Deterministic evolutionary dynamics: A unifying approach

We tackle three components of evolutionary modelling: payoffs, dynamical systems and equilibrium concepts. Firstly, we merely require that fitness functions are continuous. Secondly, we examine very general classes of dynamics. Thirdly, we give useful parallels to the Nash equilibrium and the evolutionarily stable strategy. Under (weakly) sign-compatible dynamics the change in population share of every (at least one) subgroup present in the population corresponds in sign with its relative fitness. At a saturated equilibrium, each subgroup with positive population share has highest fitness. We examine two evolutionary stability concepts: the evolutionarily stable equilibrium and the generalized evolutionarily stable state.     

Evolution in games with randomly disturbed payoffs

We consider a simple model of stochastic evolution in population games. In our model, each agent occasionally receives opportunities to update his choice of strategy. When such an opportunity arises, the agent selects a strategy that is currently optimal, but only after his payoffs have been randomly perturbed. We prove that the resulting evolutionary process converges to approximate Nash equilibrium in both the medium run and the long run in three general classes of population games: stable games, potential games, and supermodular games. We conclude by contrasting the evolutionary process studied here with stochastic fictitious play.     

Is limit pricing evolutionarily stable?

We have analyzed the question whether or not limit-pricing behavior can be viewed as a strategy, which survives the replicator dynamics in a game in which firms change strategies through imitation. Our results show that, regardless of the proportion of low cost incumbents in the population, potential entrants that play the strategy "enter" when they observe limit output are wiped out. The unique ESS consists of all low cost incumbents playing the "limit output" strategy, and all high cost incumbents playing the "monopoly output". This ESS includes potential entrants playing "stay out" after observing the limit output, and playing "enter" if monopoly output is observed in the first stage. Hence, limit pricing survives the replicator dynamics and appears in the stable state of the population.JEL Classification: B25, C73, L11We would like to thank an anonymous referee for valuable comments on an earlier version of this paper. Correspondence to: U. Soytas     

On Emissions Trading and Market Structure: Cap-and-Trade versus Intensity Standards

This paper examines the interdependence between imperfect competition and emissions trading. We particularly analyze the long run equilibrium in a two-sector (‘clean’ and ‘dirty’) model with Cournot competition among firms who face a fixed cost of production. The clean sector is defined as the sector with the highest long run cost margin on emissions. We compare the welfare implications of a cap-and-trade scheme with an emissions trading scheme based on relative intensity standards. It is shown that a firm’s long run equilibrium output in the clean or dirty sector does not depend on the emissions trading format, but only depends on the fixed cost of producing in the respective sector. Intensity standards can result in clean firms selling allowances to dirty firms, or dirty firms selling to clean firms. The former outcome yields higher welfare. It is demonstrated that cap-and-trade outperforms the intensity-based trading scheme in terms of long run welfare with free entry and exit. With intensity standards the size of the clean sector is too large.     

COMPETITIVE LONG RUN INDUSTRY EQUILIBRIUM AND FACTOR-PRICE UNCERTAINTY*

In this paper we have considered competitive long run industry equilibrium with factor-price uncertainty. We discussed the long run equilibrium output of firms with risk neutrality, output price and their responses to changes in uncertainty, factor price and industry demand. In the first part of this paper we have derived a result that, given risk neutrality, the firms operate at proper capacity, i.e. where expected long run marginal cost is equal to expected long run average cost, as shown in the case of output-price uncertainty. This result is, however, different from that obtained from Sheshinski and Dréze (1976). From the comparative static analysis we first discovered that even under risk neutrality factor-price uncertainty affects the long run industry equilibrium: that is, a mean preserving increase in uncertainty leads firm's to enter the industry, because they can decrease expected long run costs as the variability of factor price increases. Consequently, output price goes down. In contrast, firm size is kept invariable in response to its increase as long as the cost function is separable, i.e. the separability of the cost function holds when production functions are the Cobb-Douglas and CES types used commonly in empirical work, although firm size might, generally, be affected by the increase. It is an interesting fact that firm size and industry size will express different responses to a change in risk. The result that the long run industry equilibrium with cost uncertainty is explicitly affected is a sharp contrast to the result under output-price uncertainty and provides a new aspect for understanding about the behaviour of the industry with uncertainty. Secondly, increased factor-price causes the number of firms in the industry to decline and output price to rise. In addition, firm's size will expand with its increase if that factor is inferior, while the effect on firm size is ambiguous if it is normal. The firm's output, i.e. firm size, is, however, kept constant if the cost function is separable. Thirdly, the long run equilibrium output of the firm remains intact but the number of firms increases as industry demand rises. This result holds, regardless of the firm's attitude towards risk. Finally, we find throughout the paper that the functional form of the cost function plays a significant role in determining the behaviour of the industry with factor-price uncertainty.     

Schumpeterian dynamics: An evolutionary model of innovation and imitation

This paper develops a simple evolutionary model of innovation and imitation. It analyzes how dynamic interactions between equilibrating force of imitation and disequilibrating force of innovation mold the evolutionary pattern of an industry's state of technology and shows that in this Schumpeterian world the industry will never approach a neoclassical equilibrium with perfect knowlwdge even in the long run. The paper also examines the steady-state efficiency distribution of firms that characterizes the industry's long run and obtains some comparative dynamics results.     

A note on evolutionary stability of Bertrand equilibrium

It is shown that the equilibrium notion of an evolutionary stable strategy (ESS) does have predictive power for standard models of Bertrand competition. This is in contrast to a recent claim by Qin and Stuart (1997). The claim is based on the observation that the solution concept ESS behaves discontinuously when finite (discrete) action games approach an infinite (continuous) action game in the limit. Furthermore, it is argued that from a model-theoretic point of view evolutionary stability in prices (i.e. in the Bertrand model) is quite different from evolutionary stability in quantities (i.e. in the Cournot model).     

An Evolutionary Interpretation of Mixed-Strategy Equilibria

A convincing interpretation of mixed-strategy equilibria describes them as steady states in a large population in which players use pure strategies but the population as a whole mimics a mixed strategy. I study the conditions under which an evolutionary, stochastic learning process converges to the appropriate distribution over pure strategies in the population. I find that not all mixed equilibria can be justified as the result of an evolutionary process even if the equilibrium is unique. For symmetric 2 × 2 and 3 × 3 games I give necessary and sufficient conditions for convergence, which are related to the concept of an ESS, and forn × ngames I give a sufficient condition.Journal of Economic LiteratureClassification Numbers: C73, D83.     

Conjectural variations and evolutionary stability in finite populations

Recently it has been shown that consistent conjectures are evolutionarily stable. In this note we show that this finding depends on the use of the infinite population ESS (Maynard-Smith, Evolution and the theory of games, Cambridge University Press, Cambridge, UK, 1982). When applying the finite-population ESS (Schaffer, J Theor Biol 132:469–478, 1988) we show that the conjectures surviving in the long run are not consistent.      

Sophisticated imitation in cyclic games

Individuals belonging to two large populations are repeatedly randomly matched to play a cyclic game such as Matching Pennies. Between matching rounds, individuals sometimes change their strategy after observing a finite sample of other outcomes within their population. Individuals from the same population follow the same behavioral rule. In the resulting discrete time dynamics the unique Nash equilibrium is unstable. However, for sample sizes greater than one, we present an imitation rule where long run play cycles closely around the equilibrium.     

Endogenous fluctuations under evolutionary pressure in Cournot competition

An evolutionary game theoretic model of Cournot competition is investigated. Individuals choose from a finite set of different behavioral rules. Each rule specifies the quantity to be produced in the current period as a function of past quantities. Using more sophisticated rules may require extra information costs. Based upon realized payoffs, the fractions of the population choosing a certain behavioral rule are updated according to the replicator equation with noise. The long-run behavior of the evolutionary system consisting of the population dynamics coupled with the quantity dynamics of the Cournot game may be complicated and endogenous fluctuations may arise. We consider a typical example where firms can choose between two rules: the Nash rule and the best-reply rule. We show that a homoclinic tangency between the stable and unstable manifold of the equilibrium occurs as evolutionary pressure increases, implying bifurcation routes to complicated dynamics and strange attractors.     

Stochastic Replicator Dynamics

This article studies the replicator dynamics in the presence of shocks. I show that under these dynamics, strategies that do not survive the iterated deletion of strictly dominated strategies are eliminated in the long run, even in the presence of nonvanishing perturbations. I also give an example that shows that the stochastic dynamics in this article have equilibrium selection properties that differ from other dynamics in the literature.     

The macroeconomic impact of NAFTA termination

US President Trump has threatened to leave the North American Free Trade Agreement. How much would each member country gain or lose if this threat were carried out? Would trade imbalances within the region diminish? What would the transition to new production and consumption patterns look like? I provide quantitative answers to these questions using a dynamic general equilibrium model with a multi-sector input–output production structure, heterogeneous firms that make forward-looking export participation decisions, and adjustment frictions in trade and factor markets. Regional trade flows would fall dramatically, and while the US trade deficit with Canada would decline, the deficit with Mexico would grow. Welfare would fall by 0.04%, 0.12%, and 0.2% in the United States, Canada, and Mexico, respectively, and transition dynamics would significantly affect welfare in both the short run and the long run.     

Does Neutral Stability Imply Lyapunov Stability?

A central concept in evolutionary game theory is that of an evolutionarily stable strategy. However, many games of interest for economics have only strategies which meet the weaker requirement of neutral stability. While neutral stability implies Lyapunov stability under the replicator dynamics in pairwise interactions, we give here an example in which an evolutionarily stable strategy is not Lyapunov stable in a more general context. We propose to use instead the notions of (strong) uninvadability and (strong) unbeatability, refinements of evolutionary and neutral stability, respectively. Strong uninvadability implies asymptotic stability, and strong unbeatability implies Lyapunov stability under the replicator dynamics. Journal of Economic Literature Classification Number: C72.     

On the evolutionary selection of sets of Nash equilibria

It is well established for evolutionary dynamics in asymmetric games that a pure strategy combination is asymptotically stable if and only if it is a strict Nash equilibrium. We use an extension of the notion of a strict Nash equilibrium to sets of strategy combinations called ‘strict equilibrium set’ and show the following. For a large class of evolutionary dynamics, including all monotone regular selection dynamics, every asymptotically stable set of rest points that contains a pure strategy combination in each of its connected components is a strict equilibrium set. A converse statement holds for two-person games, for convex sets and for the standard replicator dynamic.     

Population dynamics and utilitarian criteria in the Lucas–Uzawa Model

This paper introduces population growth in the Uzawa–Lucas model, analyzing the implications of the choice of the welfare criterion on the model's outcome. Traditional growth theory assumes population growth to be exponential, but this is not a realistic assumption (see Brida and Accinelli, 2007). We model exogenous population change by a generic function of population size. We show that a unique non-trivial equilibrium exists and the economy converges towards it along a saddle path, independently of population dynamics. What is affected by the type of population dynamics is the dimension of the stable manifold, which can be one or two, and when the equilibrium is reached, which can happen in finite time or asymptotically. Moreover, we show that the choice of the utilitarian criterion will be irrelevant on the equilibrium of the model, if the steady state growth rate of population is null, as in the case of logistic population growth. Then, we show that a closed-form solution for the transitional dynamics of the economy (both in the case population dynamics is deterministic and stochastic) can be found for a certain parameter restriction.     

Basins of Attraction, Long-Run Stochastic Stability, and the Speed of Step-by-Step Evolution

The paper examines the behaviour of "evolutionary" models with ɛ-noise like those which have been used recently to discuss the evolution of social conventions. The paper is built around two main observations: that the"long run stochastic stability" of a convention is related tothe speed with which evolution toward and away from the convention occurs,and that evolution is more rapid (and hence more powerful) when it mayproceed via a series of small steps between intermediate steady states. Theformal analysis uses two new measures, the radius and modified coradius, tocharacterize the long run stochastically stable set of an evolutionary modeland to bound the speed with which evolutionary change occurs. Though notuniversally powerful, the result can be used to make many previous analysesmore transparent and extends them by providing results on waiting times. Anumber of applications are also discussed. The selection of the riskdominant equilibrium in 2×2 games is generalized to the selection of½-dominant equilibria in arbitrary games. Other applications involvetwo-dimensional local interaction and cycles as long run stochasticallystable sets.     

Two destabilizing strategies may be jointly stabilizing

The paper contributes to the literature on learning dynamics with heterogeneous agents, demonstrating that heterogeneity as such may be conducive to stability. The point is made in the framework of the cobweb model, where two different forecast procedures are considered. Either one destabilizes the price dynamics when it is uniformly adopted by all firms, or the price equilibrium becomes locally stable if firms are heterogenous and the two rules are suitably mixed within the population. It is also indicated that such a stabilizing composition is endogenously brought about by an adjustment process in which the population shares evolve under evolutionary pressure.     

Destruction and settlement norms as determinants of conflict: An evolutionary perspective

This paper studies conflict in a simple bargaining framework using an evolutionary game-theoretic approach. Our findings suggest that a player does not always regard the winning probability as an acceptable settlement rule. He accepts a division according to winning probability when the destruction caused to him in conflict is more than a threshold, which in turn depends on cost of conflict to the opponent and the size of the population. Further, our analysis shows that the norms with positive weight to disagreement payoff are effective against conflict, and incentivize peace. Contrary to the findings of extant studies carried out in finite population evolutionary game setting, the settlement possibility set is identical under both ESS and Nash equilibrium in our model with settlement norms.     

Optimality and Natural Selection in Markets

We ask if natural selection in markets favors profit-maximizing firms and, if so, is there a difference between the predictions of models which assume all firms are profit maximizers and the predictions of models which explicitly take account of population dynamics in the market. We show that market selection favors profit maximizing firms, but we also show that the long-run behavior of evolutionary market models is nonetheless not consistent with equilibrium models based on the profit-maximization hypothesis. Dynamic equilibrium paths with market selection are not Pareto optimal, nor even asymptotically optimal. The discrepancy arises because the dynamics created by firm evolution causes prices to vary over time and the resulting dynamical system need not have stable steady states.