Burton Simon

TOWARDS A GENERAL THEORY OF GROUP SELECTION

The longstanding debate about the importance of group (multilevel) selection suffers from a lack of formal models that describe explicit selection events at multiple levels. Here, we describe a general class of models for two‐level evolutionary processes which include birth and death events at both levels. The models incorporate the state‐dependent rates at which these events occur. The models come in two closely related forms: (1) a continuous‐time Markov chain, and (2) a partial differential equation (PDE) derived from (1) by taking a limit. We argue that the mathematical structure of this PDE is the same for all models of two‐level population processes, regardless of the kinds of events featured in the model. The mathematical structure of the PDE allows for a simple and unambiguous way to distinguish between individual‐ and group‐level events in any two‐level population model. This distinction, in turn, suggests a new and intuitively appealing way to define group selection in terms of the effects of group‐level events. We illustrate our theory of group selection by applying it to models of the evolution of cooperation and the evolution of simple multicellular organisms, and then demonstrate that this kind of group selection is not mathematically equivalent to individual‐level (kin) selection.

A network of priority queues in heavy traffic: one bottleneck station

In this paper we consider an open queueing network having multiple classes, priorities, and general service time distributions. In the case where there is a single bottleneck station we conjecture that normalized queue length and sojourn time processes converge, in the heavy traffic limit, to one-dimensional reflected Brownian motion, and present expressions for its drift and variance. The conjecture is motivated by known heavy traffic limit theorems for some special cases of the general model, and some conjectured “Heavy Traffic Principles” derived from them.Using the known stationary distribution of one-dimensional reflected Brownian motion, we present expressions for the heavy traffic limit of stationary queue length and sojourn time distributions and moments. For systems with Markov routing we are able to explicitly calculate the limits.

Hamilton's Rule in Multi-level Selection Models

Hamiltons rule is regarded as a useful tool in the understanding of social evolution, but it relies on restrictive, overly simple assumptions. Here we model more realistic situations, in which the traditional Hamiltons rule generally fails to predict the direction of selection. We offer modifications that allow accurate predictions, but also show that these Hamiltons rule type inequalities do not predict long-term outcomes. To illustrate these issues we propose a two-level selection model for the evolution of cooperation. The model describes the dynamics of a population of groups of cooperators and defectors of various sizes and compositions and contains birth-death processes at both the individual level and the group level. We derive Hamilton-like inequalities that accurately predict short-term evolutionary change, but do not reliably predict long-term evolutionary dynamics. Over evolutionary time, cooperators and defectors can repeatedly change roles as the favored type, because the amount of assortment between cooperators changes in complicated ways due to both individual-level and group-level processes. The equation that governs the dynamics of cooperator/defector assortment is a certain partial differential equation, which can be solved numerically, but whose behaviour cannot be predicted by Hamiltons rules, because Hamiltons rules only contain first-derivative information. In addition, Hamiltons rules are sensitive to demographic fitness effects such as local crowding, and hence models that assume constant group sizes are not equivalent to models like ours that relax that assumption. In the long-run, the group distribution typically reaches an equilibrium, in which case Hamiltons rules necessarily become equalities.

IMPROVING SIMULATION EFFICIENCY WITH QUASI CONTROL VARIATES

In a simulation one can often identify a random variable, Y, that is likely to be highly correlated with a random variable of interest, X. If μ Y =E(Y) is known then Y can be used as a control variate to estimate μ X =E(X) more efficiently than by a direct simulation of X. We study the asymptotic properties of a method that uses Y to potentially speed up the simulation when μ Y is not known. The method is effective when μ Y can be efficiently estimated in an auxiliary simulation that does not involve X. We call Y a quasi control variate (QCV). For a simulation of length t>0 time units, we invest pt units estimating μ Y with the auxiliary simulation, yielding Y¯ pt . The remaining qt=(1−p)t units are spent on the main simulation yielding estimates (X˜ qt ,Y˜ qt ) for (μ X ,μ Y ). The two simulations can be interleaved so they are effectively done simultaneously. For each p∈(0,1) and α∈R we have a QCV estimator for μ X , We find p and α that minimize the asymptotic variance parameter (AVP) of Xˆ t (p,α) in terms of statistics that are estimated during the simulations, and then describe an easily implemented adaptive procedure that achieves the minimum AVP. The adaptive procedure evolves into the optimal QCV procedure if it is more efficient than a direct simulation, X¯ t →μ X ; otherwise it evolves into the direct simulation. Applications in stochastic linear programming, stochastic partial differential equations (PDEs) and queuing theory are cited.

Interpolation approximations of sojourn time distributions

We present a method for approximating sojourn time distributions in open queueing systems based on light and heavy traffic limits. The method is consistent with and generalizes the interpolation approximations for moments previously presented by M. I. Reiman and B. Simon. The method is applicable to the class of systems for which both light and heavy traffic limits can be computed, which currently includes Markovian networks of priority queues with a unique bottleneck node. We illustrate the method of generating closed-form analytic approximations for the sojourn time distribution of the M/M/1 queue with Bernoulli feedback, the M/M/1 processor sharing queue, a priority queue with feedback and the M/Ek/1 queue. Empirical evidence suggests that the method works well on a large and identifiable class of priority queueing models.

HEAVY TRAFFIC APPROXIMATIONS FOR A SYSTEM OF INFINITE SERVERS WITH LOAD BALANCING

We consider an exponential queueing system with multiple stations, each of which has an infinite number of servers and a dedicated arrival stream of jobs. In addition, there is an arrival stream of jobs that choose a station based on the state of the system. In this paper we describe two heavy traffic approximations for the stationary joint probability mass function of the number of busy servers at each station. One of the approximations involves state-space collapse and is accurate for large traffic loads. The state-space in the second approximation does not collapse. It provides an accurate estimate of the stationary behavior of the system over a wide range of traffic loads.

A simple model of group selection that cannot be analyzed with inclusive fitness.

A widespread claim in evolutionary theory is that every group selection model can be recast in terms of inclusive fitness. Although there are interesting classes of group selection models for which this is possible, we show that it is not true in general. With a simple set of group selection models, we show two distinct limitations that prevent recasting in terms of inclusive fitness. The first is a limitation across models. We show that if inclusive fitness is to always give the correct prediction, the definition of relatedness needs to change, continuously, along with changes in the parameters of the model. This results in infinitely many different definitions of relatedness - one for every parameter value - which strips relatedness of its meaning. The second limitation is across time. We show that one can find the trajectory for the group selection model by solving a partial differential equation, and that it is mathematically impossible to do this using inclusive fitness.

Discrete Evolutionary Birth-Death Processes and Their Large Population Limits

We study a class of discrete-state Markovian models of evolutionary population dynamics for k types of organisms, called discrete evolutionary birth-death processes (EBDP). The organisms in a model environment interact with each other by playing a certain game. Birth rates for type i organisms at time t are determined by their expected payoff in the game against an opponent chosen randomly from the environment at time t. Death rates at time t for all the organism types are equal, and proportional to the total population at time t. A discrete EBDP is therefore a continuous-time Markov chain on the nonnegative k-dimensional integer lattice, with state transitions to neighboring vertices only. A certain system of k nonlinear ordinary differential equations (ODE) can be derived from the discrete EBDP, and used as a deterministic approximation. We prove that a properly scaled sequence of EBDPs converges (in probability, uniformly on bounded sets) to the solution of the system of ODEs. We also prove that a different scaling of the sequence converges (in distribution) to a certain system of stochastic differential equations (SDE). An example based on reactive strategies for iterated prisoners dilemma is used to illustrate population dynamics for discrete EBDPs, as well as the dynamics for the ODE and SDE limits.

Continuous-time models of group selection, and the dynamical insufficiency of kin selection models

Traditionally, the process of group selection has been described mathematically by discrete-time models, and analyzed using tools like the Price equation. This approach makes implicit assumptions about the process that are not valid in general, like the central role of synchronized mass-dispersion and group re-formation events. In many important examples (like hunter-gatherer tribes) there are no mass-dispersion events, and the group-level events that do occur, like fission, fusion, and extinction, occur asynchronously. Examples like these can be fully analyzed by the equations of two-level population dynamics (described here) so their models are dynamically sufficient. However, it will be shown that examples like these cannot be fully analyzed by kin selection (inclusive fitness) methods because kin selection versions of group selection models are not dynamically sufficient. This is a critical mathematical difference between group selection and kin selection models, which implies that the two theories are not mathematically equivalent.

Efficient Monte-Carlo simulation of a product-form model for a cellular system with dynamic resource sharing

There are many ways for users to share the radio spectrum allocated to a cell in a cellular phone system. We analyze a commonly proposed scheme wh ere the cell is divided into s sectors. Each sector has exclusive access to a certain number of channels. The remaining channels reside in a “common pool” and are shared among the sectors. The smallest unit of bandwidth that can be borrowed from the common pool is a “carrier,” which consists of c channels. When viewed as a multidimensional birth-death process, the steady-state distribution of the number of active channels in each sector has a “product form,” but because the state space is large and has a nonlinear boundary, direct calculation of quantities of interest is usually impractical. Ross and Wang have developed a Monte-Carlo technique that applies to our problem. We significantly improve the efficiency of their technique when applied to our problem by including certain (nonlinear) control variates. The kinds of control variates we use can be applied to other loss systems as well. We also explore the effect of importance sampling for our system. In many cases the variance reduction achieved from the combination of importance sampling and control variates is far greater than from either method alone. For systems with blocking probabilities in the range 0.001 to 0.1, the variance of the system-blocking probability estimator can be reduced by several orders of magnitude.