A formula is presented for designing zero-determinant(ZD) strategies of general finite games, which have $n$ players and players can have different numbers of strategies. To this end, using semi-tensor product (STP) of matrices, the profile evolutionary equation for repeated finite games is obtained. Starting from it, the ZD strategies are developed for general finite games, based on the same technique proposed by Press and Dyson \cite{pre12}. A formula is obtain to design ZD strategies for any player directly, ignoring the original ZD design process. Necessary and sufficient condition is obtained to ensure the effectiveness of the designed ZD strategies. As a consequence, it is also clear that player $i$ is able to unilaterally design $|S_i|-1$ dominating linear relations about the expected payoffs of all players. Finally, the fictitious opponent player is proposed for networked evolutionary networks (NEGs). Using it, the ZD-strategies are applied to NEGs. It is surprising that an individual in a network may use ZD strategies to conflict the whole rest network.
Repeated games have provided an explanation how mutual cooperation can be achieved even if defection is more favorable in a one-shot game in prisoner's dilemma situation. Recently found zero-determinant strategies have substantially been investigated in evolutionary game theory. The original memory-one zero-determinant strategies unilaterally enforce linear relations between average payoffs of players. Here, we extend the concept of zero-determinant strategies to memory-two strategies in repeated games. Memory-two zero-determinant strategies unilaterally enforce linear relations between correl…
Zero-determinant strategies are a class of memory-one strategies in repeated games which unilaterally enforce linear relationships between payoffs. It has long been unclear for what stage games zero-determinant strategies exist. We provide a necessary and sufficient condition for the existence of zero-determinant strategies. This condition can be interpreted as the existence of two different actions which unilaterally adjust the total value of a linear combination of payoffs. A relation between the class of stage games where zero-determinant strategies exist and other class of stage games is a…
Positive and Negative Determinant Strategies in Repeated Games with Behavior-Value Inconsistency
School of Mathematical Sciences, Beijing University of Posts and Telecommunications（北京邮电大学数学科学学院）；Key Laboratory of Mathematics and Information Networks (Beijing University of Posts and Telecommunications), Ministry of Education（教育部数学与信息网络重点实验室（北京邮电大学））；Department of Theoretical Biology, Max Planck Institute for Evolutionary Biology（马克斯·普朗克进化生物学研究所理论生物学系）；Department of Engineering, Universitat Pompeu Fabra（庞培法布拉大学工程学院）
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 an…
In two-player repeated games, Zero-Determinant (ZD) strategies enable a player to unilaterally enforce a linear payoff relation between her own and her opponent's payoff irrespective of the opponent's strategy. This manipulative nature of the ZD strategies attracted significant attention from researchers due to its close connection to controlling distributively the outcome of evolutionary games in large populations. In this paper, necessary and sufficient conditions are derived for a payoff relation to be enforceable in multiplayer social dilemmas with a finite expected number of rounds that i…
Zero-determinant strategies are memory-one strategies in repeated games which unilaterally enforce linear relations between expected payoffs of players. Recently, the concept of zero-determinant strategies was extended to the class of memory-$n$ strategies with $n\geq 1$, which enables more complicated control of payoffs by one player. However, what we can do by memory-$n$ zero-determinant strategies is still not clear. Here, we show that memory-$n$ zero-determinant strategies in repeated games can be used to control conditional expectations of payoffs. Equivalently, they can be used to contro…
Since the introduction of zero-determinant strategies, extortionate strategies have received considerable interest. While an interesting class of strategies, the definitions of extortionate strategies are algebraically rigid, apply only to memory-one strategies, and require complete knowledge of a strategy (memory-one cooperation probabilities). We describe a method to detect extortionate behaviour from the history of play of a strategy. When applied to a corpus of 204 strategies this method detects extortionate behaviour in well-known extortionate strategies as well others that do not fit the…
This paper studies the mechanisms, implications, and potential applications of the recently discovered class of Zero Determinant (ZD) strategies in iterated 2x2 games. These strategies were reported to successfully extort pure economic maximizers, and to mischievously determine the set of feasible long-term payoffs in iterated Prisoners' Dilemma by enforcing linear constraints on both players' expected average scores. These results are generalized for all symmetric 2x2 games and a general Battle of the Sexes, exemplified by four common games. Additionally, a comparison to conventional strategi…
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 evolutionary game where in each round each player will choose whether or not put his tokens into a…
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 playe…
We consider private commons for secondary sharing of licensed spectrum bands with no access coordination provided by the primary license holder. In such environments, heterogeneity in demand patterns of the secondary users can lead to constant changes in the interference levels, and thus can be a source of volatility to the utilities of the users. In this paper, we consider secondary users to be service providers that provide downlink services. We formulate the spectrum sharing problem as a non-cooperative iterated game of power control where service providers change their power levels to fix …
In a single-state repeated game, zero-determinant strategies can unilaterally force functions of the payoffs to take values in particular closed intervals. When the explicit use of a determinant is absent from the analysis, they are instead called autocratic. We extend their study to the setting of finite state games with deterministic transitions. For a given game we find that the endpoints of the intervals of enforceable values must satisfy fixed point equations. From these extreme enforceable values we show it is always possible to construct finite memory strategies to enforce a particular …
Repeated game theory has been one of the most prevailing tools for understanding the long-run relationships, which are footstones in building human society. Recent works have revealed a new set of "zero-determinant (ZD)" strategies, which is an important advance in repeated games. A ZD strategy player can exert a unilaterally control on two players' payoffs. In particular he can deterministically set the opponent's payoff, or enforce an unfair linear relationship between the players' payoffs, thereby always seizing an advantageous share of payoffs. One of the limitations of the original ZD str…
Evolutionary game theory is used to model the evolution of competing strategies in a population of players. Evolutionary stability of a strategy is a dynamic equilibrium, in which any competing mutated strategy would be wiped out from a population. If a strategy is weak evolutionarily stable, the competing strategy may manage to survive within the network. Understanding the network-related factors that affect the evolutionary stability of a strategy would be critical in making accurate predictions about the behaviour of a strategy in a real-world strategic decision making environment. In this …
Social dilemmas exist in various fields and give rise to the so-called free-riding problem, leading to collective fiascos. The difficulty of tracking individual behaviors makes egoistic incentives in large-scale systems a challenging task. However, the state-of-the-art mechanisms are either individual-based or state-dependent, resulting in low efficiency in large-scale networks. In this paper, we propose an egoistic incentive mechanism from a connected (network) perspective rather than an isolated (individual) perspective by taking advantage of the social nature of people. We make use of a zer…
The growing threat of personal data breach in data trading pinpoints an urgent need to develop countermeasures for preserving individual privacy. The state-of-the-art work either endows the data collector with the responsibility of data privacy or reports only a privacy-preserving version of the data. The basic assumption of the former approach that the data collector is trustworthy does not always hold true in reality, whereas the latter approach reduces the value of data. In this paper, we investigate the privacy leakage issue from the root source. Specifically, we take a fresh look to rever…
In a stochastic Stackelberg asymmetric security game, the strong Stackelberg equilibrium (SSE) strategy is a popular option for the defender to get the highest utility against an attacker with the best response (BR) strategy. However, the attacker may be a boundedly rational player, who adopts a combination of the BR strategy and a fixed stubborn one. In such a condition, the SSE strategy may not maintain the defensive performance due to the stubborn element. In this paper, we focus on how the defender can adopt the unilateral-control zero-determinate (ZD) strategy to confront the boundedly ra…
Two-player games have had a long and fruitful history of applications stretching across the social, biological, and physical sciences. Most applications of two-player games assume synchronous decisions or moves even when the games are iterated. But different strategies may emerge as preferred when the decisions or moves are sequential, or the games are iterated. Zero-determinant strategies developed by Press and Dyson are a new class of strategies that have been developed for synchronous two-player games, most notably the iterated prisoner's dilemma. Here we apply the Press-Dyson analysis to s…
The problem of efficient sharing of a resource is nearly ubiquitous. Except for pure public goods, each agent's use creates a negative externality; often the negative externality is so strong that efficient sharing is impossible in the short run. We show that, paradoxically, the impossibility of efficient sharing in the short run enhances the possibility of efficient sharing in the long run, even if outcomes depend stochastically on actions, monitoring is limited and users are not patient. We base our analysis on the familiar framework of repeated games with imperfect public monitoring, but we…
University of Milan（米兰大学）；Politecnico di Milano（米兰理工大学）
Security Games employ game theoretical tools to derive resource allocation strategies in security domains. Recent works considered the presence of alarm systems, even suffering various forms of uncertainty, and showed that disregarding alarm signals may lead to arbitrarily bad strategies. The central problem with an alarm system, unexplored in other Security Games, is finding the best strategy to respond to alarm signals for each mobile defensive resource. The literature provides results for the basic single-resource case, showing that even in that case the problem is computationally hard. In …
