Modelling molecular interactions with game networks' theory

We present a method to model biological systems, the theory of games networks. It extends game theory by multiplying the number of games, and by allowing agents to play several games simultaneously. Some important notions of biological systems, such as locality of interactions and modularity, can then be modelled.     

Games network and application to PAs system

In this article, we present a game theory based framework, named games network, for modeling biological interactions. After introducing the theory, we more precisely describe the methodology to model biological interactions. Then we apply it to the plasminogen activator system (PAs) which is a signal transduction pathway involved in cancer cell migration. The games network theory extends game theory by including the locality of interactions. Each game in a games network represents local interactions between biological agents. The PAs system is implicated in cytoskeleton modifications via regulation of actin and microtubules, which in turn favors cell migration. The games network model has enabled us a better understanding of the regulation involved in the PAs system.     

Evolutionarily stable sets in symmetric extensive two-person games.

Evolutionarily stable (ES) sets are characterized for evolutionary games in extensive form. It is shown that, for the normal form of games involving informational symmetries or repeated play, the standard approach of determining evolutionarily stable strategies (ESSs) often fails to predict the evolutionary outcome. The dynamic stability of ES sets is proved in both the pure strategy and mixed strategy models. ES sets are shown to also generalize the notion of direct ESSs (an earlier attempt to apply ESS theory to extensive games). The theory is illustrated by three examples of biological games in extensive form.     

Modéliser les interactions moléculaires par la théorie des réseaux de jeux

We present a method to model biological systems, the theory of games networks. It extends game theory by multiplying the number of games, and by allowing agents to play several games simultaneously. Some important notions of biological systems, such as locality of interactions and modularity, can then be modelled. To cite this article: M. Manceny et al., C. R. Biologies 329 (2006).     

Exploring the effects of working for endowments on behaviour in standard economic games

In recent years, significant advances have been made in understanding the adaptive (ultimate) and mechanistic (proximate) explanations for the evolution and maintenance of cooperation. Studies of cooperative behaviour in humans invariably use economic games. These games have provided important insights into the mechanisms that maintain economic and social cooperation in our species. However, they usually rely on the division of monetary tokens which are given to participants by the investigator. The extent to which behaviour in such games may reflect behaviour in the real world of biological markets--where money must be earned and behavioural strategies incur real costs and benefits--is unclear. To provide new data on the potential scale of this problem, we investigated whether people behaved differently in two standard economic games (public goods game and dictator game) when they had to earn their monetary endowments through the completion of dull or physically demanding tasks, as compared with simply being given the endowment. The requirement for endowments to be 'earned' through labour did not affect behaviour in the dictator game. However, the requirement to complete a dull task reduced cooperation in the public goods game among the subset of participants who were not familiar with game theory. There has been some effort to test whether the conclusions drawn from standard, token-based cooperation games adequately reflect cooperative behaviour 'in the wild.' However, given the almost total reliance on such games to study cooperation, more exploration of this issue would be welcome. Our data are not unduly worrying, but they do suggest that further exploration is needed if we are to make general inferences about human behaviour from the results of structured economic games.     

Individual differences in plasticity and sampling when playing behavioural games

When engaged in behavioural games, animals can adjust their use of alternative tactics until groups reach stable equilibria. Recent theory on behavioural plasticity in games predicts that individuals should differ in their plasticity or responsiveness and hence in their degree of behavioural adjustment. Moreover, individuals are predicted to be consistent in their plasticity within and across biological contexts. These predictions have yet to be tested empirically and so we examine the behavioural adjustment of individual nutmeg mannikins (Lonchura punctulata), gregarious ground-feeding passerines, when playing two different social foraging games: producer-scrounger (PS) and patch-choice (PC) games. We found: (i) significant individual differences in plasticity and sampling behaviour in each of the two games, (ii) individual differences in sampling behaviour were consistent over different test conditions within a game (PC) and over a six month period (PS), (iii) but neither individual plasticity nor sampling behaviour was correlated from one social foraging game to another. The rate at which birds sampled alternative tactics was positively associated with seed intake in PS trials but negatively associated in PC trials. These results suggest that games with frequency dependence of pay-offs can maintain differences in behavioural plasticity but that an important component of this plasticity is group- and/or context-specific.     

Paradoxical persistence through mixed-system dynamics: towards a unified perspective of reversal behaviours in evolutionary ecology

Counterintuitive dynamics of various biological phenomena occur when composite system dynamics differ qualitatively from that of their component systems. Such composite systems typically arise when modelling situations with time-varying biotic or abiotic conditions, and examples range from metapopulation dynamics to population genetic models. These biological, and related physical, phenomena can often be modelled as simple financial games, wherein capital is gained and lost through gambling. Such games have been developed and used as heuristic devices to elucidate the processes at work in generating seemingly paradoxical outcomes across a spectrum of disciplines, albeit in a field-specific, ad hoc fashion. Here, we propose that studying these simple games can provide a much deeper understanding of the fundamental principles governing paradoxical behaviours in models from a diversity of topics in evolution and ecology in which fluctuating environmental effects, whether deterministic or stochastic, are an essential aspect of the phenomenon of interest. Of particular note, we find that, for a broad class of models, the ecological concept of equilibrium reactivity provides an intuitive necessary condition that must be satisfied in order for environmental variability to promote population persistence. We contend that further investigations along these lines promise to unify aspects of the study of a range of topics, bringing questions from genetics, species persistence and coexistence and the evolution of bet-hedging strategies, under a common theoretical purview.     

The Shapley value of phylogenetic trees

Every weighted tree corresponds naturally to a cooperative game that we call a tree game; it assigns to each subset of leaves the sum of the weights of the minimal subtree spanned by those leaves. In the context of phylogenetic trees, the leaves are species and this assignment captures the diversity present in the coalition of species considered. We consider the Shapley value of tree games and suggest a biological interpretation. We determine the linear transformation M that shows the dependence of the Shapley value on the edge weights of the tree, and we also compute a null space basis of M. Both depend on the split counts of the tree. Finally, we characterize the Shapley value on tree games by four axioms, a counterpart to Shapley’s original theorem on the larger class of cooperative games. We also include a brief discussion of the core of tree games. Research of Francis Edward Su was partially supported by NSF Grants DMS-0301129 and DMS-0701308.     

Dizeez: An Online Game for Human Gene-Disease Annotation

Structured gene annotations are a foundation upon which many bioinformatics and statistical analyses are built. However the structured annotations available in public databases are a sparse representation of biological knowledge as a whole. The rate of biomedical data generation is such that centralized biocuration efforts struggle to keep up. New models for gene annotation need to be explored that expand the pace at which we are able to structure biomedical knowledge. Recently, online games have emerged as an effective way to recruit, engage and organize large numbers of volunteers to help address difficult biological challenges. For example, games have been successfully developed for protein folding (Foldit), multiple sequence alignment (Phylo) and RNA structure design (EteRNA). Here we present Dizeez, a simple online game built with the purpose of structuring knowledge of gene-disease associations. Preliminary results from game play online and at scientific conferences suggest that Dizeez is producing valid gene-disease annotations not yet present in any public database. These early results provide a basic proof of principle that online games can be successfully applied to the challenge of gene annotation. Dizeez is available at http://genegames.org.     

Interaction rates,vital rates,background fitness and replicator dynamics: how to embed evolutionary game structure into realistic population dynamics

In this paper we are concerned with how aggregated outcomes of individual behaviours, during interactions with other individuals (games) or with environmental factors, determine the vital rates constituting the growth rate of the population. This approach needs additional elements, namely the rates of event occurrence (interaction rates). Interaction rates describe the distribution of the interaction events in time, which seriously affects the population dynamics, as is shown in this paper. This leads to the model of a population of individuals playing different games, where focal game affected by the considered trait can be extracted from the general model, and the impact on the dynamics of other events (which is not neutral) can be described by an average background fertility and mortality. This leads to a distinction between two types of background fitness, strategically neutral elements of the focal games (correlated with the focal game events) and the aggregated outcomes of other interactions (independent of the focal game). The new approach is useful for clarification of the biological meaning of concepts such as weak selection. Results are illustrated by a Hawk–Dove example.     

On equilibrium properties of evolutionary multi-player games with random payoff matrices

The analysis of equilibrium points in biological dynamical systems has been of great interest in a variety of mathematical approaches to biology, such as population genetics, theoretical ecology or evolutionary game theory. The maximal number of equilibria and their classification based on stability have been the primary subjects of these studies, for example in the context of two-player games with multiple strategies. Herein, we address a different question using evolutionary game theory as a tool. If the payoff matrices are drawn randomly from an arbitrary distribution, what are the probabilities of observing a certain number of (stable) equilibria? We extend the domain of previous results for the two-player framework, which corresponds to a single diploid locus in population genetics, by addressing the full complexity of multi-player games with multiple strategies. In closing, we discuss an application and illustrate how previous results on the number of equilibria, such as the famous Feldman-Karlin conjecture on the maximal number of isolated fixed points in a viability selection model, can be obtained as special cases of our results based on multi-player evolutionary games. We also show how the probability of realizing a certain number of equilibria changes as we increase the number of players and number of strategies.     

Evolution of cooperation in spatial public goods games with common resource dynamics

Investment in a common resource shared by all players is difficult to evolve despite higher returns because a non-investor (free-rider) always receives more than an investor (altruist). This situation is referred to as the Tragedy of the Commons and is often observed in various biological systems including environmental problems of human society. Punishment and reputation are effective mechanisms but require cooperator's ability to identify free-riders. Volunteering can work in anonymous public goods games but this requires voluntary participation, which is not always the case. Here, we show that the evolution of altruism is possible in anonymous and obligate public goods games if we consider the spatiotemporal dynamics of the common resource that incorporate spatial diffusion and internal dynamics of the commons. The investors' strategy to counter free-riders is to increase population density and to outnumber them with the common resource level kept as low as that of the free-riders.     

An Application of Evolutionary Game Theory to Social Dilemmas: The Traveler's Dilemma and the Minimum Effort Coordination Game

The Traveler''s Dilemma game and the Minimum Effort Coordination game are two social dilemmas that have attracted considerable attention due to the fact that the predictions of classical game theory are at odds with the results found when the games are studied experimentally. Moreover, a direct application of deterministic evolutionary game theory, as embodied in the replicator dynamics, to these games does not explain the observed behavior. In this work, we formulate natural variants of these two games as smoothed continuous-strategy games. We study the evolutionary dynamics of these continuous-strategy games, both analytically and through agent-based simulations, and show that the behavior predicted theoretically is in accord with that observed experimentally. Thus, these variants of the Traveler''s Dilemma and the Minimum Effort Coordination games provide a simple resolution of the paradoxical behavior associated with the original games.     

Unpredictability induced by unfocused games in evolutionary game dynamics

Evolutionary game theory is a basis of replicator systems and has applications ranging from animal behavior and human language to ecosystems and other hierarchical network systems. Most studies in evolutionary game dynamics have focused on a single game, but, in many situations, we see that many games are played simultaneously. We construct a replicator equation with plural games by assuming that a reward of a player is a simple summation of the reward of each game. Even if the numbers of the strategies of the games are different, its dynamics can be described in one replicator equation. We here show that when players play several games at the same time, the fate of a single game cannot be determined without knowing the structures of the whole other games. The most absorbing fact is that even if a single game has a ESS (evolutionary stable strategy), the relative frequencies of strategies in the game does not always converge to the ESS point when other games are played simultaneously.     

Gradients for the evolution of bimatrix games

The evolutionary dynamics of bimatrix games is studied for rescaled partnership games and zero sum games. The former case leads to gradient systems. The selection equations for sexual and asexual reproduction of genotypes corresponding to mixed strategies are analysed. As examples, the origin of anisogamy and cyclic chases for predator-prey coevolution are studied.     

Game On,Science - How Video Game Technology May Help Biologists Tackle Visualization Challenges

The video games industry develops ever more advanced technologies to improve rendering, image quality, ergonomics and user experience of their creations providing very simple to use tools to design new games. In the molecular sciences, only a small number of experts with specialized know-how are able to design interactive visualization applications, typically static computer programs that cannot easily be modified. Are there lessons to be learned from video games? Could their technology help us explore new molecular graphics ideas and render graphics developments accessible to non-specialists? This approach points to an extension of open computer programs, not only providing access to the source code, but also delivering an easily modifiable and extensible scientific research tool. In this work, we will explore these questions using the Unity3D game engine to develop and prototype a biological network and molecular visualization application for subsequent use in research or education. We have compared several routines to represent spheres and links between them, using either built-in Unity3D features or our own implementation. These developments resulted in a stand-alone viewer capable of displaying molecular structures, surfaces, animated electrostatic field lines and biological networks with powerful, artistic and illustrative rendering methods. We consider this work as a proof of principle demonstrating that the functionalities of classical viewers and more advanced novel features could be implemented in substantially less time and with less development effort. Our prototype is easily modifiable and extensible and may serve others as starting point and platform for their developments. A webserver example, standalone versions for MacOS X, Linux and Windows, source code, screen shots, videos and documentation are available at the address: http://unitymol.sourceforge.net/.     

The evolution of n-player cooperation-threshold games and ESS bifurcations

An evolutionary game of individuals cooperating to obtain a collective benefit is here modelled as an n-player Prisoner's Dilemma game. With reference to biological situations, such as group foraging, we introduce a threshold condition in the number of cooperators required to obtain the collective benefit. In the simplest version, a three-player game, complex behaviour appears as the replicator dynamics exhibits a catastrophic event separating a parameter region allowing for coexistence of cooperators and defectors and a region of pure defection. Cooperation emerges through an ESS bifurcation, and cooperators only thrive beyond a critical point in cost-benefit space. Moreover, a repelling fixed point of the dynamics acts as a barrier to the introduction of cooperation in defecting populations. The results illustrate the qualitative difference between two-player games and multiple player games and thus the limitations to the generality of conclusions from two-player games. We present a procedure to find the evolutionarily stable strategies in any n-player game with cost and benefit depending on the number of cooperators. This was previously done by Motro [1991. Co-operation and defection: playing the field and the ESS. J. Theor. Biol. 151, 145-154] in the special cases of convex and concave benefit functions and constant cost.     

A theory for the evolutionary game

We present an evolutionary game theory. This theory differs in several respects from current theories related to Maynard Smith's pioneering work on evolutionary stable strategies (ESS). Most current work deals with two person matrix games. For these games the strategy set is finite. We consider evolutionary games which are defined over a continuous strategy set and which permit any number of players. Matrix games are included as a bilinear continuous game. However, under our definition, such games will not posses an ESS on the interior of the strategy set. We extend previous work on continuous games by developing an ESS definition which permits the ESS to be composed of a coalition of several strategies. This definition requires that the coalition must not only be stable with respect to perturbations in strategy frequencies which comprise the coalition, but the coalition must also satisfy the requirement that no mutant strategies can invade. Ecological processes are included in the model by explicitly considering population size and density dependent selection.     

DILEMMAS OF PARTIAL COOPERATION

Related to the often applied cooperation models of social dilemmas, we deal with scenarios in which defection dominates cooperation, but an intermediate fraction of cooperators, that is, “partial cooperation,” would maximize the overall performance of a group of individuals. Of course, such a solution comes at the expense of cooperators that do not profit from the overall maximum. However, because there are mechanisms accounting for mutual benefits after repeated interactions or through evolutionary mechanisms, such situations can constitute “dilemmas” of partial cooperation. Among the 12 ordinally distinct, symmetrical 2 × 2 games, three (barely considered) variants are correspondents of such dilemmas. Whereas some previous studies investigated particular instances of such games, we here provide the unifying framework and concisely relate it to the broad literature on cooperation in social dilemmas. Complementing our argumentation, we study the evolution of partial cooperation by deriving the respective conditions under which coexistence of cooperators and defectors, that is, partial cooperation, can be a stable outcome of evolutionary dynamics in these scenarios. Finally, we discuss the relevance of such models for research on the large biodiversity and variation in cooperative efforts both in biological and social systems.