Chaos and language

Human language is a complex communication system with unlimited expressibility. Children spontaneously develop a native language by exposure to linguistic data from their speech community. Over historical time, languages change dramatically and unpredictably by accumulation of small changes and by interaction with other languages. We have previously developed a mathematical model for the acquisition and evolution of language in heterogeneous populations of speakers. This model is based on game dynamical equations with learning. Here, we show that simple examples of such equations can display complex limit cycles and chaos. Hence, language dynamical equations mimic complicated and unpredictable changes of languages over time. In terms of evolutionary game theory, we note that imperfect learning can induce chaotic switching among strict Nash equilibria.     

Evolutionary game dynamics in a Wright-Fisher process

Evolutionary game dynamics in finite populations can be described by a frequency dependent, stochastic Wright-Fisher process. We consider a symmetric game between two strategies, A and B. There are discrete generations. In each generation, individuals produce offspring proportional to their payoff. The next generation is sampled randomly from this pool of offspring. The total population size is constant. The resulting Markov process has two absorbing states corresponding to homogeneous populations of all A or all B. We quantify frequency dependent selection by comparing the absorption probabilities to the corresponding probabilities under random drift. We derive conditions for selection to favor one strategy or the other by using the concept of total positivity. In the limit of weak selection, we obtain the 1/3 law: if A and B are strict Nash equilibria then selection favors replacement of B by A, if the unstable equilibrium occurs at a frequency of A which is less than 1/3.     

Nash equilibrium and evolutionary stability in large- and finite-population "playing the field" models

This paper studies the correspondence between Nash equilibrium and evolutionary stability in large- and finite-population "playing the field" models. Whenever the fitness function is sufficiently continuous, any large-population ESS corresponds to a symmetric Nash equilibrium in the game that describes the simultaneous interaction of the individuals in the population, and any strict, symmetric Nash equilibrium in that game corresponds to a large-population ESS. This correspondence continues to hold, approximately, in finite populations; and it holds exactly for strict pure-strategy equilibria in sufficiently large finite populations. By contrast, a sequence of (mixed-strategy) finite-population ESSs can converge, as the population grows, to a limit that is not a large-population ESS, and a large-population ESS need not be the limit of any sequence of finite-population ESSs.     

On Nash Equilibrium and Evolutionarily Stable States That Are Not Characterised by the Folk Theorem

In evolutionary game theory, evolutionarily stable states are characterised by the folk theorem because exact solutions to the replicator equation are difficult to obtain. It is generally assumed that the folk theorem, which is the fundamental theory for non-cooperative games, defines all Nash equilibria in infinitely repeated games. Here, we prove that Nash equilibria that are not characterised by the folk theorem do exist. By adopting specific reactive strategies, a group of players can be better off by coordinating their actions in repeated games. We call it a type-k equilibrium when a group of k players coordinate their actions and they have no incentive to deviate from their strategies simultaneously. The existence and stability of the type-k equilibrium in general games is discussed. This study shows that the sets of Nash equilibria and evolutionarily stable states have greater cardinality than classic game theory has predicted in many repeated games.     

Evolutionary game dynamics with non-uniform interaction rates

The classical setting of evolutionary game theory, the replicator equation, assumes uniform interaction rates. The rate at which individuals meet and interact is independent of their strategies. Here we extend this framework by allowing the interaction rates to depend on the strategies. This extension leads to non-linear fitness functions. We show that a strict Nash equilibrium remains uninvadable for non-uniform interaction rates, but the conditions for evolutionary stability need to be modified. We analyze all games between two strategies. If the two strategies coexist or exclude each other, then the evolutionary dynamics do not change qualitatively, only the location of the equilibrium point changes. If, however, one strategy dominates the other in the classical setting, then the introduction of non-uniform interaction rates can lead to a pair of interior equilibria. For the Prisoner's Dilemma, non-uniform interaction rates allow the coexistence between cooperators and defectors. For the snowdrift game, non-uniform interaction rates change the equilibrium frequency of cooperators.     

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.     

Nash Equilibria in Multi-Agent Motor Interactions

Social interactions in classic cognitive games like the ultimatum game or the prisoner''s dilemma typically lead to Nash equilibria when multiple competitive decision makers with perfect knowledge select optimal strategies. However, in evolutionary game theory it has been shown that Nash equilibria can also arise as attractors in dynamical systems that can describe, for example, the population dynamics of microorganisms. Similar to such evolutionary dynamics, we find that Nash equilibria arise naturally in motor interactions in which players vie for control and try to minimize effort. When confronted with sensorimotor interaction tasks that correspond to the classical prisoner''s dilemma and the rope-pulling game, two-player motor interactions led predominantly to Nash solutions. In contrast, when a single player took both roles, playing the sensorimotor game bimanually, cooperative solutions were found. Our methodology opens up a new avenue for the study of human motor interactions within a game theoretic framework, suggesting that the coupling of motor systems can lead to game theoretic solutions.     

Mating strategies in primates: a game theoretical approach to infanticide

Infanticide by newly immigrated or newly dominant males is reported among a variety of taxa, such as birds, rodents, carnivores and primates. Here we present a game theoretical model to explain the presence and prevalence of infanticide in primate groups. We have formulated a three-player game involving two males and one female and show that the strategies of infanticide on the males' part and polyandrous mating on the females' part emerge as Nash equilibria that are stable under certain conditions. Moreover, we have identified all the Nash equilibria of the game and arranged them in a novel hierarchical scheme. Only in the subspace spanned by the males are the Nash equilibria found to be strict, and hence evolutionarily stable. We have therefore proposed a selection mechanism informed by adaptive dynamics to permit the females to transition to, and remain in, optimal equilibria after successive generations. Our model concludes that polyandrous mating by females is an optimal strategy for the females that minimizes infanticide and that infanticide confers advantage to the males only in certain regions of parameter space. We have shown that infanticide occurs during turbulent changes accompanying male immigration into the group. For changes in the dominance hierarchy within the group, we have shown that infanticide occurs only in primate groups where the chance for the killer to sire the next infant is high. These conclusions are confirmed by observations in the wild. This model thus has enabled us to pinpoint the fundamental processes behind the reproductive decisions of the players involved, which was not possible using earlier theoretical studies.     

The Worst-Case Weighted Multi-Objective Game with an Application to Supply Chain Competitions

In this paper, we propose a worst-case weighted approach to the multi-objective n-person non-zero sum game model where each player has more than one competing objective. Our “worst-case weighted multi-objective game” model supposes that each player has a set of weights to its objectives and wishes to minimize its maximum weighted sum objectives where the maximization is with respect to the set of weights. This new model gives rise to a new Pareto Nash equilibrium concept, which we call “robust-weighted Nash equilibrium”. We prove that the robust-weighted Nash equilibria are guaranteed to exist even when the weight sets are unbounded. For the worst-case weighted multi-objective game with the weight sets of players all given as polytope, we show that a robust-weighted Nash equilibrium can be obtained by solving a mathematical program with equilibrium constraints (MPEC). For an application, we illustrate the usefulness of the worst-case weighted multi-objective game to a supply chain risk management problem under demand uncertainty. By the comparison with the existed weighted approach, we show that our method is more robust and can be more efficiently used for the real-world applications.     

Bistability of Evolutionary Stable Vaccination Strategies in the Reinfection SIRI Model

We use the reinfection SIRI epidemiological model to analyze the impact of education programs and vaccine scares on individuals decisions to vaccinate or not. The presence of the reinfection provokes the novelty of the existence of three Nash equilibria for the same level of the morbidity relative risk instead of a single Nash equilibrium as occurs in the SIR model studied by Bauch and Earn (PNAS 101:13391–13394, 2004). The existence of three Nash equilibria, with two of them being evolutionary stable, introduces two scenarios with relevant and opposite features for the same level of the morbidity relative risk: the low-vaccination scenario corresponding to the evolutionary stable vaccination strategy, where individuals will vaccinate with a low probability; and the high-vaccination scenario corresponding to the evolutionary stable vaccination strategy, where individuals will vaccinate with a high probability. We introduce the evolutionary vaccination dynamics for the SIRI model and we prove that it is bistable. The bistability of the evolutionary dynamics indicates that the damage provoked by false scares on the vaccination perceived morbidity risks can be much higher and much more persistent than in the SIR model. Furthermore, the vaccination education programs to be efficient they need to implement a mechanism to suddenly increase the vaccination coverage level.     

Dynamics to Equilibrium in Network Games: Individual Behavior and Global Response

Various social contexts can be depicted as games of strategic interactions on networks, where an individual’s welfare depends on both her and her partners’ actions. Whereas much attention has been devoted to Bayes-Nash equilibria in such games, here we look at strategic interactions from an evolutionary perspective. To this end, we present the results of a numerical simulations program for these games, which allows us to find out whether Nash equilibria are accessible by adaptation of player strategies, and in general to identify the attractors of the evolution. Simulations allow us to go beyond a global characterization of the cooperativeness at equilibrium and probe into individual behavior. We find that when players imitate each other, evolution does not reach Nash equilibria and, worse, leads to very unfavorable states in terms of welfare. On the contrary, when players update their behavior rationally, they self-organize into a rich variety of Nash equilibria, where individual behavior and payoffs are shaped by the nature of the game, the social network’s structure and the players’ position within the network. Our results allow to assess the validity of mean-field approaches we use to describe the dynamics of these games. Interestingly, our dynamically-found equilibria generally do not coincide with (but show qualitatively the same features of) those resulting from theoretical predictions in the context of one-shot games under incomplete information.     

Natural selection of the critical period for language acquisition

The language acquisition period in humans lasts about 13 years. After puberty it becomes increasingly difficult to learn a language. We explain this phenomenon by using an evolutionary framework. We present a dynamical system describing competition between language acquisition devices, which differ in the length of the learning period. There are two selective forces that play a role in determining the critical learning period: (i) having a longer learning period increases the accuracy of language acquisition; (ii) learning is associated with certain costs that affect fitness. As a result, there exists a limited learning period which is evolutionarily stable. This result is obtained analytically by means of a Nash equilibrium analysis of language acquisition devices. Interestingly, the evolutionarily stable learning period does not maximize the average fitness of the population.     

An SIS epidemiology game with two subpopulations

There is significant current interest in the application of game theory to problems in epidemiology. Most mathematical analyses of epidemiology games have studied populations where all individuals have the same risks and interests. This paper analyses the rational-expectation equilibria in an epidemiology game with two interacting subpopulations of equal size where decisions change the prevalence and transmission patterns of an infectious disease. The transmission dynamics are described by an SIS model and individuals are only allowed to invest in daily prevention measures like hygiene. The analysis shows that disassortative mixing may lead to multiple Nash equilibria when there are two interacting subpopulations affecting disease prevalence. The dynamic stability of these equilibria is analysed under the assumption that strategies change slowly in the direction of self-interest. When mixing is disassortative, interior Nash equilibria are always unstable. When mixing is positively assortative, there is a unique Nash equilibrium that is globally stable.     

An SIS epidemiology game with two subpopulations

There is significant current interest in the application of game theory to problems in epidemiology. Most mathematical analyses of epidemiology games have studied populations where all individuals have the same risks and interests. This paper analyses the rational-expectation equilibria in an epidemiology game with two interacting subpopulations of equal size where decisions change the prevalence and transmission patterns of an infectious disease. The transmission dynamics are described by an SIS model and individuals are only allowed to invest in daily prevention measures like hygiene. The analysis shows that disassortative mixing may lead to multiple Nash equilibria when there are two interacting subpopulations affecting disease prevalence. The dynamic stability of these equilibria is analysed under the assumption that strategies change slowly in the direction of self-interest. When mixing is disassortative, interior Nash equilibria are always unstable. When mixing is positively assortative, there is a unique Nash equilibrium that is globally stable.     

A game theoretical approach to the evolution of structured communication codes

Structured meaning-signal mappings, i.e., mappings that preserve neighborhood relationships by associating similar signals with similar meanings, are advantageous in an environment where signals are corrupted by noise and sub-optimal meaning inferences are rewarded as well. The evolution of these mappings, however, cannot be explained within a traditional language evolutionary game scenario in which individuals meet randomly because the evolutionary dynamics is trapped in local maxima that do not reflect the structure of the meaning and signal spaces. Here we use a simple game theoretical model to show analytically that when individuals adopting the same communication code meet more frequently than individuals using different codes—a result of the spatial organization of the population—then advantageous linguistic innovations can spread and take over the population. In addition, we report results of simulations in which an individual can communicate only with its K nearest neighbors and show that the probability that the lineage of a mutant that uses a more efficient communication code becomes fixed decreases exponentially with increasing K. These findings support the mother tongue hypothesis that human language evolved as a communication system used among kin, especially between mothers and offspring.     

An error limit for the evolution of language

On the evolutionary trajectory that led to human language there must have been a transition from a fairly limited to an essentially unlimited communication system. The structure of modern human languages reveals at least two steps that are required for such a transition: in all languages (i) a small number of phonemes are used to generate a large number of words; and (ii) a large number of words are used to a produce an unlimited number of sentences. The first (and simpler) step is the topic of the current paper. We study the evolution of communication in the presence of errors and show that this limits the number of objects (or concepts) that can be described by a simple communication system. The evolutionary optimum is achieved by using only a small number of signals to describe a few valuable concepts. Adding more signals does not increase the fitness of a language. This represents an error limit for the evolution of communication. We show that this error limit can be overcome by combining signals (phonemes) into words. The transition from an analogue to a digital system was a necessary step toward the evolution of human language.     

Old World monkeys are more similar to humans than New World monkeys when playing a coordination game

There is much debate about how humans' decision-making compares with that of other primates. One way to explore this is to compare species' performance using identical methodologies in games with strategical interactions. We presented a computerized Assurance Game, which was either functionally simultaneous or sequential, to investigate how humans, rhesus monkeys and capuchin monkeys used information in decision-making. All species coordinated via sequential play on the payoff-dominant Nash equilibrium, indicating that information about the partner's choice improved decisions. Furthermore, some humans and rhesus monkeys found the payoff-dominant Nash equilibrium in the simultaneous game, even when it was the first condition presented. Thus, Old World primates solved the task without any external cues to their partner's choice. Finally, when not explicitly prohibited, humans spontaneously used language to coordinate on the payoff-dominant Nash equilibrium, indicating an alternative mechanism for converting a simultaneous move game into a sequential move game. This phylogenetic distribution implies that no single mechanism drives coordination decisions across the primates, while humans' ability to spontaneously use language to change the structure of the game emphasizes that multiple mechanisms may be used even within the same species. These results provide insight into the evolution of decision-making strategies across the primates.     

Regular systems of inbreeding

Regular systems of inbreeding with discrete, nonoverlapping generations and the same number of individuals and mating pattern in every generation are studied. The matrix Q that specifies the recursion relations satisfied by the probabilities of identity is expressed in terms of the matrix M that describes the mating system. Necessary and sufficient conditions for convergence to genetic uniformity are given, and it is determined which probabilities of identity approach one. If the mating system has certain symmetries and these are imposed initially, then a matrix R, of lower dimension than Q, specifies the recursion relations. For such a mating system, for generic initial conditions, the condensed matrix R suffices for determining whether convergence to uniformity occurs and which probabilities of identity approach one. If Q is irreducible, the maximal eigenvalue of R is the same as that of Q. If Q is also aperiodic, this implies that the asymptotic rate of convergence to homogeneity of the condensed system is the same as that of the complete one. The above results apply to autosomal loci in monoecious (with or without selfing) and dioecious populations and to X-linked loci. As an example, all the eigenvalues and right and left eigenvectors of Q for circular mating are found.Supported by National Science Foundation Grant BSR-8512844     

The evolutionary language game.

We explore how evolutionary game dynamics have to be modified to accomodate a mathematical framework for the evolution of language. In particular, we are interested in the evolution of vocabulary, that is associations between signals and objects. We assume that successful communication contributes to biological fitness: individuals who communicate well leave more offspring. Children inherit from their parents a strategy for language learning (a language acquisition device). We consider three mechanisms whereby language is passed from one generation to the next: (i) parental learning: children learn the language of their parents; (ii) role model learning: children learn the language of individuals with a high payoff; and (iii) random learning: children learn the language of randomly chosen individuals. We show that parental and role model learning outperform random learning. Then we introduce mistakes in language learning and study how this process changes language over time. Mistakes increase the overall efficacy of parental and role model learning: in a world with errors evolutionary adaptation is more efficient. Our model also provides a simple explanation why homonomy is common while synonymy is rare.     

Games of age-dependent prevention of chronic infections by social distancing

Epidemiological games combine epidemic modelling with game theory to assess strategic choices in response to risks from infectious diseases. In most epidemiological games studied thus-far, the strategies of an individual are represented with a single choice parameter. There are many natural situations where strategies can not be represented by a single dimension, including situations where individuals can change their behavior as they age. To better understand how age-dependent variations in behavior can help individuals deal with infection risks, we study an epidemiological game in an SI model with two life-history stages where social distancing behaviors that reduce exposure rates are age-dependent. When considering a special case of the general model, we show that there is a unique Nash equilibrium when the infection pressure is a monotone function of aggregate exposure rates, but non-monotone effects can appear even in our special case. The non-monotone effects sometimes result in three Nash equilibria, two of which have local invasion potential simultaneously. Returning to a general case, we also describe a game with continuous age-structure using partial-differential equations, numerically identify some Nash equilibria, and conjecture about uniqueness.