Infinite Population Limit Research Articles

On a class of linear quadratic Gaussian quantilized mean field games

An energy provider faced with energy generation risks and a large homogeneous pool of customers designs its energy price as a time-varying function of a risk-related quantile of the total energy demand, which generalizes pricing through the mean of the total energy demand. In the infinite population limit, we model the pricing problem with a class of linear quadratic Gaussian quantilized mean field games. For these quantilized mean field games, we show existence and uniqueness of an equilibrium which reveals the price trajectory, as well as an approximate Nash property when the quantilized mean field game’s feedback control functions are applied to the large but finite game and the rate of convergence of the Nash deviation to zero as a function of the population size and the quantile is provided. Finally, the use of this class of quantilized mean field games is illustrated in the context of equivalent thermal parameter models for households heater and an energy provider using solar generation.

Evolutionary Game-Theoretic Approach to the Population Dynamics of Early Replicators.

The population dynamics of early replicators has revealed numerous puzzles, highlighting the difficulty of transitioning from simple template-directed replicating molecules to complex biological systems. The resolution of these puzzles has set the research agenda on prebiotic evolution since the seminal works of Manfred Eigen in the 1970s. Here, we study the effects of demographic noise on the population dynamics of template-directed (non-enzymatic) and protein-mediated (enzymatic) replicators. We borrow stochastic algorithms from evolutionary game theory to simulate finite populations of two types of replicators. These algorithms recover the replicator equation framework in the infinite population limit. For large but finite populations, we use finite-size scaling to determine the probability of fixation and the mean time to fixation near a threshold that delimits the regions of dominance of each replicator type. Since enzyme-producing replicators cannot evolve in a well-mixed population containing replicators that benefit from the enzyme but do not encode it, we study the evolution of enzyme-producing replicators in a finite population structured in temporarily formed random groups of fixed size n. We argue that this problem is identical to the weak-altruism version of the n-player prisoner's dilemma, and show that the threshold is given by the condition that the reward for altruistic behavior is equal to its cost.

[formula omitted]-player game formulation of the majority-vote model of opinion dynamics

From a self-centered perspective, it can be assumed that people only hold opinions that can benefit them. If opinions have no intrinsic value, and acquire their value when held by the majority of individuals in a discussion group, then we have a situation that can be modeled as an N-player game. Here we explore the dynamics of (binary) opinion formation using a game-theoretic framework to study an N-player game version of Galam’s local majority-vote model. The opinion dynamics is modeled by a stochastic imitation dynamics in which the individuals copy the opinion of more successful peers. In the infinite population limit, this dynamics is described by the classical replicator equation of evolutionary game theory. The equilibrium solution shows a threshold separating the initial frequencies that lead to the fixation of one opinion or the other. A comparison with Galam’s deterministic model reveals contrasting results, especially in the presence of inflexible individuals, who never change their opinions. In particular, the N-player game predicts a polarized equilibrium consisting only of extremists. Using finite-size scaling analysis, we evaluate the critical exponents that determine the population size dependence of the opinion’s fixation probability and mean fixation times near the threshold. The results underscore the usefulness of combining evolutionary game theory with opinion dynamics and the importance of statistical physics tools to summarize the results of Monte Carlo simulations.

Zero-Sum Games between Mean-Field Teams: Reachability-Based Analysis under Mean-Field Sharing

This work studies the behaviors of two large-population teams competing in a discrete environment. The team-level interactions are modeled as a zero-sum game while the agent dynamics within each team is formulated as a collaborative mean-field team problem. Drawing inspiration from the mean-field literature, we first approximate the large-population team game with its infinite-population limit. Subsequently, we construct a fictitious centralized system and transform the infinite-population game to an equivalent zero-sum game between two coordinators. Via a novel reachability analysis, we study the optimality of coordination strategies, which induce decentralized strategies under the original information structure. The optimality of the resulting strategies is established in the original finite-population game, and the theoretical guarantees are verified by numerical examples.

The probability of epidemic burnout in the stochastic SIR model with vital dynamics

We present an approach to computing the probability of epidemic "burnout," i.e., the probability that a newly emergent pathogen will go extinct after a major epidemic. Our analysis is based on the standard stochastic formulation of the Susceptible-Infectious-Removed (SIR) epidemic model including host demography (births and deaths) and corresponds to the standard SIR ordinary differential equations (ODEs) in the infinite population limit. Exploiting a boundary layer approximation to the ODEs and a birth-death process approximation to the stochastic dynamics within the boundary layer, we derive convenient, fully analytical approximations for the burnout probability. We demonstrate-by comparing with computationally demanding individual-based stochastic simulations and with semi-analytical approximations derived previously-that our fully analytical approximations are highly accurate for biologically plausible parameters. We show that the probability of burnout always decreases with increased mean infectious period. However, for typical biological parameters, there is a relevant local minimum in the probability of persistence as a function of the basic reproduction number [Formula: see text]. For the shortest infectious periods, persistence is least likely if [Formula: see text]; for longer infectious periods, the minimum point decreases to [Formula: see text]. For typical acute immunizing infections in human populations of realistic size, our analysis of the SIR model shows that burnout is almost certain in a well-mixed population, implying that susceptible recruitment through births is insufficient on its own to explain disease persistence.

Large population limit of the spectrum of killed birth-and-death processes

We consider a general class of birth-and-death processes with state space {0,1,2,3,…} which describes the size of a population going eventually to extinction with probability one. We obtain the complete spectrum of the generator of the process killed at 0 in the large population limit, that is, we scale the process by a parameter K, and take the limit K→+∞. We assume that the differential equation dx/dt=b(x)−d(x) describing the infinite population limit (in any finite-time interval) has a repulsive fixed point at 0, and an attractive fixed point x⁎>0. We prove that, asymptotically, the spectrum is the superposition of two spectra. One is the spectrum of the generator of an Ornstein-Uhlenbeck process, which is n(b′(x⁎)−d′(x⁎)), n≥0. The other one is the spectrum of a continuous-time binary branching process conditioned on non-extinction, and is given by n(d′(0)−b′(0)), n≥1. A major difficulty is that different scales and function spaces are involved. We work at the level of the eigenfunctions that we split over different regions, and study their asymptotic dependence on K in each region. In particular, we prove that the spectral gap goes to min⁡{b′(0)−d′(0),d′(x⁎)−b′(x⁎)}. This work complements a previous work of ours in which we studied the approximation of the quasi-stationary distribution and of the mean time to extinction.

Partially Observed Discrete-Time Risk-Sensitive Mean Field Games

In this paper, we consider discrete-time partially observed mean-field games with the risk-sensitive optimality criterion. We introduce risk-sensitivity behavior for each agent via an exponential utility function. In the game model, each agent is weakly coupled with the rest of the population through its individual cost and state dynamics via the empirical distribution of states. We establish the mean-field equilibrium in the infinite-population limit using the technique of converting the underlying original partially observed stochastic control problem to a fully observed one on the belief space and the dynamic programming principle. Then, we show that the mean-field equilibrium policy, when adopted by each agent, forms an approximate Nash equilibrium for games with sufficiently many agents. We first consider finite-horizon cost function and then discuss extension of the result to infinite-horizon cost in the next-to-last section of the paper.

Large deviations principle for discrete-time mean-field games

In this paper, we establish a large deviations principle (LDP) for interacting particle systems that arise from state and action dynamics of discrete-time mean-field games under the equilibrium policy of the infinite-population limit. The LDP is proved under weak Feller continuity of state and action dynamics. The proof is based on transferring LDP for empirical measures of initial states and noise variables under setwise topology to the original game model via contraction principle, which was first suggested by Delarue, Lacker, and Ramanan to establish LDP for continuous-time mean-field games under common noise. We also compare our work with LDP results established in prior literature for interacting particle systems, which are in a sense uncontrolled versions of mean-field games.

On inferring evolutionary stability in finite populations using infinite population models

Models of evolution by natural selection often make the simplifying assumption that populations are infinitely large. In this infinite population limit, rare mutations that are selected against always go extinct, whereas in finite populations they can persist and even reach fixation. Nevertheless, for mutations of arbitrarily small phenotypic effect, it is widely believed that in sufficiently large populations, if selection opposes the invasion of rare mutants, then it also opposes their fixation. Here, we identify circumstances under which infinite-population models do or do not accurately predict evolutionary outcomes in large, finite populations. We show that there is no population size above which considering only invasion generally suffices: for any finite population size, there are situations in which selection opposes the invasion of mutations of arbitrarily small effect, but favours their fixation. This is not an unlikely limiting case; it can occur when fitness is a smooth function of the evolving trait, and when the selection process is biologically sensible. Nevertheless, there are circumstances under which opposition of invasion does imply opposition of fixation: in fact, for the n-player snowdrift game (a common model of cooperation) we identify sufficient conditions under which selection against rare mutants of small effect precludes their fixation—in sufficiently large populations—for any selection process. We also find conditions under which—no matter how large the population—the trait that fixes depends on the selection process, which is important because any particular selection process is only an approximation of reality.

Approximate Markov-Nash Equilibria for Discrete-Time Risk-Sensitive Mean-Field Games

In this paper, we study a class of discrete-time mean-field games under the infinite-horizon risk-sensitive optimality criterion. Risk sensitivity is introduced for each agent (player) via an exponential utility function. In this game model, each agent is coupled with the rest of the population through the empirical distribution of the states, which affects both the agent’s individual cost and its state dynamics. Under mild assumptions, we establish the existence of a mean-field equilibrium in the infinite-population limit as the number of agents (N) goes to infinity, and we then show that the policy obtained from the mean-field equilibrium constitutes an approximate Nash equilibrium when N is sufficiently large.

Propagation of chaos for stochastic spatially structured neuronal networks with delay driven by jump diffusions

Spatially structured neural networks driven by jump diffusion noise with monotone coefficients, fully path dependent delay and with a disorder parameter are considered. Well-posedness for the associated McKean-Vlasov equation and a corresponding propagation of chaos result in the infinite population limit are proven. Our existence result for the McKean-Vlasov equation is based on the Euler approximation, that is applied to this type of equation for the first time.

Discrete-time average-cost mean-field games on Polish spaces

In stochastic dynamic games, when the number of players is sufficiently large and the interactions between agents depend on empirical state distribution, one way to approximate the original game is to introduce infinitepopulation limit of the problem. In the infinite population limit, a generic agent is faced with a so-called mean-field game. In this paper, we study discrete-time mean-field games with average-cost criteria. Using average cost optimality equation and Kakutani's fixed point theorem, we establish the existence of Nash equilibria for mean-field games under drift and minorization conditions on the dynamics of each agent. Then, we show that the equilibrium policy in the mean-field game, when adopted by each agent, is an approximate Nash equilibrium for the corresponding finite-agent game with sufficiently many agents.

Phase Transitions in the Evolution Model with Phenotypic Variation via Learning

The Baldwin effect describes how phenotypic variations such as the learned behaviors among clones (individuals with identical genes, i.e., isogenic individuals) can generally affect the course of evolution. We investigate the effects of phenotypic variation via learning on the evolutionary dynamics of a population. In a population, we assume the individuals to be haploid with two genotypes: one genotype shows phenotypic variation via learning and the other does not. We use an individual-based Moran model in which the evolutionary dynamics is formulated in terms of a master equation and is approximated as the Fokker-Planck equation (FPE) and the coupled nonlinear stochastic differential equations (SDEs) with multiplicative noise. In the infinite population limit, we analyze the deterministic part of the SDEs to obtain the fixed points and determine the stability of each fixed point. We find discrete phase transitions in the population distribution when the set of probabilities of reproducing the fitter and the unfit isogenic individuals is equal to the set of critical values determined by the stability of the fixed points. We also carry out numerical simulations in the form of the Gillespie algorithm and find that the results of the simulations are consistent with the analytic predictions.

Approximate Nash Equilibria in Partially Observed Stochastic Games with Mean-Field Interactions

Establishing the existence of Nash equilibria for partially observed stochastic dynamic games is known to be quite challenging, with the difficulties stemming from the noisy nature of the measurements available to individual players (agents) and the decentralized nature of this information. When the number of players is sufficiently large and the interactions among agents is of the mean-field type, one way to overcome this challenge is to investigate the infinite-population limit of the problem, which leads to a mean-field game. In this paper, we consider discrete-time partially observed mean-field games with infinite-horizon discounted-cost criteria. Using the technique of converting the original partially observed stochastic control problem to a fully observed one on the belief space and the dynamic programming principle, we establish the existence of Nash equilibria for these game models under very mild technical conditions. Then, we show that the mean-field equilibrium policy, when adopted by each agent, forms an approximate Nash equilibrium for games with sufficiently many agents.

The Evolving Moran Genealogy

We study the evolution of the population genealogy in the classic neutral Moran Model of finite size n∈N and in discrete time. The stochastic transformations that shape a Moran population can be realized directly on its genealogy and give rise to a process on a state space consisting of n-sized binary increasing trees. We derive a number of properties of this process, and show that they are in agreement with existing results on the infinite-population limit of the Moran Model. Most importantly, this process admits time reversal, which makes it possible to simplify the mechanisms determining state changes, and allows for a thorough investigation of the Most Recent Common Ancestorprocess.

Mean Field Games with Partial Observation

Subject to reasonable conditions, in large population stochastic dynamics games, where the agents are coupled by the system's mean field (i.e., the state distribution of the generic agent) through their nonlinear dynamics and their nonlinear cost functions, it can be shown that a best response control action for each agent exists which (i) depends only upon the individual agent's state observations and the mean field, and (ii) achieves an $\epsilon$-Nash equilibrium for the system. In this work we formulate a class of problems where each agent has only partial observations on its individual state. We employ nonlinear filtering theory and the separation principle in order to analyze the game in the asymptotically infinite population limit. The main result is that the $\epsilon$-Nash equilibrium property holds where the best response control action of each agent depends upon the conditional density of its own state generated by a nonlinear filter, together with the system's mean field. Finally, comparing this MFG problem with state estimation to that found in the literature with a major agent whose partially observed state process is independent of the control action of any individual agent, it is seen that, in contrast, the partially observed state process of any agent in this work depends upon that agent's control action.

Phase Transitions and Phase Diagram of the Island Model with Migration

We investigate the evolutionary dynamics and the phase transitions of the island model which consists of subdivided populations of individuals confined to two islands. In the island model, the population is subdivided so that migration acts to determine the evolutionary dynamics along with selection and genetic drift. The individuals are assumed to be haploid and to be one of two species, X or Y. They reproduce according to their fitness values, die at random, and migrate between the islands. The evolutionary dynamics of an individual based model is formulated in terms of a master equation and is approximated by using the diffusion method as the multidimensional Fokker-Planck equation (FPE) and the coupled non-linear stochastic differential equations (SDEs) with multiplicative noise. We analyze the infinite population limit to find the phase transitions from the monomorphic state of one type to the polymorphic state to the monomorphic state of the other type as we vary the ratio of the fitness values in two islands and complete the phase diagram of our island model.

Dynamic properties in the four-state haploid coupled discrete-time mutation-selection model with an infinite population limit

The dynamic properties, such as the crossing time and time-dependence of the relative density of the four-state haploid coupled discrete-time mutation-selection model, were calculated with the assumption that μ ij = μ ji , where μ ij denotes the mutation rate between the sequence elements, i and j. The crossing time for s = 0 and r 23 = r 42 = 1 in the four-state model became saturated at a large fitness parameter when r 12 > 1, was scaled as a power law in the fitness parameter when r 12 = 1, and diverged when the fitness parameter approached the critical fitness parameter when r 12 < 1, where r ij = μ ij /μ 14.

Moran-type bounds for the fixation probability in a frequency-dependent Wright-Fisher model.

We study stochastic evolutionary game dynamics in a population of finite size. Individuals in the population are divided into two dynamically evolving groups. The structure of the population is formally described by a Wright-Fisher type Markov chain with a frequency dependent fitness. In a strong selection regime that favors one of the two groups, we obtain qualitatively matching lower and upper bounds for the fixation probability of the advantageous population. In the infinite population limit we obtain an exact result showing that a single advantageous mutant can invade an infinite population with a positive probability. We also give asymptotically sharp bounds for the fixation time distribution.

Convergence of a Moran model to Eigen's quasispecies model

We prove that a Moran model converges in probability to Eigen's quasispecies model in the infinite population limit. We show further that the invariant probability measure of the Moran model converges to the unique stationary solution of Eigen's quasispecies model.