Extortion under uncertainty: Zero-determinant strategies in noisy games

Conditional cooperation with longer memory

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 the cost-to-benefit ratio of cooperation, the error rate, and the strength of selection. Within the strategy sets we consider, we find that longer memory tends to promote cooperation. This positive effect of memory is particularly pronounced when individuals take into account the precise sequence of moves.

Outlearning extortioners: unbending strategies can foster reciprocal fairness and cooperation

Recent theory shows that extortioners taking advantage of the zero-determinant (ZD) strategy can unilaterally claim an unfair share of the payoffs in the Iterated Prisoner's Dilemma. It is thus suggested that against a fixed extortioner, any adapting coplayer should be subdued with full cooperation as their best response. In contrast, recent experiments demonstrate that human players often choose not to accede to extortion out of concern for fairness, actually causing extortioners to suffer more loss than themselves. In light of this, here we reveal fair-minded strategies that are unbending to extortion such that any payoff-maximizing extortioner ultimately will concede in their own interest by offering a fair split in head-to-head matches. We find and characterize multiple general classes of such unbending strategies, including generous ZD strategies and Win-Stay, Lose-Shift (WSLS) as particular examples. When against fixed unbending players, extortioners are forced with consequentially increasing losses whenever intending to demand a more unfair share. Our analysis also pivots to the importance of payoff structure in determining the superiority of ZD strategies and in particular their extortion ability. We show that an extortionate ZD player can be even outperformed by, for example, WSLS, if the total payoff of unilateral cooperation is smaller than that of mutual defection. Unbending strategies can be used to outlearn evolutionary extortioners and catalyze the evolution of Tit-for-Tat-like strategies out of ZD players. Our work has implications for promoting fairness and resisting extortion so as to uphold a just and cooperative society.

The evolutionary extortion game of multiple groups in hypernetworks

As a type of zero-determinant strategies, the extortion strategy was found to be an evolutionarily stable strategy in structural groups. However, instead of complex networks structure, this paper focus on a multi-group game in hypernetworks, using the framework of a gift giving game driven by replicator-like dynamics. We find that the extortion is evolutionarily stable in the hypernetwork structure. The extortion game in hypernetworks can promote the emergence of the cooperative behavior compared to the traditional dual-strategy game and the extortion game in complex networks. The results show that the cooperation behavior attracts most of the groups for the smaller benefit value. With the increase of benefit value, cooperators turn into defectors and extortioners, but cooperation behavior still survives in hypernetworks under extreme conditions. Moreover, small-scale groups are more conducive to cooperation.

Evolution of cooperation through cumulative reciprocity

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 predictive of human behaviour than several classical strategies. The basic principle of cumulative reciprocity is versatile and can be extended to a range of social dilemmas.

Mitigation of block withholding attack based on zero-determinant strategy

This article focuses on the mining dilemma of block withholding attack between the mining pools in the bitcoin system. In order to obtain the higher revenue, the rational mining pool usually chooses an infiltration attack, that is, the pool will falls into the mining dilemma of the PoW consensus algorithm. Thus the article proposes to apply zero-determinant strategies for optimizing the behavior selection of the mining pool under PoW consensus mechanism to increase the total revenues of the system, so as to solve the mining dilemma. After theoretically studying the set and extortionate strategy of zero-determinant, the article devises an adaptive zero-determinant strategy that the pool can change the corporation probability of the next round based on its previous revenues. To verify the effectiveness of zero-determinant strategies, based on the actual revenue of the mining pool defined and deduced in the paper, it simulates 30 sets of game strategies to illustrate the revenue variation of the mining pools. The simulation results show that the three zero-determinant strategies can effectively improve the convergence rate of cooperation, mitigate block withholding attack and maximize the total revenues of the system. Compared with the set and extortionate strategy, the adaptive strategy can ensure more stability and more revenue.

Adapting paths against zero-determinant strategies in repeated prisoner's dilemma games

Long-term cooperation, competition, or exploitation among individuals can be modeled through repeated games. In repeated games, Press and Dyson discovered zero-determinant (ZD) strategies that enforce a special relationship between two players. This special relationship implies that a ZD player can unilaterally impose a linear payoff relationship to the opponent regardless of the opponent's strategies. A ZD player also has a property that can lead the opponent to an unconditional cooperation if the opponent tries to improve its payoff. This property has been mathematically confirmed by Chen and Zinger. Humans often underestimate a payoff obtained in the future. However, such discounting was not considered in their analysis. Here, we mathematically explored whether a ZD player can lead the opponent to an unconditional cooperation even if a discount factor is incorporated. Specifically, we represented the expected payoff with a discount factor as the form of determinants and calculated whether the values obtained by partially differentiating each factor in the strategy vector become positive. As a result, we proved that the strategy vector ends up as an unconditional cooperation even when starting from any initial strategy. This result was confirmed through numerical calculations. We extended the applicability of ZD strategies to real world problems.

Direct reciprocity between individuals that use different strategy spaces

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.

Misperception influence on zero-determinant strategies in iterated Prisoner's Dilemma

Zero-determinant (ZD) strategies have attracted wide attention in Iterated Prisoner's Dilemma (IPD) games, since the player equipped with ZD strategies can unilaterally enforce the two players' expected utilities subjected to a linear relation. On the other hand, uncertainties, which may be caused by misperception, occur in IPD inevitably in practical circumstances. To better understand the situation, we consider the influence of misperception on ZD strategies in IPD, where the two players, player X and player Y, have different cognitions, but player X detects the misperception and it is believed to make ZD strategies by player Y. We provide a necessary and sufficient condition for the ZD strategies in IPD with misperception, where there is also a linear relationship between players' utilities in player X's cognition. Then we explore bounds of players' expected utility deviation from a linear relationship in player X's cognition with also improving its own utility.

Cooperation in alternating interactions with memory constraints

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 different strategies to maintain cooperation in noisy environments. Moreover, none of the respective strategies are evolutionarily stable.

In many instances of reciprocity, individuals cooperate in turns. Here, the authors analyze the equilibria and the dynamics of such alternating games, and in particular describe all strategies with one-round memory that maintain cooperation.

A unified framework of direct and indirect reciprocity

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. Our results draw unexpected connections between direct and indirect reciprocity while highlighting important differences regarding their evolvability.

Conditions for the existence of zero-determinant strategies under observation errors in repeated games

Repeated games are useful models to analyze long term interactions of living species and complex social phenomena. Zero-determinant (ZD) strategies in repeated games discovered by Press and Dyson in 2012 enforce a linear payoff relationship between a focal player and the opponent. This linear relationship can be set arbitrarily by a ZD player. Hence, a subclass of ZD strategies can fix the opponent's expected payoff and another subclass of the strategies can exceed the opponent for the expected payoff. Since this discovery, theories for ZD strategies are extended to cope with various natural situations. It is especially important to consider the theory of ZD strategies for repeated games with a discount factor and observation errors because it allows the theory to be applicable in the real world. Recent studies revealed their existence of ZD strategies even in repeated games with both factors. However, the conditions for the existence has not been sufficiently analyzed. Here, we mathematically analyzed the conditions in repeated games with both factors. First, we derived the thresholds of a discount factor and observation errors which ensure the existence of Equalizer and positively correlated ZD (pcZD) strategies, which are well-known subclasses of ZD strategies. We found that ZD strategies exist only when a discount factor remains high as the error rates increase. Next, we derived the conditions for the expected payoff of the opponent enforced by Equalizer as well as the conditions for the slope and base line payoff of linear lines enforced by pcZD. As a result, we found that, as error rates increase or a discount factor decreases, the conditions for the linear line that Equalizer or pcZD can enforce become strict.

Memory-two zero-determinant strategies in repeated games

Repeated games have provided an explanation of how mutual cooperation can be achieved even if defection is more favourable in a one-shot game in the Prisoner's Dilemma situation. Recently found zero-determinant (ZD) strategies have substantially been investigated in evolutionary game theory. The original memory-one ZD strategies unilaterally enforce linear relationships between average pay-offs of players. Here, we extend the concept of ZD strategies to memory-two strategies in repeated games. Memory-two ZD strategies unilaterally enforce linear relationships between correlation functions of pay-offs and pay-offs of the previous round. Examples of memory-two ZD strategy in the repeated Prisoner's Dilemma game are provided, some of which generalize the tit-for-tat strategy to a memory-two case. Extension of ZD strategies to memory-n case with n ≥ ~2 is also straightforward.

Zero-determinant strategies under observation errors in repeated games

Zero-determinant (ZD) strategies are a novel class of strategies in the repeated prisoner's dilemma (RPD) game discovered by Press and Dyson. This strategy set enforces a linear payoff relationship between a focal player and the opponent regardless of the opponent's strategy. In the RPD game, games with discounting and observation errors represent an important generalization, because they are better able to capture real life interactions which are often noisy. However, they have not been considered in the original discovery of ZD strategies. In some preceding studies, each of them has been considered independently. Here, we analytically study the strategies that enforce linear payoff relationships in the RPD game considering both a discount factor and observation errors. As a result, we first reveal that the payoffs of two players can be represented by the form of determinants as shown by Press and Dyson even with the two factors. Then, we search for all possible strategies that enforce linear payoff relationships and find that both ZD strategies and unconditional strategies are the only strategy sets to satisfy the condition. We also show that neither Extortion nor Generous strategies, which are subsets of ZD strategies, exist when there are errors. Finally, we numerically derive the threshold values above which the subsets of ZD strategies exist. These results contribute to a deep understanding of ZD strategies in society.

Linear algebraic structure of zero-determinant strategies in repeated games

Zero-determinant (ZD) strategies, a recently found novel class of strategies in repeated games, has attracted much attention in evolutionary game theory. A ZD strategy unilaterally enforces a linear relation between average payoffs of players. Although existence and evolutional stability of ZD strategies have been studied in simple games, their mathematical properties have not been well-known yet. For example, what happens when more than one players employ ZD strategies have not been clarified. In this paper, we provide a general framework for investigating situations where more than one players employ ZD strategies in terms of linear algebra. First, we theoretically prove that a set of linear relations of average payoffs enforced by ZD strategies always has solutions, which implies that incompatible linear relations are impossible. Second, we prove that linear payoff relations are independent of each other under some conditions. These results hold for general games with public monitoring including perfect-monitoring games. Furthermore, we provide a simple example of a two-player game in which one player can simultaneously enforce two linear relations, that is, simultaneously control her and her opponent's average payoffs. All of these results elucidate general mathematical properties of ZD strategies.

Synergistic third-party rewarding and punishment in the public goods game

We study the evolution of cooperation in the spatial public goods game in the presence of third-party rewarding and punishment. The third party executes public intervention, punishing groups where cooperation is weak and rewarding groups where cooperation is strong. We consider four different scenarios to determine what works best for cooperation, in particular, neither rewarding nor punishment, only rewarding, only punishment or both rewarding and punishment. We observe strong synergistic effects when rewarding and punishment are simultaneously applied, which are absent if neither of the two incentives or just each individual incentive is applied by the third party. We find that public cooperation can be sustained at comparatively low third-party costs under adverse conditions, which is impossible if just positive or negative incentives are applied. We also examine the impact of defection tolerance and application frequency, showing that the higher the tolerance and the frequency of rewarding and punishment, the more cooperation thrives. Phase diagrams and characteristic spatial distributions of strategies are presented to corroborate these results, which will hopefully prove useful for more efficient public policies in support of cooperation in social dilemmas.

Single equalizer strategy with no information transfer for conflict escalation

In an iterated two-person game, for instance prisoner's dilemma or the snowdrift game, there exist strategies that force the payoffs of the opponents to be equal. These equalizer strategies form a subset of the more general zero-determinant strategies that unilaterally set the payoff of an opponent. A challenge in the attempts to understand the role of these strategies in the evolution of animal behavior is the lack of iterations in the fights for mating opportunities or territory control. We show that an arbitrary two-parameter strategy may possess a corresponding equalizer strategy which produces the same result: statistics of the fight outcomes in the contests with competitors are the same for each of these two strategies. Therefore, analyzing only the equalizer strategy space may be sufficient to predict animal behavior if nature, indeed, reduces (marginalizes) complex strategies to equalizer strategy space. The work's main finding is that there is a unique equalizer strategy that predicts fight outcomes without symmetric cooperation responses. The lack of symmetric cooperation responses is a common trait in conflict escalation contests that generally require a clear winner. In addition, this unique strategy does not assess information of the opponent's state. The method bypasses the standard analysis of evolutionary stability. The results fit well the observations of combat between male bowl and doily spiders and support an empirical assumption of the war of attrition model that the species use only information regarding their own state during conflict escalation.

Divide-and-conquer Tournament on Social Networks

In social gaming networks, previous studies extensively investigated the influence of a variety of strategies on reciprocal behaviors in the prisoner's dilemma game. The studied frameworks range from the case that an individual uniformly cooperates or defects with all social contacts, to the recently reported divide-and-conquer games, where an individual can choose a particular move to play with each neighbor. In this paper, we investigate a divide-and-conquer tournament among 14 well-known strategies on social gaming networks. In the tournament, an individual's fitness is measured by accumulated and average payoff aggregated for a certain number of rounds. On the base of their fitness, the evolution of the population follows a local learning mechanism. Our observation indicates that the distribution of individuals adopting a strategy in degree ranking fundamentally changes the frequency of the strategy. In the divide-and-conquer gaming networks, our result suggests that the connectivity in social networks and strategy are two key factors that govern the evolution of the population.

Asymmetric Power Boosts Extortion in an Economic Experiment

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-extortionate players under the same conditions indicates a substantial net gain to extortion. Our results thus highlight how power asymmetries can endanger mutually beneficial interactions, and transform them into exploitative relationships. In particular, our results indicate that the extortionate strategies predicted from ZD theory could play a more prominent role in our daily interactions than previously thought.

Extortion can outperform generosity in the iterated prisoner's dilemma

Zero-determinant (ZD) strategies, as discovered by Press and Dyson, can enforce a linear relationship between a pair of players' scores in the iterated prisoner's dilemma. Particularly, the extortionate ZD strategies can enforce and exploit cooperation, providing a player with a score advantage, and consequently higher scores than those from either mutual cooperation or generous ZD strategies. In laboratory experiments in which human subjects were paired with computer co-players, we demonstrate that both the generous and the extortionate ZD strategies indeed enforce a unilateral control of the reward. When the experimental setting is sufficiently long and the computerized nature of the opponent is known to human subjects, the extortionate strategy outperforms the generous strategy. Human subjects' cooperation rates when playing against extortionate and generous ZD strategies are similar after learning has occurred. More than half of extortionate strategists finally obtain an average score higher than that from mutual cooperation.

The zero-determinant (ZD) strategies discovered by Press and Dyson overturned several decades of consensus about the iterated prisoner's dilemma. Here, the authors provide the first empirical evidence in support of Press and Dyson's theory, by showing that knowledge of the opponent and the length of the interaction can facilitate the Generous and Extortionate ZD strategies as predicted.

Unfavorable Individuals in Social Gaming Networks

In social gaming networks, the current research focus has been on the origin of widespread reciprocal behaviors when individuals play non-cooperative games. In this paper, we investigate the topological properties of unfavorable individuals in evolutionary games. The unfavorable individuals are defined as the individuals gaining the lowest average payoff in a round of game. Since the average payoff is normally considered as a measure of fitness, the unfavorable individuals are very likely to be eliminated or change their strategy updating rules from a Darwinian perspective. Considering that humans can hardly adopt a unified strategy to play with their neighbors, we propose a divide-and-conquer game model, where individuals can interact with their neighbors in the network with appropriate strategies. We test and compare a series of highly rational strategy updating rules. In the tested scenarios, our analytical and simulation results surprisingly reveal that the less-connected individuals in degree-heterogeneous networks are more likely to become the unfavorable individuals. Our finding suggests that the connectivity of individuals as a social capital fundamentally changes the gaming environment. Our model, therefore, provides a theoretical framework for further understanding the social gaming networks.

Zero-Determinant Strategies in Iterated Public Goods Game

Recently, Press and Dyson have proposed a new class of probabilistic and conditional strategies for the two-player iterated Prisoner’s Dilemma, so-called zero-determinant strategies. A player adopting zero-determinant strategies is able to pin the expected payoff of the opponents or to enforce a linear relationship between his own payoff and the opponents’ payoff, in a unilateral way. This paper considers zero-determinant strategies in the iterated public goods game, a representative multi-player game where in each round each player will choose whether or not to put his tokens into a public pot, and the tokens in this pot are multiplied by a factor larger than one and then evenly divided among all players. The analytical and numerical results exhibit a similar yet different scenario to the case of two-player games: (i) with small number of players or a small multiplication factor, a player is able to unilaterally pin the expected total payoff of all other players; (ii) a player is able to set the ratio between his payoff and the total payoff of all other players, but this ratio is limited by an upper bound if the multiplication factor exceeds a threshold that depends on the number of players.