Spatial Evolutionary Games Research Articles

Cancer heterogeneity and multilayer spatial evolutionary games.

BackgroundEvolutionary game theory (EGT) has been widely used to simulate tumour processes. In almost all studies on EGT models analysis is limited to two or three phenotypes. Our model contains four main phenotypes. Moreover, in a standard approach only heterogeneity of populations is studied, while cancer cells remain homogeneous. A multilayer approach proposed in this paper enables to study heterogeneity of single cells.MethodIn the extended model presented in this paper we consider four strategies (phenotypes) that can arise by mutations. We propose multilayer spatial evolutionary games (MSEG) played on multiple 2D lattices corresponding to the possible phenotypes. It enables simulation and investigation of heterogeneity on the player-level in addition to the population-level. Moreover, it allows to model interactions between arbitrary many phenotypes resulting from the mixture of basic traits.ResultsDifferent equilibrium points and scenarios (monomorphic and polymorphic populations) have been achieved depending on model parameters and the type of played game. However, there is a possibility of stable quadromorphic population in MSEG games for the same set of parameters like for the mean-field game.ConclusionThe model assumes an existence of four possible phenotypes (strategies) in the population of cells that make up tumour. Various parameters and relations between cells lead to complex analysis of this model and give diverse results. One of them is a possibility of stable coexistence of different tumour cells within the population, representing almost arbitrary mixture of the basic phenotypes.ReviewersThis article was reviewed by Tomasz Lipniacki, Urszula Ledzewicz and Jacek Banasiak.

Critical Behavior of Spatial Evolutionary Game with Altruistic to Spiteful Preferences on Two-Dimensional Lattices**Supported by the National Natural Science Foundation of China under Grant Nos. 11121403 and 11504384

Self-questioning mechanism which is similar to single spin-flip of Ising model in statistical physics is introduced into spatial evolutionary game model. We propose a game model with altruistic to spiteful preferences via weighted sums of own and opponent's payoffs. This game model can be transformed into Ising model with an external field. Both interaction between spins and the external field are determined by the elements of payoff matrix and the preference parameter. In the case of perfect rationality at zero social temperature, this game model has three different phases which are entirely cooperative phase, entirely non-cooperative phase and mixed phase. In the investigations of the game model with Monte Carlo simulation, two paths of payoff and preference parameters are taken. In one path, the system undergoes a discontinuous transition from cooperative phase to non-cooperative phase with the change of preference parameter. In another path, two continuous transitions appear one after another when system changes from cooperative phase to non-cooperative phase with the prefenrence parameter. The critical exponents v, β, and γ of two continuous phase transitions are estimated by the finite-size scaling analysis. Both continuous phase transitions have the same critical exponents and they belong to the same universality class as the two-dimensional Ising model.

Playing with Sybil

Recommender systems have become quite popular recently. However, such systems are vulnerable to several types of attacks that target user ratings. One such attack is the Sybil attack where an entity masquerades as several identities with the intention of diverting user ratings. In this work, we propose classical and spatial evolutionary game theory as possible solutions to the Sybil attack in recommender systems. We model the attack using the two techniques and use replicator dynamics in the classical model to solve for evolutionary stable strategies. Results from both models agree that under conditions that are easily achievable by a system administrator, the probability of an attack strategy drops to zero implying degraded fitness for Sybils that eventually die out.

Reciprocity in spatial evolutionary public goods game on double-layered network.

Spatial evolutionary games have mainly been studied on a single, isolated network. However, in real world systems, many interaction topologies are not isolated but many different types of networks are inter-connected to each other. In this study, we investigate the spatial evolutionary public goods game (SEPGG) on double-layered random networks (DRN). Based on the mean-field type arguments and numerical simulations, we find that SEPGG on DRN shows very rich interesting phenomena, especially, depending on the size of each layer, intra-connectivity, and inter-connected couplings, the network reciprocity of SEPGG on DRN can be drastically enhanced through the inter-connected coupling. Furthermore, SEPGG on DRN can provide a more general framework which includes the evolutionary dynamics on multiplex networks and inter-connected networks at the same time.

Cooperation in spatial evolutionary games with historical payoffs

The most common of strategy adoption in evolutionary games relies on players' payoffs of the last round. While a rational player usually fixes the coming strategy by comprehensively considering certain amount of payoff information within its memory length. Here, we explore several measures of historical payoffs in getting the weighted average payoff. Then, player sets the strategy by comparing the weighted average payoff of neighbour's and itself. We show that, cooperators can resist the invasion by referring to the most payoff information, when strategy and measure coevolve. In contrast, strategy adoption of defectors only relies on the nearest one round. Especially, our results suggest that, excessive attention of past payoffs is not favorable to spread cooperative behaviors.

The role of mixed strategies in spatial evolutionary games

Abstract We study three-strategy evolutionary games on a square lattice when the third strategy is a mixed strategy of the first and second ones. It is shown that the resultant three-strategy game is a potential game as well as its two-strategy version. Evaluating the potential we derive a phase diagram on a two-dimensional plane of rescaled payoff parameters that is valid in the zero noise limit of the logit dynamical rule. The mixed strategy is missing in this phase diagram. The effects of two different dynamical rules are analyzed by Monte Carlo simulations and the results of imitation dynamics indicate the dominance of the mixed strategy within the region of the hawk–dove game where it is an evolutionarily stable strategy. The effects and consequences of the different dynamical rules on the final stationary states and phase transitions are discussed.

Promotion of cooperation induced by discriminators in the spatial multi-player donor–recipient game

Abstract Although the two-player donor–recipient game has been used extensively in studying cooperation in social dilemmas, the scenario in which a donor can simultaneously donate resources to multiple recipients is also common in human societies, economic systems, and social networks. This paper formulates a model of the multi-player donor–recipient game considering a multi-recipient scenario. The promotion of cooperation is also studied by introducing a discriminative cooperation strategy into the game, which donates resources to recipients in proportion to their previous donations with a cost for the collection of information. The evolutionary dynamics of individual strategies are explored in homogeneous and heterogeneous scenarios by leveraging spatial evolutionary game theory. The results show that in a homogeneous scenario, defectors can dominate the network at the equilibrium state only when the cost-to-benefit ratio ( R ) of donated resources is large. In a heterogeneous scenario, three strategies can coexist all the time within the range of R that was studied, and the promotion of cooperation is more effective when the values of R are smaller. Results from a single node evolution and the formation of local patterns of interaction are provided, and it is analytically shown that discriminators can maintain fairness in resource donation and guarantee long-term cooperation when R is not too large.

Development of a population of cancer cells: Observation and modeling by a Mixed Spatial Evolutionary Games approach

Living cells, like whole living organisms during evolution, communicate with their neighbors, interact with the environment, divide, change their phenotypes, and eventually die. The development of specific ways of communication (through signaling molecules and receptors) allows some cellular subpopulations to survive better, to coordinate their physiological status, and during embryonal development to create tissues and organs or in some conditions to become tumors. Populations of cells cultured in vitro interact similarly, also competing for space and nutrients and stimulating each other to better survive or to die. The results of these intercellular interactions of different types seem to be good examples of biological evolutionary games, and have been the subjects of simulations by the methods of evolutionary game theory where individual cells are treated as players. Here we present examples of intercellular contacts in a population of living human cancer HeLa cells cultured in vitro and propose an evolutionary game theory approach to model the development of such populations. We propose a new technique termed Mixed Spatial Evolutionary Games (MSEG) which are played on multiple lattices corresponding to the possible cellular phenotypes which gives the possibility of simulating and investigating the effects of heterogeneity at the cellular level in addition to the population level. Analyses performed with MSEG suggested different ways in which cellular populations develop in the case of cells communicating directly and through factors released to the environment.

Extension of a spatial evolutionary coordination game with neutral options

The multiagent evolutionary games on a lattice are equivalent to a kinetic Ising model if the uniform pair interactions are defined by a two-strategy coordination game and the logit rule controls the strategy updates. Now we extend this model by allowing the players to use additional neutral strategies that provide zero payoffs for both players if one of them selects one of the neutral strategies. In the resulting n-strategy evolutionary games the analytical methods and numerical simulations indicate continuous order-disorder phase transitions when increasing the noise level if n does not exceed a threshold value. For larger n the system exhibits a first order phase transition at a critical noise level decreasing asymptotically as 2/ln(n).

Dynamical Patterns of Coexisting Strategies in a Hybrid Discrete-continuum Spatial Evolutionary Game Model

We present a novel hybrid modelling framework that takes into account two aspects which have been largely neglected in previous models of spatial evolutionary games: random motion and chemotaxis. A stochastic individual-based model is used to describe the player dynamics, whereas the evolution of the chemoattractant is governed by a reaction-diffusion equation. The two models are coupled by deriving individual movement rules via the discretisa- tion of a taxis-diffusion equation which describes the evolution of the local number of players. In this framework, individuals occupying the same position can engage in a two-player game, and are awarded a payoff, in terms of reproductive fitness, according to their strategy. As an example, we let individuals play the Hawk-Dove game. Numerical simulations illustrate how random motion and chemotactic response can bring about self-generated dynamical patterns that create favourable conditions for the coexistence of hawks and doves in situations in which the two strategies cannot coexist otherwise. In this sense, our work offers a new perspective of research on spatial evolutionary games, and provides a general formalism to study the dynamics of spatially-structured populations in biological and social contexts where individual motion is likely to affect natural selection of behavioural traits.

Percolation in spatial evolutionary prisoner's dilemma game on two-dimensional lattices.

We study the spatial evolutionary prisoner's dilemma game with updates of imitation max on triangular, hexagonal, and square lattices. We use the weak prisoner's dilemma game with a single parameter b. Due to the competition between the temptation value b and the coordination number z of the base lattice, a greater variety of percolation properties is expected to occur on the lattice with the larger z. From the numerical analysis, we find six different regimes on the triangular lattice (z=6). Regardless of the initial densities of cooperators and defectors, cooperators always percolate in the steady state in two regimes for small b. In these two regimes, defectors do not percolate. In two regimes for the intermediate value of b, both cooperators and defectors undergo percolation transitions. The defector always percolates in two regimes for large b. On the hexagonal lattice (z=3), there exist two distinctive regimes. For small b, both the cooperators and the defectors undergo percolation transitions while only defectors always percolate for large b. On the square lattice (z=4), there exist three regimes. Combining with the finite-size scaling analyses, we show that all the observed percolation transitions belong to the universality class of the random percolation. We also show how the detailed growth mechanism of cooperator and defector clusters decides each regime.

Topological chaos of the spatial prisoner׳s dilemma game on regular networks

The spatial version of evolutionary prisoner׳s dilemma on infinitely large regular lattice with purely deterministic strategies and no memories among players is investigated in this paper. Based on the statistical inferences, it is pertinent to confirm that the frequency of cooperation for characterizing its macroscopic behaviors is very sensitive to the initial conditions, which is the most practically significant property of chaos. Its intrinsic complexity is then justified on firm ground from the theory of symbolic dynamics; that is, this game is topologically mixing and possesses positive topological entropy on its subsystems. It is demonstrated therefore that its frequency of cooperation could not be adopted by simply averaging over several steps after the game reaches the equilibrium state. Furthermore, the chaotically changing spatial patterns via empirical observations can be defined and justified in view of symbolic dynamics. It is worth mentioning that the procedure proposed in this work is also applicable to other deterministic spatial evolutionary games therein.

Multidimensional extended spatial evolutionary games

The goal of this paper is to study the classical hawk–dove model using mixed spatial evolutionary games (MSEG). In these games, played on a lattice, an additional spatial layer is introduced for dependence on more complex parameters and simulation of changes in the environment. Furthermore, diverse polymorphic equilibrium points dependent on cell reproduction, model parameters, and their simulation are discussed. Our analysis demonstrates the sensitivity properties of MSEGs and possibilities for further development. We discuss applications of MSEGs, particularly algorithms for modelling cell interactions during the development of tumours.

Payoff components and their effects in a spatial three-strategy evolutionary social dilemma.

We study a three-strategy spatial evolutionary prisoner's dilemma game with imitation and logit update rules. Players can follow the always-cooperating, always-defecting or the win-stay-lose-shift (WSLS) strategies and gain their payoff from games with their direct neighbors on a square lattice. The friendliness parameter of the WSLS strategy-characterizing its cooperation probability in the first round-tunes the cyclic component of the game determining whether the game can be characterized by a potential. We measured and calculated the phase diagrams of the system for a wide range of parameters. When the game is a potential game and the logit rule is applied, the theoretically predicted phase diagram agrees very well with the simulation results. Surprisingly, this phase diagram can be accurate even in the nonpotential case if there are only two surviving strategies in the stationary state; this result harmonizes with the fact that all 2×2 games are potential games. For the imitation dynamics, we found that the effects of spatiality combined with the presence of two cooperative strategies are so strong that they suppress even substantial changes in the payoff matrix, thus the phase diagrams are independent of the cyclic component's intensity. At the same time, this type of strategy update mechanism supports the formation of cooperative clusters that results in a cooperative society in a wider parameter range compared to the logit dynamics.

Frontier technologies of trust computing and network security

The increasing complexity of computer systems and communication networks induces tremendous requirements for trust and security. This special issue includes topics on trusted computing, risk and reputation management, network security and survivable computer systems/networks. These issues have evolved into an active and important area of research and development. The past decade has witnessed a proliferation of concurrency and computation systems for practice of highly trust, security and privacy, which has become a key subject in determining future research and development activities in many academic and industrial branches. This special issue aims to present and discuss advances of current research and development in all aspects of trusted computing and network security. In addition, this special issue provides snapshots of contemporary academia work in the field of network trusted computing. We prepared and organized this special issue to record state-of-the-art research, novel development and trends for future insight in this domain. In this special issue, 14 papers have been accepted for publication, which demonstrate novel and original work in this field. A detailed overview of the selected works is given below.

Spatial evolutionary public goods game on complete graph and dense complex networks.

We study the spatial evolutionary public goods game (SEPGG) with voluntary or optional participation on a complete graph (CG) and on dense networks. Based on analyses of the SEPGG rate equation on finite CG, we find that SEPGG has two stable states depending on the value of multiplication factor r, illustrating how the “tragedy of the commons” and “an anomalous state without any active participants” occurs in real-life situations. When r is low (), the state with only loners is stable, and the state with only defectors is stable when r is high (). We also derive the exact scaling relation for r*. All of the results are confirmed by numerical simulation. Furthermore, we find that a cooperator-dominant state emerges when the number of participants or the mean degree, 〈k〉, decreases. We also investigate the scaling dependence of the emergence of cooperation on r and 〈k〉. These results show how “tragedy of the commons” disappears when cooperation between egoistic individuals without any additional socioeconomic punishment increases.

Network topology control strategy based on spatial evolutionary public goods game

It is often the case that rational individuals will adjust their connectivity in commercial or social activities for maximizing their payoffs. In this process, we can observe that individuals always gather around a leader or a competitive individual who can bring them more benefits. Inspired by this, we propose a strategy that impels nodes of network to connect with a specific node that we have specified with the perspective of spatial evolutionary public goods game. Thus a node is specified and given a larger enhancement factor which reflects his advantage over others. Then we employ a payoff-oriented preferential rewire strategy that individual will sever a neighbor who provides him with the lowest benefit and then link others randomly. The results illustrate that this strategy not only ensures the promotion of cooperation but also increases the degree of the specified node. Furthermore, we analyze the effect of two relevant parameters: enhancement factor and rewire frequency. We find that if the control strategy expects to work effectively, these two parameters have to ensure an evolution environment where cooperators can prevail defectors. We also conclude that a relatively low rewire frequency contributes to increasing the degree of the specified node. Meanwhile we attempt to present our interpretations for these phenomena.

Sensitivity of Mixed Spatial Evolutionary Games - Hawk-Dove Study

The main goal of this paper is to study properties of a classical hawk-dove model using Mixed Spatial Evolutionary Games (MSEG). These games are played on a multiple lattices corresponding to the possible phenotypes and give possibility to simulate and investigate heterogeneity on the player-level in addition to the population-level. Furthermore, diverse polymorphic equilibrium points dependent on individuals reproduction, model parameters and their simulation are discussed. The analysis demonstrates the sensitivity properties of MSEGs and the possibility for further development of spatial games.

Fourier decomposition of payoff matrix for symmetric three-strategy games.

In spatial evolutionary games the payoff matrices are used to describe pair interactions among neighboring players located on a lattice. Now we introduce a way how the payoff matrices can be built up as a sum of payoff components reflecting basic symmetries. For the two-strategy games this decomposition reproduces interactions characteristic to the Ising model. For the three-strategy symmetric games the Fourier components can be classified into four types representing games with self-dependent and cross-dependent payoffs, variants of three-strategy coordinations, and the rock-scissors-paper (RSP) game. In the absence of the RSP component the game is a potential game. The resultant potential matrix has been evaluated. The general features of these systems are analyzed when the game is expressed by the linear combinations of these components.

Improving QoS in Cooperative Communication Networks: An Evolutionary Game Approach

In Cooperative Communication Networks (CCN), the wireless terminals use relay-cooperation techniques to improve the quality of communication, which is becoming a new trend in recent wireless network researches. However, the relay process always consumes time or energy, and the wireless terminals usually prefer to act as ”free riders.” In this paper, we model the problem of cooperation dynamics in CCN into a spatial evolutionary game framework, and within this model, we propose a special kind of cooperator, i.e., the punisher, who punishes the defector with fine β at the cost of γ. Simulations show the evolution of CCN is jointly affected by β, γ and system's initial state. These results are explained by average total payoff, transition possibility and evolutionary snapshots. We also explore the influence of population heterogeneity on cooperation rate, and find that the heterogeneity of population distribution can affect the emergence and persistence of cooperation.