemma. We find that this asymmetric replacement opportunity generally promotes cooperation, but often the resulting payoff distribution reflects the underlying power structure. Almost half of the subjects in a better strategic position turn into extortioners, who quickly proceed to exploit their peers. By adapting their cooperation probabilities consistent with ZD theory, extortioners force their co-players to cooperate without being similarly cooperative themselves. Comparison to non …
Cited by:29
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Comparing reactive and memory-one strategies of direct reciprocity
2016
Seung Ki Baek, Hyeong-Chai Jeong, Christian Hilbe, Martin A Nowak
Scientific reports 6 (1), 25676
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Authors:Seung Ki Baek, Hyeong-Chai Jeong, Christian Hilbe, Martin A Nowak
Publication date:05/10/2016
Journal:Scientific reports
Volume:6
Issue:1
Pages:25676
Publisher:Nature Publishing Group UK
Description:Direct reciprocity is a mechanism for the evolution of cooperation based on repeated interactions. When individuals meet repeatedly, they can use conditional strategies to enforce cooperative outcomes that would not be feasible in one-shot social dilemmas. Direct reciprocity requires that individuals keep track of their past interactions and find the right response. However, there are natural bounds on strategic complexity: Humans find it difficult to remember past interactions accurately, especially over long timespans. Given these limitations, it is natural to ask how complex strategies need to be for cooperation to evolve. Here, we study stochastic evolutionary game dynamics in finite populations to systematically compare the evolutionary performance of reactive strategies, which only respond to the co-player’s previous move, and memory-one strategies, which take into account the own and the co-player’s previous …
Cited by:84
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Coevolutionary interactions between farmers and mafia induce host acceptance of avian brood parasites
2016
Maria Abou Chakra, Christian Hilbe, Arne Traulsen
Royal Society Open Science 3 (5), 160036
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Authors:Maria Abou Chakra, Christian Hilbe, Arne Traulsen
Publication date:05/04/2016
Journal:Royal Society Open Science
Volume:3
Issue:5
Pages:160036
Publisher:The Royal Society
Description:Brood parasites exploit their host in order to increase their own fitness. Typically, this results in an arms race between parasite trickery and host defence. Thus, it is puzzling to observe hosts that accept parasitism without any resistance. The ‘mafia’ hypothesis suggests that these hosts accept parasitism to avoid retaliation. Retaliation has been shown to evolve when the hosts condition their response to mafia parasites, who use depredation as a targeted response to rejection. However, it is unclear if acceptance would also emerge when ‘farming’ parasites are present in the population. Farming parasites use depredation to synchronize the timing with the host, destroying mature clutches to force the host to re-nest. Herein, we develop an evolutionary model to analyse the interaction between depredatory parasites and their hosts. We show that coevolutionary cycles between farmers and mafia can still induce host …
Cited by:7
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Dynamics of prebiotic RNA reproduction illuminated by chemical game theory
2016
Jessica AM Yeates, Christian Hilbe, Martin Zwick, Martin A Nowak, Niles Lehman
Proceedings of the National Academy of Sciences 113 (18), 5030-5035
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Authors:Jessica AM Yeates, Christian Hilbe, Martin Zwick, Martin A Nowak, Niles Lehman
Publication date:05/03/2016
Journal:Proceedings of the National Academy of Sciences
Volume:113
Issue:18
Pages:5030-5035
Publisher:National Academy of Sciences
Description:Many origins-of-life scenarios depict a situation in which there are common and potentially scarce resources needed by molecules that compete for survival and reproduction. The dynamics of RNA assembly in a complex mixture of sequences is a frequency-dependent process and mimics such scenarios. By synthesizing Azoarcus ribozyme genotypes that differ in their single-nucleotide interactions with other genotypes, we can create molecules that interact among each other to reproduce. Pairwise interplays between RNAs involve both cooperation and selfishness, quantifiable in a 2 × 2 payoff matrix. We show that a simple model of differential equations based on chemical kinetics accurately predicts the outcomes of these molecular competitions using simple rate inputs into these matrices. In some cases, we find that mixtures of different RNAs reproduce much better than each RNA type alone, reflecting a …
Cited by:40
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Humans choose representatives who enforce cooperation in social dilemmas through extortion
2016
Manfred Milinski, Christian Hilbe, Dirk Semmann, Ralf Sommerfeld, Jochem Marotzke
Nature communications 7 (1), 10915
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Authors:Manfred Milinski, Christian Hilbe, Dirk Semmann, Ralf Sommerfeld, Jochem Marotzke
Publication date:03/07/2016
Journal:Nature communications
Volume:7
Issue:1
Pages:10915
Publisher:Nature Publishing Group UK
Description:Social dilemmas force players to balance between personal and collective gain. In many dilemmas, such as elected governments negotiating climate-change mitigation measures, the decisions are made not by individual players but by their representatives. However, the behaviour of representatives in social dilemmas has not been investigated experimentally. Here inspired by the negotiations for greenhouse-gas emissions reductions, we experimentally study a collective-risk social dilemma that involves representatives deciding on behalf of their fellow group members. Representatives can be re-elected or voted out after each consecutive collective-risk game. Selfish players are preferentially elected and are hence found most frequently in the ‘representatives’ treatment. Across all treatments, we identify the selfish players as extortioners. As predicted by our mathematical model, their steadfast strategies enforce …
Cited by:73
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Cooperate without looking in a non-repeated game
2015
Christian Hilbe, Moshe Hoffman, Martin A Nowak
Games 6 (4), 458-472
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Authors:Christian Hilbe, Moshe Hoffman, Martin A Nowak
Publication date:09/30/2015
Journal:Games
Volume:6
Issue:4
Pages:458-472
Publisher:MDPI
Description:We propose a simple model for why we have more trust in people who cooperate without calculating the associated costs. Intuitively, by not looking at the payoffs, people indicate that they will not be swayed by high temptations to defect, which makes them more attractive as interaction partners. We capture this intuition using a simple four-stage game. In the first stage, nature draws the costs and benefits of cooperation according to a commonly-known distribution. In the second stage, Player 1 chooses whether or not to look at the realized payoffs. In the third stage, Player 2 decides whether to exit or let Player 1 choose whether or not to cooperate in the fourth stage. Using backward induction, we provide a complete characterization for when we expect Player 1 to cooperate without looking. Moreover, we show with numerical simulations how cooperating without looking can emerge through simple evolutionary processes.
Cited by:18
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Partners or rivals? Strategies for the iterated prisoner’s dilemma
2015
Christian Hilbe, Arne Traulsen, Karl Sigmund
Games and economic behavior 92, 41-52
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Authors:Christian Hilbe, Arne Traulsen, Karl Sigmund
Publication date:07/01/2015
Journal:Games and economic behavior
Volume:92
Pages:41-52
Publisher:Academic Press
Description:Within the class of memory-one strategies for the iterated Prisoner's Dilemma, we characterize partner strategies, competitive strategies and zero-determinant strategies. If a player uses a partner strategy, both players can fairly share the social optimum; but a co-player preferring an unfair solution will be penalized by obtaining a reduced payoff. A player using a competitive strategy never obtains less than the co-player. A player using a zero-determinant strategy unilaterally enforces a linear relation between the two players' payoffs. These properties hold for every strategy used by the co-player, whether memory-one or not.
Cited by:151
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Evolutionary performance of zero-determinant strategies in multiplayer games
2015
Christian Hilbe, Bin Wu, Arne Traulsen, Martin A Nowak
Journal of theoretical biology 374, 115
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Authors:Christian Hilbe, Bin Wu, Arne Traulsen, Martin A Nowak
Publication date:06/07/2015
Journal:Journal of theoretical biology
Volume:374
Pages:115
Description:Repetition is one of the key mechanisms to maintain cooperation. In long-term relationships, in which individuals can react to their peers׳ past actions, evolution can promote cooperative strategies that would not be stable in one-shot encounters. The iterated prisoner׳s dilemma illustrates the power of repetition. Many of the key strategies for this game, such as ALLD, ALLC, Tit-for-Tat, or generous Tit-for-Tat, share a common property: players using these strategies enforce a linear relationship between their own payoff and their co-player׳s payoff. Such strategies have been termed zero-determinant (ZD). Recently, it was shown that ZD strategies also exist for multiplayer social dilemmas, and here we explore their evolutionary performance. For small group sizes, ZD strategies play a similar role as for the repeated prisoner׳s dilemma: extortionate ZD strategies are critical for the emergence of cooperation, whereas …
Cited by:89
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Modelling the dynamics of crime and punishment. Comment on” Statistical physics of crime: A review” by MR D’Orsogna and M. Perc
2015
Maria Abou Chakra, Christian Hilbe
Physics of Life Reviews 12, 22-23
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Authors:Maria Abou Chakra, Christian Hilbe
Publication date:03/01/2015
Journal:Physics of Life Reviews
Volume:12
Pages:22-23
Description:The causes and consequences of crime have fascinated humans for centuries. What makes people break the law, and how does crime affect societies? To approach such questions, it may seem most natural to take the viewpoint of sociology or psychology. However, starting with the pioneering work of Nobel prize laureate Gary S. Becker [1], it has been increasingly emphasized that crime does not need to be the result of insane minds. Instead, committing a crime may be quite rational, given the right circumstances. Whether we are willing to break the law depends on the environment and on the incentives at stake: What are the chances of being caught after committing a certain felony? And how severe is the punishment? From the perspective of economics, crime can often be understood as a result of conscious decisions, as an equilibrium in a strategic game.
Cited by:1
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Reconciling different views on responsible leadership: A rationality-based approach
2014
Christof Miska, Christian Hilbe, Susanne Mayer
Journal of Business Ethics 125, 349-360
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Authors:Christof Miska, Christian Hilbe, Susanne Mayer
Publication date:12/01/2014
Journal:Journal of Business Ethics
Volume:125
Pages:349-360
Publisher:Springer Netherlands
Cited by:184
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Cooperation and control in multiplayer social dilemmas
2014
Christian Hilbe, Bin Wu, Arne Traulsen, Martin A Nowak
Proceedings of the National Academy of Sciences 111 (46), 16425-16430
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Authors:Christian Hilbe, Bin Wu, Arne Traulsen, Martin A Nowak
Publication date:11/18/2014
Journal:Proceedings of the National Academy of Sciences
Volume:111
Issue:46
Pages:16425-16430
Publisher:National Academy of Sciences
Description:Direct reciprocity and conditional cooperation are important mechanisms to prevent free riding in social dilemmas. However, in large groups, these mechanisms may become ineffective because they require single individuals to have a substantial influence on their peers. However, the recent discovery of zero-determinant strategies in the iterated prisoner’s dilemma suggests that we may have underestimated the degree of control that a single player can exert. Here, we develop a theory for zero-determinant strategies for iterated multiplayer social dilemmas, with any number of involved players. We distinguish several particularly interesting subclasses of strategies: fair strategies ensure that the own payoff matches the average payoff of the group; extortionate strategies allow a player to perform above average; and generous strategies let a player perform below average. We use this theory to describe strategies that …
Cited by:203
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Sanctions as honest signals–The evolution of pool punishment by public sanctioning institutions
2014
Sarah Schoenmakers, Christian Hilbe, Bernd Blasius, Arne Traulsen
Journal of Theoretical Biology 356, 36-46
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Authors:Sarah Schoenmakers, Christian Hilbe, Bernd Blasius, Arne Traulsen
Publication date:09/07/2014
Journal:Journal of Theoretical Biology
Volume:356
Pages:36-46
Publisher:Academic Press
Description:In many species, mutual cooperation is stabilized by forms of policing and peer punishment: if cheaters are punished, there is a strong selective pressure to cooperate. Most human societies have complemented, and sometimes even replaced, such peer punishment mechanisms with pool punishment, where punishment is outsourced to central institutions such as the police. Even before free-riding occurs, such institutions require investments, which could serve as costly signals. Here, we show with a game theoretical model that this signaling effect in turn can be crucial for the evolution of punishment institutions: In the absence of such signals, pool punishment is only stable with second-order punishment and can only evolve when individuals have the freedom not to take part in any interaction. With such signals, individuals can opportunistically adjust their behavior, which promotes the evolution of stable pool …
Cited by:79
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Extortion subdues human players but is finally punished in the prisoner’s dilemma
2014
Christian Hilbe, Torsten Röhl, Manfred Milinski
Nature communications 5 (1), 3976
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Authors:Christian Hilbe, Torsten Röhl, Manfred Milinski
Publication date:05/29/2014
Journal:Nature communications
Volume:5
Issue:1
Pages:3976
Publisher:Nature Publishing Group UK
Description:Extortion is the practice of obtaining advantages through explicit forces and threats. Recently, it was demonstrated that even the repeated prisoner’s dilemma, one of the key models to explain mutual cooperation, allows for implicit forms of extortion. According to the theory, extortioners demand and receive an excessive share of any surplus, which allows them to outperform any adapting co-player. To explore the performance of such strategies against humans, we have designed an economic experiment in which participants were matched either with an extortioner or with a generous co-player. Although extortioners succeeded against each of their human opponents, extortion resulted in lower payoffs than generosity. Human subjects showed a strong concern for fairness: they punished extortion by refusing to fully cooperate, thereby reducing their own, and even more so, the extortioner’s gains. Thus, the prospects …
Cited by:120
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Plastic behaviors in hosts promote the emergence of retaliatory parasites
2014
Maria Abou Chakra, Christian Hilbe, Arne Traulsen
Scientific reports 4 (1), 4251
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Authors:Maria Abou Chakra, Christian Hilbe, Arne Traulsen
Publication date:03/04/2014
Journal:Scientific reports
Volume:4
Issue:1
Pages:4251
Publisher:Nature Publishing Group UK
Cited by:29
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Democratic decisions establish stable authorities that overcome the paradox of second-order punishment
2014
Christian Hilbe, Arne Traulsen, Torsten Röhl, Manfred Milinski
Proceedings of the National Academy of Sciences 111 (2), 752-756
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Authors:Christian Hilbe, Arne Traulsen, Torsten Röhl, Manfred Milinski
Publication date:01/14/2014
Journal:Proceedings of the National Academy of Sciences
Volume:111
Issue:2
Pages:752-756
Publisher:National Academy of Sciences
Description:Individuals usually punish free riders but refuse to sanction those who cooperate but do not punish. This missing second-order peer punishment is a fundamental problem for the stabilization of cooperation. To solve this problem, most societies today have implemented central authorities that punish free riders and tax evaders alike, such that second-order punishment is fully established. The emergence of such stable authorities from individual decisions, however, creates a new paradox: it seems absurd to expect individuals who do not engage in second-order punishment to strive for an authority that does. Herein, we provide a mathematical model and experimental results from a public goods game where subjects can choose between a community with and without second-order punishment in two different ways. When subjects can migrate continuously to either community, we identify a bias toward institutions that …
Cited by:114
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Adaptive dynamics of extortion and compliance
2013
Christian Hilbe, Martin A Nowak, Arne Traulsen
PloS one 8 (11), e77886
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Authors:Christian Hilbe, Martin A Nowak, Arne Traulsen
Publication date:11/01/2013
Journal:PloS one
Volume:8
Issue:11
Pages:e77886
Publisher:Public Library of Science
Description:Direct reciprocity is a mechanism for the evolution of cooperation. For the iterated prisoner’s dilemma, a new class of strategies has recently been described, the so-called zero-determinant strategies. Using such a strategy, a player can unilaterally enforce a linear relationship between his own payoff and the co-player’s payoff. In particular the player may act in such a way that it becomes optimal for the co-player to cooperate unconditionally. In this way, a player can manipulate and extort his co-player, thereby ensuring that the own payoff never falls below the co-player’s payoff. However, using a compliant strategy instead, a player can also ensure that his own payoff never exceeds the co-player’s payoff. Here, we use adaptive dynamics to study when evolution leads to extortion and when it leads to compliance. We find a remarkable cyclic dynamics: in sufficiently large populations, extortioners play a transient role, helping the population to move from selfish strategies to compliance. Compliant strategies, however, can be subverted by altruists, which in turn give rise to selfish strategies. Whether cooperative strategies are favored in the long run critically depends on the size of the population; we show that cooperation is most abundant in large populations, in which case average payoffs approach the social optimum. Our results are not restricted to the case of the prisoners dilemma, but can be extended to other social dilemmas, such as the snowdrift game. Iterated social dilemmas in large populations do not lead to the evolution of strategies that aim to dominate their co-player. Instead, generosity succeeds.
Cited by:112
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The evolution of strategic timing in collective-risk dilemmas
2013
Christian Hilbe, Maria Abou Chakra, Philipp M Altrock, Arne Traulsen
PloS one 8 (6), e66490
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Authors:Christian Hilbe, Maria Abou Chakra, Philipp M Altrock, Arne Traulsen
Publication date:06/14/2013
Journal:PloS one
Volume:8
Issue:6
Pages:e66490
Publisher:Public Library of Science
Description:In collective-risk dilemmas, a group needs to collaborate over time to avoid a catastrophic event. This gives rise to a coordination game with many equilibria, including equilibria where no one contributes, and thus no measures against the catastrophe are taken. In this game, the timing of contributions becomes a strategic variable that allows individuals to interact and influence one another. Herein, we use evolutionary game theory to study the impact of strategic timing on equilibrium selection. Depending on the risk of catastrophe, we identify three characteristic regimes. For low risks, defection is the only equilibrium, whereas high risks promote equilibria with sufficient contributions. Intermediate risks pose the biggest challenge for cooperation. In this risk regime, the option to interact over time is critical; if individuals can contribute over several rounds, then the group has a higher chance to succeed, and the expected welfare increases. This positive effect of timing is of particular importance in larger groups, where successful coordination becomes increasingly difficult.
Cited by:51
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Emergence of responsible sanctions without second order free riders, antisocial punishment or spite
2012
Christian Hilbe, Arne Traulsen
Scientific reports 2 (1), 458
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Authors:Christian Hilbe, Arne Traulsen
Publication date:06/13/2012
Journal:Scientific reports
Volume:2
Issue:1
Pages:458
Publisher:Nature Publishing Group UK
Description:While empirical evidence highlights the importance of punishment for cooperation in collective action, it remains disputed how responsible sanctions targeted predominantly at uncooperative subjects can evolve. Punishment is costly; in order to spread it typically requires local interactions, voluntary participation, or rewards. Moreover, theory and experiments indicate that some subjects abuse sanctioning opportunities by engaging in antisocial punishment (which harms cooperators), spiteful acts (harming everyone) or revenge (as a response to being punished). These arguments have led to the conclusion that punishment is maladaptive. Here, we use evolutionary game theory to show that this conclusion is premature: If interactions are non-anonymous, cooperation and punishment evolve even if initially rare and sanctions are directed towards non-cooperators only. Thus, our willingness to punish free riders is …
Cited by:166
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Local replicator dynamics: a simple link between deterministic and stochastic models of evolutionary game theory
2011
Christian Hilbe
Bulletin of Mathematical Biology 73 (9), 2068-2087
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Authors:Christian Hilbe
Publication date:09/01/2011
Journal:Bulletin of Mathematical Biology
Volume:73
Issue:9
Pages:2068-2087
Publisher:Springer New York
Description:Classical replicator dynamics assumes that individuals play their games and adopt new strategies on a global level: Each player interacts with a representative sample of the population and if a strategy yields a payoff above the average, then it is expected to spread. In this article, we connect evolutionary models for infinite and finite populations: While the population itself is infinite, interactions and reproduction occurs in random groups of size N. Surprisingly, the resulting dynamics simplifies to the traditional replicator system with a slightly modified payoff matrix. The qualitative results, however, mirror the findings for finite populations, in which strategies are selected according to a probabilistic Moran process. In particular, we derive a one-third law that holds for any population size. In this way, we show that the deterministic replicator equation in an infinite population can be used to study the Moran process …
Cited by:71
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Darwin and the evolution of human cooperation
2011
Karl Sigmund, Christian Hilbe
Principles of Evolution: From the Planck Epoch to Complex Multicellular Life, 331-347
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Authors:Karl Sigmund, Christian Hilbe
Publication date:02/22/2011
Journal:Principles of Evolution: From the Planck Epoch to Complex Multicellular Life
Pages:331-347
Publisher:Springer Berlin Heidelberg
Description:Humans are characterized by a high propensity for cooperation. The emergence and stability of this trait is one of the hottest topics in evolutionary game theory, and leads to a wide variety of models offering a rich source of complex dynamics based on social interactions. This chapter offers an overview of different approaches to this topic (such as kin selection, group selection, direct and indirect reciprocity) and relates it to some of the views that Darwin expressed over 150 years ago. It turns out that, in many cases, Darwin displayed a remarkably lucid intuition of the major issues affecting the complex mechanisms promoting the evolution of cooperation.
Cited by:4
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Equilibrium notions and framing effects
2010
Christian Hilbe
arXiv preprint arXiv:1012.1188
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Authors:Christian Hilbe
Publication date:12/03/2010
Journal:arXiv preprint arXiv:1012.1188
Description:Experimental economics has repeatedly demonstrated that the Nash equilibrium makes inaccurate predictions for a vast set of games. Instead, several alternative theoretical concepts predict behavior that is much more in tune with observed data, with the quantal response equilibrium as the most prominent example. However, here we show that this equilibrium notion itself, like any other concept that varies smoothly with the payoffs, is necessarily subject to framing effects: If the same economic problem is represented in a different but equivalent way, the predicted results will differ. As a consequence, we argue that tools and methods that are successful in explaining human behavior in laboratory experiments may be unsuitable for doing theory.
Cited by:3
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Incentives and opportunism: from the carrot to the stick
2010
Christian Hilbe, Karl Sigmund
Proceedings of the Royal Society B: Biological Sciences 277 (1693), 2427-2433
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Authors:Christian Hilbe, Karl Sigmund
Publication date:08/22/2010
Journal:Proceedings of the Royal Society B: Biological Sciences
Volume:277
Issue:1693
Pages:2427-2433
Publisher:The Royal Society
Description:Cooperation in public good games is greatly promoted by positive and negative incentives. In this paper, we use evolutionary game dynamics to study the evolution of opportunism (the readiness to be swayed by incentives) and the evolution of trust (the propensity to cooperate in the absence of information on the co-players). If both positive and negative incentives are available, evolution leads to a population where defectors are punished and players cooperate, except when they can get away with defection. Rewarding behaviour does not become fixed, but can play an essential role in catalysing the emergence of cooperation, especially if the information level is low.
Cited by:211
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Contrition does not ensure cooperation in the Iterated Prisoner’s Dilemma
2009
Christian Hilbe
International Journal of Bifurcation and Chaos 19 (11), 3877-3885
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Authors:Christian Hilbe
Publication date:11/01/2009
Journal:International Journal of Bifurcation and Chaos
Volume:19
Issue:11
Pages:3877-3885
Publisher:World Scientific Publishing Company
Description:In search of strategies that boost cooperation in the iterated prisoner's dilemma game (IPD), Contrite Tit for Tat is a promising candidate. It combines the necessary rigor in handling selfish strategies with contrition whenever the player has exploited its coplayer inadvertently. This paper shows that, contrary to conventional wisdom, Contrite Tit for Tat is not evolutionarily stable in general. In a relatively noisy environment with a low benefit of mutual cooperation it pays to be a defector. Additionally, errors in perception impede altruism notably.
Cited by:6
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