Tit For Tat Strategy Research Articles

Fairness Matters

The proof-of-work (PoW) based blockchains are more secure nowadays since profit-oriented miners contribute more computing powers in exchange for fair revenues. This virtuous circle only works under an incentive-compatible consensus, which is found to be fragile under selfish mining attacks. Specifically, selfish miners can conceal and reveal blocks strategically to earn unfairly higher revenue compared to honest behaviors. Previous countermeasures either require incompatible modifications or fail to consider the asynchronous network and multiple honest nodes setting in reality. In this paper, we introduce the unfairness measurement based on the KL-divergence from the computing power distribution to the revenue distribution of miners. To improve fairness with the existence of selfish miners, we propose a novel block promotion strategy namely Tit-for-Tat (TFT), for honest miners. In particular, based on a miner's local observation of forks, we design the suspicious probability measurement of other nodes. Rather than promoting a fresh block instantly, miners withhold it for different time periods according to others' suspicious probability before delivery. Meanwhile, to minimize the attacker's unfair revenue, we formulate the delay vector (DV) problem for honest miners to determine the optimal withholding time. We prove that DV problem is nonconvex, and thus propose two approximation algorithms that yield ϵ-suboptimal solutions. In addition, we extend TFT strategy to support dynamic networks. Extensive experiments validate the efficiency and effectiveness of our strategy and algorithms to reduce unfairness by 54.62% within bounded withholding time.

Probabilistic memory-one strategies to dominate the iterated prisoner’s dilemma over networks

The Iterated Prisoner’s Dilemma (IPD) has been a classical game theoretical scenario used to model behaviour interactions among agents. From the famous Axelrod’s tournament, and the successful results obtained by the Tit for Tat strategy, to the introduction of the zero-determinant strategies in the last decade, the game theory community has been exploring the performance of multiple strategies for years. This article grounds on such previous work, studying probabilistic memory-one strategies (PMO) and using evolutionary game theory, to analyse the criteria to find the most successful set of strategies in networked topologies. The results are nearly deterministic in discrete PMO scenarios. However, results become much more complex when moving to continuous ones, and there is no optimal strategy for a given scenario. Finally, this article describes how, using machine learning and evolutionary techniques; a cluster of agents, playing synchronously and adaptively, is able to dominate the rest of the population.

A Theoretical Model of Sequential Combinatorial Games of Subsidies and Penalties: From Waste to Renewable Energy

Subsidies and penalties are two main regulation methods adopted by authorities to promote the development of renewable energy. Due to the possibility of subsidy fraud, it is necessary to explore effective ways to combine these two policies. In this article, subsidy and penalty policies are incorporated into a sequential game theory model to explore the impact of different regulatory mechanisms on the promotion of renewable energy from recycled resources. We take biodiesel production from used cooking oil (UCO) as an example. UCO can be converted into environmentally friendly biodiesel or mixed with fresh cooking oil, resulting in inferior cooking oil containing harmful carcinogens but with huge profits. There are two mechanisms in the sequential combination model, spot checks after subsidy and subsidy after spot checks. Under both cases, fines are imposed if fraud is found during spot checks. The amounts of subsidies and fines also need to be determined. We show that the effects of subsidies depend on the implementation of the timing. The ex-ante subsidies have no effect. When spot checks are performed first, the larger subsidies will increase the probability of producing inferior cooking oil due to lower probability of spot checks. While combined with penalties, the ex-post subsidies have a positive effect on biodiesel production, that is, there exists synergy effect of penalty and subsidy on renewable energy production. In an infinitely repeated game, the shutdown threat of a grim trigger strategy (GTS) is much easier to induce biodiesel production than the penalty threat of a tit-for-tat strategy (TFT). When penalties are large enough, TFT can achieve the same goal of legal production effectively as GTS. The sooner illegal production is observed, the lower penalties are required to induce the processor to produce legally. Compared to subsidies, penalties are more effective in encouraging processors to produce renewable energy rather than illegal products. Moreover, our simulation results suggest that higher fines or profits from legal production are more likely to stimulate renewable energy production than subsidies. Our findings enrich our knowledge of the link between government regulations and the promotion of renewable energy.

A Secure Trust Method for Multi-Agent System in Smart Grids Using Blockchain

This paper proposes a blockchain based trust management method for agents in a multi-agent system (MAS). In this work, three objectives are achieved: trust, cooperation and privacy. The trust of agents depends on the credibility of trust evaluators, which is verified using the proposed methods of trust distortion, consistency and reliability. To enhance the cooperation between agents, a tit-3-for-tat (T3FT) repeated game strategy is developed. The strategy is more forgiving than the existing tit-for-tat (TFT) strategy. It encourages cheating agents to re-establish their trust by cooperating for three consecutive rounds of play. Also, a proof-of-cooperation consensus protocol is proposed to improve agents' cooperation while creating and validating blocks. The privacy of agents is preserved in this work using the publicly verifiable secret sharing mechanism. The proposed methods are implemented using MATLAB R2018a while the MAS is simulated using Java Agent DEvelopment framework (JADE). Simulation results validate the effectiveness of the proposed work. From the simulation results, the proposed trust method outperforms an existing fuzzy logic trust method in terms of detecting the cheating behavior of agents in the system. Besides, the proposed T3FT strategy is effective as compared to the existing tit-for-2-tat and TFT strategies in the literature. Moreover, the security analysis of the proposed method is performed. The analysis shows that the proposed work is safe from bad-mouthing and on-off trust related attacks.

Analysis of optimal piece flow in tit-for-tat-based P2P streaming

BitTorrent, which is one of the successful Peer-to-Peer (P2P) file distribution systems, adopts the tit-for-tat (TFT) strategy in game theory to encourage cooperation among peers, i.e., each peer has to provide fragments of the original file, called pieces, to others so as to retrieve its demanding pieces from them. Because the TFT strategy can restrict free riding behavior of peers, there are also several TFT-based P2P streaming systems and the performance of such existing systems has been analyzed. However, optimal piece flow in TFT-based P2P streaming has not been revealed yet. In this paper, a discrete-time model of TFT-based P2P streaming is first developed and integer linear programming (ILP) is formulated to determine the optimal piece flow where the average play-out delay is minimized. By solving the ILP using existing solver, i.e., CPLEX, we can obtain numerical examples of optimal piece flow. The analysis of obtained optimal piece flow reveals that (1) optimal piece selection is based on the balance between in-order piece retrieving and the rarest-first piece retrieving, (2) optimal peer selection depends on the upload capacities of peers and the stage of streaming, (3) the number of pieces does not affect the system performance, (4) the maximum play-out delay can be bounded by the ratio of the number of peers to the server’s upload capacity, and (5) how the relaxation of TFT constraint can improve the system performance.

An evolutionary approach to low-level conversational cooperation

The cooperative character of language is an empirical fact and one of the key tenets in linguistics. However, this cooperative character is what makes it evolutionarily suspect: under normal circumstances sharing honest information with biologically unrelated individuals and without any obvious costs is not an ESS (evolutionarily stable strategy) and is not expected to evolve. Here we approach this problem in the context of prototypical language use (also evolutionarily), that is face-to-face conversation, a multimodal interaction unfolding in real time that involves a complex interplay between embodied agents in spatial proximity. We focus our review on low-level coordinative processes – examples being proxemic alignment or postural mirroring – to a large degree continuous with what is found in non-human primates. We aim at categorisation and terminological clarification of these processes, which in turn helps us evaluate their role in initiating and maintaining cooperatively oriented communicative interaction. We conclude with suggesting ways in which such low-level coordinative processes might have formed an evolutionary basis for cooperation, with a focus on the possible role of TFT (tit-for-tat) and TFT-like strategies.

An Empirical Game-Theoretic Analysis of the Dynamics of Cooperation in Small Groups

Many models of the evolution of cooperation have shown the importance of direct reciprocity (for example tit for tat strategies) or alternatively indirect reciprocity (conspicuous altruism based on a reputa- tion or image score). In the latter case many models make the implicit assumption that group sizes are large relativetotheexpectednumberofinteractions,whichmakestheiranalysismoretractableinseveralways,not least by allowing us to ignore any strategic interaction between the direct and indirect classes of reciprocation strategy. However, in smaller groups the possibility arises that both classes of strategy will play a role in de- termining the equilibrium behaviour. Therefore we introduce a replicator dynamics model which incorporates both direct and indirect reciprocity, and use simulation and numerical methods to quantitatively assess how the level of cooperation in equilibrium is aected by changes in the group size and the frequency with which othergroupmembersareencountered. Ouranalysisshowsthat,forintermediategroupsizes,directreciprocity persists in equilibrium alongside indirect reciprocity. In contrast to previous simulation studies, we provide a sound game-theoretic underpinning to our analysis, and examine the precise conditions which give rise to a mix of both forms of reciprocity.

Reciprocity in Human-Robot Interaction: A Quantitative Approach Through the Prisoner’s Dilemma and the Ultimatum Game

Reciprocity is an important factor in human-human interaction, so it can be expected that it should also play a major role in human-robot interaction (HRI). Participants in our study played the Repeated Prisoner's Dilemma Game (RPDG) and the mini Ultimatum Game (mUG) with robot and human agents, with the agents using either Tit for Tat (TfT) or Random strategies. As part of the study we also measured the perceived personality traits in the agents using the TIPI test after every round of RPDG and mUG. The results show that the participants collaborated more with humans than with a robot, however they tended to be equally reciprocal with both agents. The experiment also showed the TfT strategy as the most profitable strategy; affecting collaboration, reciprocation, profit and joint profit in the game. Most of the participants tended to be fairer with the human agent in mUG. Furthermore, robots were perceived as less open and agreeable than humans. Consciousness, extroversion and emotional stability were perceived roughly the same in humans and robots. TfT strategy became associated with an extroverted and agreeable personality in the agents. We could observe that the norm of reciprocity applied in HRI has potential implications for robot design.

Catalysts of Cooperation in System of Systems: The Role of Diversity and Network Structure

Cooperation and competition are critical aspects of complex systems that intrigue researchers and practitioners alike. Recently, there has been great interest in how structure affects these cooperative behaviors and how they evolve. In this paper, we utilize the methods of evolutionary game theory on graphs to develop a model of the structured evolution of adaptive agents. Rather than simple memoryless agents, we employ adaptive agents that take on a diverse set of strategies including the cooperative catalyst tit-for-tat (TFT) and its generous and suspicious variants. We also employ all-defect, unconditional cooperation, and random strategies. Agents use these strategies during their evolution in order to maximize their own relative fitness, measured by their performance in a repeated Prisoner's Dilemma game versus their structural neighbors. We test the model with a variety of regular graph structures with variable degree, randomness, and size. Our parallel-execution agent-based simulations show that each strategy evolves toward its own structural niche with suspicious TFT maintaining the highest survival rate over all structures. We also show that increases in network density make the TFT strategies even more dominant, and connectivity randomness encourages cooperation in sparse networks.

Probability distribution of the number of deceits in collective robotics

The benefit obtained by a selfish robot by cheating in a real multirobotic system can be represented by the random variable Xn,q: the number of cheating interactions needed before all the members in a cooperative team of robots, playing a TIT FOR TAT strategy, recognize the selfish robot. Stability of cooperation depends on the ratio between the benefit obtained by selfish and cooperative robots. In this paper, we establish the probability model for Xn,q. If the values of the parameters n and q are known, then this model allows us to make predictions about the stability of cooperation. Moreover, if these parameters are modifiable, it is possible to tune them to guarantee the viability of cooperation.

Probability distribution of the number of deceits in collective robotics

The benefit obtained by a selfish robot by cheating in a real multirobotic system can be represented by the random variable X n,q : the number of cheating interactions needed before all the members in a cooperative team of robots, playing a TIT FOR TAT strategy, recognize the selfish robot. Stability of cooperation depends on the ratio between the benefit obtained by selfish and cooperative robots. In this paper, we establish the probability model for X n,q . If the values of the parameters n and q are known, then this model allows us to make predictions about the stability of cooperation. Moreover, if these parameters are modifiable, it is possible to tune them to guarantee the viability of cooperation.

Effects of initial choices in the prisoner's dilemma

AbstractIn the two‐person prisoner's dilemma game (PDG), there is considerable evidence that the tit‐for‐tat (TFT) strategy is most effective in inducing the other person to cooperate. One of the important features of the TFT strategy is that it cooperates on the first trial. We varied the initial choices of a simulated other, and tested the effects of (a) initial choices (cooperation or defection), and (b) persistence of initial choices: unilateral cooperation (or defection) on the first trial, first two trials, or first four trials. Results showed that a cooperative strategy—one that starts with cooperative choices—induced greater cooperation than a strategy that started with defections. The results of this study clearly show that (a) a cooperative strategy—one that initiates unilateral cooperation at the outset and then adopts a TFT strategy—is very effective in inducing subsequent cooperation from the other party, (b) the effectiveness of a cooperative strategy varies directly with the cooperative orientation of the other party (a cooperative strategy is more effective against a cooperative than a competitive person), and (c) initial cooperation is more effective if it is repeated more than once. Copyright © 2002 John Wiley & Sons, Ltd.

Certain similarities between the evolution of cooperation and phase transitions in physics

AbstractThe theory of evolution of cooperation presents a number of striking similarities to the theory of phase transitions in physics. A system that is in a higher energy phase P+ will make a transition to a lower energy phase P−, given enough time, as long as the intervening energy barrier is not too high. The phase transition is realized through the nucleation of a sufficiently large bubble of the energetically preferred phase P− within the higher energy phase P+. Similarly, if we think of the strategies of Defection and of Tit for Tat as the phases P+ and P− respectively, a population using the ‘energetically unfavourable’ Defection strategy will switch over to the ‘energetically favourable’ Tit for Tat strategy, provided the players interact continuously and substantially. The presence of a large enough Tit for Tat concentration is necessary for the onset of this change. Copyright © 2001 John Wiley & Sons, Ltd.

ON LOCAL PRISONER'S DILEMMA GAME WITH PARETO UPDATING RULE

Prisoner's Dilemma games with two and three strategies are studied. The corresponding replicator equations, their steady states and their asymptotic stability are discussed. Local Prisoner's Dilemma games are studied using Pareto optimality. As in the case with Nash updating rule, the existence of tit for tat strategy is crucial to imply cooperation even in one dimension. Pareto updating implies less erratic behavior since the steady state configurations are mostly fixed points or at most 2-cycle. Finally, Prisoner's Dilemma game is simulated on small-world networks which are closer to real systems than regular lattices. There are no significant changes compared to the results of the regular lattice.

No Strategy is Evolutionarily Stable in the Repeated Prisoner's Dilemma

Following the influential work of Axelrod, the repeated Prisoner's Dilemma game has become the theoretical gold standard for understanding the evolution of co-operative behavior among unrelated individuals. Using the game, several authors have found that a reciprocal strategy known as Tit for Tat ( TFT) has done quite well in a wide range of environments. TFT strategists start out co-operating and then do what the other player did on the previous move. Despite the success of TFT and similar strategies in experimental studies of the game, Boyd & Lorberbaum (1987, Nature, Lond. 327, 58) have shown that no pure strategy, including TFT, is evolutionarily stable in the sense that each can be invaded by the joint effect of two invading strategies when long-term interaction occurs in the repeated game and future moves are discounted. Farrell & Ware (1989, Theor. Popul. Biol. 36 , 161) have since extended these results to include finite mixes of pure strategies as well. Here, it is proven that no strategy is evolutionarily stable when long-term relationships are maintained in the repeated Prisoner's Dilemma and future moves are discounted. Namely, it is shown each completely probabilistic strategy (i.e. one that both co-operates and defects with positive probability after every sequence of behavior) may be invaded by a single deviant strategy. This completes the proof started by Boyd and Lorberbaum and extended by Farrell and Ware. This paper goes on to prove that no reactive strategy with a memory restricted to the opponent's preceding move is evolutionarily stable when there is no discounting of future moves. This is true despite the success of a more forgiving variant of TFT called GTFT in a recent tournament among reactive strategies conducted by Nowak & Sigmund (1992, Nature 355 , 250) where future moves were not discounted. GTFT, for example, may be invaded by a pair of reactive mutants. Since no strategy is evolutionarily stable when future moves are discounted in the repeated game, the restriction of strategy types to those actually maintained by mutation and phenotypic and environmental variability in natural populations may be the key to understanding the evolution of co-operation. However, the result presented here that the somewhat realistic reactive strategies are also not evolutionarily stable at least in the non-discounted game suggests something else may be going on. For one, the proof that no reactive strategy is evolutionarily stable ironically shows the robustness of TFT -like strategies. Also, a significant invasion of strategies like GTFT (among reactive strategies) and TFT which can only be invaded by the joint effect of two deviant strategies relies on the maintenance of at least one of the mutants by recurrent mutation or other process. This weakens this paper's results somewhat. Further, it may be possible that some completely probabilistic strategies which are close to their pure counterparts may only be invadible by strategies with less "errors" in them: thus, a population may tend toward a pure strategy. Such findings could help explain the persistence of certain strategy types in the repeated PD when long-term interaction is possible.

Reciprocal allogrooming in impala, Aepyceros melampus

Adult female and male impala engage in a type of allogrooming in which partners alternately deliver bouts of oral grooming to the head and neck. These grooming encounters comprise typically six to 12 tit-for-tat-like exchanges of bouts and are highly reciprocal among adult females, adult bachelor males and subadult males. Although allogrooming among females could be between related individuals, that occurring among adult males would appear to be between unrelated individuals. Unlike allogrooming reported for some primate species and other ungulates, the dominant impala received no more grooming than the subordinate. It is proposed that one function of impala allogrooming is to reduce the ectoparasite load on body areas an animal cannot reach with its own mouth. The impala reciprocal allogrooming system is unique among free-ranging antelope and other ungulates and may be a candidate for the tit-fortat strategy of evolved cooperation.

Brand-Name Investment of Candidates and District Homogeneity: An Ordinal Response Model

Advertising is often viewed as a sunk investment that will receive a return only if firms do not cheat their customers on quality, much like a security bond that will be lost for non-performance of a contract. In the last period a firm will cheat if there is no future return from the investment [15]. Knowing that a firm is in its last period will then reduce sales in the last period to zero, so the firms will begin to cheat the period before the last. This is the usual iterated prisoner's dilemma (PD) game with a finite life; cheating begins in the first period and continues. Demsetz [10] argues that since firms are transferable through voluntary trade, the possibly infinite life of a firm transforms this finite, iterated PD game with a known end into an iterated PD game with no known end. As Axelrod [1] suggests, the best of the known strategies for this game is the titfor-tat strategy. In this game, firms live up to quality claims if customers continue to pay for their product. Demsetz also argues that brand name cannot serve as a barrier to entry for alienable firms because the firm, with its brand name intact, can be purchased by buyers who are more efficient producers than the current producer. Since professionals and politicians who have made investments in brand name have namebrand capital that is inalienable, these markets may suffer from the last-period problem [12; 17; 18; 19]. If the seller knows the last period because he intends to quit, cheating cannot be prevented

Dynamics of the TIT FOR TAT strategy during predator inspection in the guppy (Poecilia reticulata)

One well-known solution to the iterated Prisoner's Dilemma is the TIT FOR TAT strategy. This strategy has three “characteristics” associated with it. TIT FOR TAT is nice (cooperates on the first move of a game), retaliatory (plays defect against an individual that defected on the prior move), and forgiving (cooperates with an individual which has defected in the past but cooperates in the present). Predator inspection behavior in guppies (Poecilia reticulata) was examined in order to determine whether guppies displayed these three characteristics. Results indicate that while it can be quite difficult to translate the abstract concepts of niceness, retaliation, and forgiveness into measurable behaviors, the data support the hypothesis that guppies display the three characteristics associated with the TIT FOR TAT strategy. The “selfish herd” and “dilution effect” are discussed as possible alternative explanations for the observed behavior. These hypotheses alone were insufficient to explain the results.

Reciprocity and Cooperation in Social Dilemmas

In computer simulation studies of the prisoner's dilemma conducted by Axelrod (1984), the tit-for-tat (TFT) strategy was found to be most effective. Two important properties of the TFT strategy are based on the reciprocity norm: (1) It is “provocable” and immediately retaliates if the other person defects; and (2) it is “forgiving” and immedicately reciprocates cooperation if the other returns to cooperation after defection. Delay of reciprocity for the two types of TFT properties were varied: immediate versus one trial delay. In two experiments, a single naive subject (undergraduate male student) was led to believe that he was playing a two-person prisoner's dilemma, but he actually played against a programmed strategy. The results of both experiments yielded significant effects for delay of forgiveness: Mean proportion of cooperative choices was greater when cooperative overtures were reciprocated immediately than when they were delayed. Two other properties of TFT, “niceness” and clarity, were also assessed. The results are interpreted in terms of operant conditioning principles and in terms of Osgood's GRIT strategy.

Tit-For-Tat in guppies (Poecilia reticulata): the relative nature of cooperation and defection during predator inspection

The introduction of game-theoretical thinking into evolutionary biology has laid the groundwork for a heuristic view of animal behaviour in which individuals employ ‘strategies’ — rules that instruct them how to behave in a given circumstance to maximize relative fitness. Axelrod and Hamilton (1981) found that a strategy called Tit-For-Tat (TFT) is one robust cooperative solution to the iterated Prisoner's Dilemma game. There exists, however, little empirical evidence that animals employ TFT. Predator inspection in fish provides one ecological context in which to examine the use of the TFT strategy. Here we describe a study in which guppies were tested in multiple predator inspection trials. An individual was tested with its mirror images as well as a series of live conspecifics. Results indicate that guppies are capable of recognizing and remembering their partner's behaviour and seem to employ TFT-like strategies over the course of many inspection visits. In addition, significant differences exist between individuals in the degree to which they will inspect a predator, suggesting that ‘cooperator’ and ‘defector’ may be relative terms rather than discrete categories of behaviour.