An ESS maximum principle for matrix games

Previous work has demonstrated that for games defined by differential or difference equations with a continuum of strategies, there exists a G-function, related to individual fitness, that must take on a maximum with respect to a virtual variable v whenever v is one of the vectors in the coalition of vectors which make up the evolutionarily stable strategy (ESS). This result, called the ESS maximum principle, is quite useful in determining candidates for an ESS. This principle is reformulated here, so that it may be conveniently applied to matrix games. In particular, we define a matrix game to be one in which fitness is expressed in terms of strategy frequencies and a matrix of expected payoffs. It is shown that the G-function in the matrix game setting must again take on a maximum value at all the strategies which make up the ESS coalition vector. The reformulated maximum principle is applicable to both bilinear and nonlinear matrix games. One advantage in employing this principle to solve the traditional bilinear matrix game is that the same G-function is used to find both pure and mixed strategy solutions by simply specifying an appropriate strategy space. Furthermore we show how the theory may be used to solve matrix games which are not in the usual bilinear form. We examine in detail two nonlinear matrix games: the game between relatives and the sex ratio game. In both of these games an ESS solution is determined. These examples not only illustrate the usefulness of this approach to finding solutions to an expanded class of matrix games, but aids in understanding the nature of the ESS as well.     

Evolutionarily stable strategies in differential and difference equation models

Summary A definition for an evolutionarily stable strategy (ESS) is given which is applicable to a general differential equation population model and two difference equation analogs. With the introduction of a fitnessgenerating function, it is possible to develop necessary conditions for the determination of an ESS for each of these systems. In most situations, an ESS for one system will also be an ESS for the other. Necessary conditions for an ESS are obtained. Under certain restrictions, they are shown to be valid, even under an unstable equilibrium in population density. the results are illustrated with an example which has the same ESS solution whether a continuous or discrete model is used. The behavior of the ESS for the discrete model is then examined under unstable equilibrium conditions in population density.     

Evolutionary stability: States and strategies

Different aspects and modifications of the definition of an evolutionarily stable (ES) strategy that have been considered in the literature can be incorporated in a unifying concept which regards the population context. This concept of evolutionary stability will generally characterize population states in both pure- and mixed-strategist models. In particular, it includes ES strategies, represented as a phenotype unique in an ES population. For an important class of mixed-strategist models, no strict ESS can exist. This will be the case whenever the success of an individual strategy is considered to follow as an average from the successes of its behavioural components. Instead, ESS results may be obtained from what will be called a “degenerate” form of the model, which is simply an ESS model on the level of elementary actions. Then, however, the correct interpretation of an ESS is not an individual phenotype but rather a population mixture of elementary actions. If an ES state exists in a mixed-strategist model it may be determined by an equilibrium condition; if there is an ES strategy, a different approach—mainly maximum considerations—is needed for finding it. An equilibrium condition does not hold for the components of an ES strategy straightforwardly; but it can be derived in terms of an auxiliary ESS model that considers first-order effects of the components. Several examples illustrate the significance of these results. Particularly, two models on “Games between Relatives” are reconsidered in order to display both their formal interrelation and the different meaning of their results in the context of mixed-strategist models.     

Evolutionarily stable strategies for a finite population and a variable contest size

This paper presents a generalization of Maynard Smith's concept of an evolutionarily stable strategy (ESS) to cover the cases of a finite population and a variable contest size. Both equilibrium and stability conditions are analysed. The standard Maynard Smith ESS with an infinite population and a contest size of two (pairwise contests) is shown to be a special case of this generalized ESS. An important implication of the generalized ESS is that in finite populations the behaviour of an ESS player is "spiteful", in the sense that an ESS player acts not only to increase his payoff but also to decrease the payoffs of his competitors. The degree of this "spiteful" behaviour is shown to increase with a decrease in the population size, and so is most likely to be observed in small populations. The paper concludes with an extended example: a symmetric two-pure-strategies two-player game for a finite population. It is shown that a mixed strategy ESS is globally stable against invasion by any one type of mutant strategist. The condition for the start of simultaneous invasion by two types of mutant is also given.     

Bounds on the number of ESSs of a matrix game

It is well known that for any evolutionary game there may be more than one evolutionarily stable strategy (ESS). In general, the more ESSs there are, the more difficult it is to work out how the population will behave (unless there are no ESSs at all). If a matrix game has an ESS which allows all possible pure strategies to be played, referred to as an internal ESS, then no other ESS can exist. In fact, the number of ESSs possible is highly dependent upon how many of the pure strategies each allow to be played, their support size. It is shown that if alpha is the ratio of the mean support size to the number of pure strategies n, then as n tends to infinity the greatest number of ESSs can be represented by a continuous function f(alpha) with useful regularity properties, and bounds are found for both f(alpha) and the value alpha(*), where it attains its maximum. Thus we can obtain a limit on the complexity of any particular system as a function of its mean support size.     

On evolutionarily stable sets

As an extension of the concept of an evolutionarily stable strategy (ESS) evolutionarily stable sets are introduced, i.e. sets of equilibrium strategies (EQS) which have much of the properties of an ESS. They are primarily used with evolutionary game models that allow a continuum of EQSs, none of which can be an ESS, but also include common ESSs as a special case. For a large class even of nonlinear models it can be shown that the standard dynamics converge towards some equilibrium point in an ES set if started within a neighbourhood of the set. Important applications of ES sets include e.g. mixed-strategist models and evolutionary game models in sexual populations.     

The evolution of anisogamy: the adaptive significance of damage, repair and mortality

Classic theory on the evolution of anisogamy focuses on the trade-off between gamete productivity and provisioning and mechanisms associated with post-zygotic survival. In this article, the role of mortality acting on both zygotes and gametes is explored as a factor influencing the evolution of different sized gametes. In particular, variable mortality through differential survival or metabolic damage is shown to affect the persistence of isogamy, the evolution of more than two sexes and the evolution of anisogamy. Evolutionary stable isogamous states are shown to be locally unstable and disruptive selection can induce the evolution of anisogamy. Analysis of both the isogamous and anisogamous ESS points reveals that the persistence of either of these conditions is not always assured. The implications of variable survival on the evolution of anisogamy are discussed.     

Contributions to the theory of active transport

A system consisting of two solutions separated by a membrane may be in one of four possible states: (1) transient, (2) steady, (3) equilibrium, or (4) pseudo-equilibrium. The latter state denotes that in the solutions the net flow of all components is zero but at least one of the components is not in thermodynamic equilibrium. Transient and steady-state systems may or may not have active transport. Thus only systems in either equilibrium or pseudo-equilibrium are considered in this paper, since the former indicates that there is no active transport, whereas in the latter case there always is active transport. This simplifies the problem of finding whether a system does or does not have an active transport mechanism, since it is frequently fairly easy to determine experimentally whether a system is in equilibrium or pseudo-equilibrium. The assumption that electric neutrality exists within very thin membranes is shown not to be valid. However, electric neutrality does exist in the solutions in a system in a pseudo-equilibrium state with fixed charges and impermeative ions. It is then shown how the presence and sign of an electric potential may be found by use of electroneutrality. The mechanism of active transport may be due to a general force acting on all particles of a particular component or to an individual force acting on the individual particles of a particular component. A general solvent flow or a diffusion drag force illustrates the first mechanism while the second is accomplished by either a carrier or a Maxwell Demon. The general type of active transport has been extensively treated in the literature, while the individual type has not been treated in a generalized form. Therefore, the individual type of active transport is discussed at length, and a simple illustrative model is intensively analyzed. Following this, there is a discossion of the Maxwell Demon and some models of it are presented. This research was supported by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command under contract No. AF 18(600) 1454.     

Kleptoparasitic Melees��Modelling Food Stealing Featuring Contests with Multiple Individuals

Kleptoparasitism is the stealing of food by one animal from another. This has been modelled in various ways before, but all previous models have only allowed contests between two individuals. We investigate a model of kleptoparasitism where individuals are allowed to fight in groups of more than two, as often occurs in real populations. We find the equilibrium distribution of the population amongst various behavioural states, conditional upon the strategies played and environmental parameters, and then find evolutionarily stable challenging strategies. We find that there is always at least one ESS, but sometimes there are two or more, and discuss the circumstances when particular ESSs occur, and when there are likely to be multiple ESSs.     

An explicit approach to evolutionarily stable dispersal strategies: no cost of dispersal

The evolution of dispersal is examined by looking at evolutionarily stable strategies (ESS) for dispersal parameters in discrete time multisite models without any cost of dispersal. ESS are investigated analytically, based on explicit results on sensitivity analysis of matrix models. The basic model considers an arbitrary number of sites and a single age class. An ESS for dispersal parameters is obtained when the spatial reproductive values, calculated at the density-dependent population equilibrium, are equal across sites. From this basic formulation, one derives equivalently that all local populations should be at equilibrium in the absence of migration, and that dispersal between sites should be balanced, i.e., the numbers of individuals arriving to and leaving a site are equal. These results are then generalized to a model with several age classes. Equal age-specific reproductive values do not however imply balanced dispersal in this case. Our results generalize to any number of sites and age classes those available ?M. Doebeli, Dispersal and dynamics, Theoret. Popul. 47 (1995) 82 for two sites and one age class.     

Red Queens and ESS: the coevolution of evolutionary rates

Summary The Red Queen principle states that a set of interacting species reaches an evolutionary equilibrium at which all their rates of coevolution exactly balance each other. The lag-load model, which is one way of searching for Red Queens, has, by itself, previously predicted that they do not exist. But this model has assumed that infinite maladaptedness is possible. The lag-load model is improved by assuming that once the lag load of all but one species is determined, so is that of the final species. This assumption eliminates the possibility of infinite maladaptedness. Its result is to allow the lag-load model to yield Red Queen coevolution. It does this whether or not speciation and extinction rates are included. Thus the lag-load model is harmonized with the earlier Red Queen model derived from studies of predation.Because of the intercorrelation of phenotypic traits, the predatory model concluded that the eventual stable rate of coevolution must be zero (except for intermittent bursts after some correlation or compromise is successfully broken). Another model that predicts stable coevolutionary rates of zero is that of evolutionarily stable strategies (ESS).Red Queen assumes that the more extreme a phenotypic trait is, the better it is, and that there are no constraints on the growth of such a phenotypic trait value. Such traits are the key to the Red Queen prediction of progressive coevolution. ESS models make no such assumptions. Eliminating unbounded traits from the model of predator-victim evolution changed its prediction from progressive coevolution to stasis. Before this paper, no model had dealt simultaneously with both unbounded and constrained traits.To handle both sorts of phenotypic traits at the same time in the same model, we abandoned lag load as a measure of evolutionary rate (lag loads do not uniquely determine phenotype). Instead, we used the traditional assumption that rate is proportional to the slope of the adaptive landscape. A model, relying on continuous evolutionary game theory, was developed and simulated under various conditions in two or three species sets, with up to five independent traits coevolving simultaneously. The results were: (1) there was always a set of equilibrium densities eventually achieved by coevolution; if the population interaction represented by this stable coevolutionary state is also stable, then the system should persist whether it evolves further or not; (2) whenever traits were present which were unbounded and best at their most extreme values, then a Red Queen emerged; (3) whenever traits were present which were correlated with each other or constrained below infinity, then an ESS emerged; (4) if both types were present, both results occurred: Red Queen in the unbounded traits and ESS in the constrained ones.Because unbounded traits may not exist, the Red Queen may have no domain. But the domain of ESS is real. ESS should lead to the evolutionary pattern called punctuated equilibrium. The changes in design rules which punctuate stasis should lead to an ever-expanding independence of traits from each other, i.e. to more and more refined differentiation. A single set of design rules which governs a set of species is called a fitness-generating function. Such functions may help to define the concepts of adaptive zone and ecological guild.     

Equilibrium structure and stability in a frequency-dependent,two-population diploid model

We investigate the equilibrium structure for an evolutionary genetic model in discrete time involving two monoecious populations subject to intraspecific and interspecific random pairwise interactions. A characterization for local stability of an equilibrium is found, related to the proximity of this equilibrium with evolutionarily stable strategies (ESS). This extends to a multi-population framework a principle initially proposed for single populations, which states that the mean population strategy at a locally stable equilibrium is as close as possible to an ESS.     

Periodic metabolic systems: Oscillations in multiple-loop negative feedback biochemical control networks

Summary For a general multiple loop feedback inhibition system in which the end product can inhibit any or all of the intermediate reactions it is shown that biologically significant behaviour is always confined to a bounded region of reaction space containing a unique equilibrium. By explicit construction of a Liapunov function for the general n dimensional differential equation it is shown that some values of reaction parameters cause the concentration vector to approach the equilibrium asymptotically for all physically realizable initial conditions. As the parameter values change, periodic solutions can appear within the bounded region. Some information about these periodic solutions can be obtained from the Hopf bifurcation theorem. Alternatively, if specific parameter values are known a numerical method can be used to find periodic solutions and determine their stability by locating a zero of the displacement map. The single loop Goodwin oscillator is analysed in detail. The methods are then used to treat an oscillator with two feedback loops and it is found that oscillations are possible even if both Hill coefficients are equal to one.     

Are there really no evolutionarily stable strategies in the iterated prisoner's dilemma?

The evolutionary form of the iterated prisoner's dilemma (IPD) is a repeated game where players strategically choose whether to cooperate with or exploit opponents and reproduce in proportion to game success. It has been widely used to study the evolution of cooperation among selfish agents. In the past 15 years, researchers proved over a series of papers that there is no evolutionarily stable strategy (ESS) in the IPD when players maintain long-term relationships. This makes it difficult to make predictions about what strategies can actually persist as prevalent in a population over time. Here, we show that this no ESS finding may be a mathematical technicality, relying on implausible players who are "too perfect" in that their probability of cooperating on any move is arbitrarily close to either 0 or 1. Specifically, in the no ESS proof, all strategies were allowed, meaning that after a strategy X experiences any history H, X cooperates with an unrestricted probability p (X, H) where 0< or =p (X, H)< or =1. Here, we restrict strategies to the set S in which X is a member of S [corrected] if after any H, X cooperates with a restricted probability p (X, H) where e< or =p (X, H)< or =1-e and 0相似文献   

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.     

Optimization of a weak 3' splice site counteracts the function of a bovine papillomavirus type 1 exonic splicing suppressor in vitro and in vivo

Alternative splicing is a critical component of the early to late switch in papillomavirus gene expression. In bovine papillomavirus type 1 (BPV-1), a switch in 3' splice site utilization from an early 3' splice site at nucleotide (nt) 3225 to a late-specific 3' splice site at nt 3605 is essential for expression of the major capsid (L1) mRNA. Three viral splicing elements have recently been identified between the two alternative 3' splice sites and have been shown to play an important role in this regulation. A bipartite element lies approximately 30 nt downstream of the nt 3225 3' splice site and consists of an exonic splicing enhancer (ESE), SE1, followed immediately by a pyrimidine-rich exonic splicing suppressor (ESS). A second ESE (SE2) is located approximately 125 nt downstream of the ESS. We have previously demonstrated that the ESS inhibits use of the suboptimal nt 3225 3' splice site in vitro through binding of cellular splicing factors. However, these in vitro studies did not address the role of the ESS in the regulation of alternative splicing. In the present study, we have analyzed the role of the ESS in the alternative splicing of a BPV-1 late pre-mRNA in vivo. Mutation or deletion of just the ESS did not significantly change the normal splicing pattern where the nt 3225 3' splice site is already used predominantly. However, a pre-mRNA containing mutations in SE2 is spliced predominantly using the nt 3605 3' splice site. In this context, mutation of the ESS restored preferential use of the nt 3225 3' splice site, indicating that the ESS also functions as a splicing suppressor in vivo. Moreover, optimization of the suboptimal nt 3225 3' splice site counteracted the in vivo function of the ESS and led to preferential selection of the nt 3225 3' splice site even in pre-mRNAs with SE2 mutations. In vitro splicing assays also showed that the ESS is unable to suppress splicing of a pre-mRNA with an optimized nt 3225 3' splice site. These data confirm that the function of the ESS requires a suboptimal upstream 3' splice site. A surprising finding of our study is the observation that SE1 can stimulate both the first and the second steps of splicing.     

Modes and cuts in metabolic networks: complexity and algorithms

Constraint-based approaches recently brought new insight into our understanding of metabolism. By making very simple assumptions such as that the system is at steady-state and some reactions are irreversible, and without requiring kinetic parameters, general properties of the system can be derived. A central concept in this methodology is the notion of an elementary mode (EM for short) which represents a minimal functional subsystem. The computation of EMs still forms a limiting step in metabolic studies and several algorithms have been proposed to address this problem leading to increasingly faster methods. However, although a theoretical upper bound on the number of elementary modes that a network may possess has been established, surprisingly, the complexity of this problem has never been systematically studied. In this paper, we give a systematic overview of the complexity of optimisation problems related to modes. We first establish results regarding network consistency. Most consistency problems are easy, i.e., they can be solved in polynomial time. We then establish the complexity of finding and counting elementary modes. We show in particular that finding one elementary mode is easy but that this task becomes hard when a specific EM (i.e. an EM containing some specified reactions) is sought. We then show that counting the number of elementary modes is musical sharpP-complete. We emphasize that the easy problems can be solved using currently existing software packages. We then analyse the complexity of a closely related task which is the computation of so-called minimum reaction cut sets and we show that this problem is hard. We then present two positive results which both allow to avoid computing EMs as a prior to the computation of reaction cuts. The first one is a polynomial approximation algorithm for finding a minimum reaction cut set. The second one is a test for verifying whether a set of reactions constitutes a reaction cut; this test can be readily included in existing algorithms to improve their performance. Finally, we discuss the complexity of other cut-related problems.     

Frequency-dependent selection in sexual family-structured populations

In this paper, a two-phenotype, single-locus, n -allele matrix game diploid model incorporating interactions between full sibs influencing personal fitness is investigated. Necessary and sufficient conditions for an ESS are given. We show that if a strategy is an ESS for this model with the payoff matrix A, then it must be an ESS for the standard game formulation with payoff matrix A+(r/2) A(T) where r is the probability to interact with a sib, but it is also possible that no ESS exists. Moreover, under the assumption of weak selection, the partial change in phenotype frequencies brings the population closer to an ESS when it exists.     

Spatial patterns in population conflicts

We consider a dynamical model for evolutionary games, and enquire how the introduction of diffusion may lead to the formation of stationary spatially inhomogeneous solutions, that is patterns. For the model equations being used it is already known that if there is an evolutionarily stable strategy (ESS), then it is stable. Equilibrium solutions which are not ESS's and which are stable with respect to spatially constant perturbations may be unstable for certain choices of the dispersal rates. We prove by a bifurcation technique that under appropriate conditions the instability leads to patterns. Computations using a curve-following technique show that the bifurcations exhibit a rich structure with loops joined by symmetry-breaking branches.     

Biological signals as handicaps

An ESS model of Zahavi's handicap principle is constructed. This allows a formal exposition of how the handicap principle works, and shows that its essential elements are strategic. The handicap model is about signalling, and it is proved under fairly general conditions that if the handicap principle's conditions are met, then an evolutionarily stable signalling equilibrium exists in a biological signalling system, and that any signalling equilibrium satisfies the conditions of the handicap principle. Zahavi's major claims for the handicap principle are thus vindicated. The place of cheating is discussed in view of the honesty that follows from the handicap principle. Parallel signalling models in economics are discussed. Interpretations of the handicap principle are compared. The models are not fully explicit about how females use information about male quality, and, less seriously, have no genetics. A companion paper remedies both defects in a model of the handicap principle at work in sexual selection.