A differentiable homotopy to compute Nash equilibria of n-person games

Summary. The literature on the computation of Nash equilibria in n-person games is dominated by simplicial methods. This paper is the first to introduce a globally convergent algorithm that fully exploits the differentiability present in the problem. It presents an everywhere differentiable homotopy to do the computations. The homotopy path can therefore be followed by several numerical techniques. Moreover, instead of computing some Nash equilibrium, the algorithm is constructed in such a way that it computes the Nash equilibrium selected by the tracing procedure of Harsanyi and Selten. As a by-product of our proofs it follows that for a generic game the tracing procedure defines a unique feasible path. The numerical performance of the algorithm is illustrated by means of several examples. Received: December 21, 1999; revised version: December 27, 2000     

The power of ESS: An experimental study

Abstract. Our experimental design mimics a traditional evolutionary game framework where players are matched pairwise to play a symmetric 33 bimatrix game that has two Nash equilibria. One equilibrium is an evolutionary stable state, or ESS; the other is an equilibrium in dominated strategies. Our primary experimental result is the observation that the ESS becomes extremely attractive when subjects have minimal information about the payoff functions, although the dominated equilibrium assures the highest equilibrium payoff. The attractiveness of the ESS is only moderate when players are completely informed about the 33 payoff matrix. Correspondence to: S.K. Berninghaus     

A note on control costs and logit rules for strategic games

We consider best response dynamics with endogenous noise based on a finite game in strategic form. A player can reduce the noise level by expending an extra effort and incurring some disutility or control costs. We specify control costs that result in logit adjustment rules. The stochastically stable states of the dynamic process are partial Nash configurations, that is, states where at least one player plays a best response against the others. If the game has a potential, then the stochastically stable states coincide with the Nash equilibria on which the potential is maximized. RID="*" ID="*" Instructive comments of a referee are gratefully acknowledged. Correspondence to:H. Haller     

Quantitative growth effects of subsidies in a search theoretic R&D model

Should government subsidize R&D and does it matter how these subsidies are allocated? We examine these questions in a dynamic model where R&D is described as sequential sampling from a distribution of new ideas. Successful discoveries affect future available resources and incentives for further R&D. Consequently, there may be under-investment in R&D. We study the effect of government interventions aimed at fostering growth through R&D. Calibrating the model with aggregate data from the Israeli business sector allows us to quantitatively compare two forms of support resembling those actually used to encourage R&D in the Israeli business sector: (i) an unrestricted subsidy that may be used at the recipients' discretion to finance R&D or other investments, (ii) a subsidy earmarked by the government for R&D activities only. While there is no theoretical way to determine which of the two subsidies will have a greater impact on search for new ideas and growth, we find that in the calibrated economy both subsidies have a significant but similar impact on the economy's output and TFP growth rates. Accordingly, in the case of the Israeli business sector, the incentives to conduct R&D were sufficiently strong, and no R&D-specific encouragement was needed. However, a sensitivity analysis reveals that for economies characterized by other parameter values this result may not be true. Correspondence to: B. Bental     

Competitive selection, self-organisation and Joseph A. Schumpeter

Post-Schumpeterians have tended to use biological analogies to understand economic evolution, in contrast to Schumpeter himself. In this paper it is argued that the biological analogies used tend to be outdated and that Schumpeter espoused an intuitive understanding of the evolutionary economic process that is closely related to modern conceptions of self-organisation, suitably adapted for application in socioeconomic systems. Using a self-organisation approach, competition can be understood without recourse to biological analogy, in terms of general systemic principles that operate in the presence of variety. Viewing economic evolution in terms of complex adaptation in self-organising systems yields nonequilibrium and nonlinear perspectives that parallel Schumpeter's own intuitions, reinvigorating them as the basis of evolutionary economic thinking in the new Millennium.     

Variety,growth and demand

A dynamic model of demand compatible with a changing composition of the economic system is presented in this paper. Consumers are not expected to have completely formed preferences for radically new objects of consumption. Consumers adopt new goods or services ,created by innovation, only if three barriers are overcome: 1) a critical (minimum) level of income, 2) critical human capital, 3) critical fitness. However, even a new good or service with a fitness higher than that of pre-existing ones, will not be immediately adopted. Consumers'.limited knowledge will slow down the rate of adoption of any new good or service.     

Knowledge, consumption, and endogenous growth

In neoclassical theory, knowledge generates increasing returns – and therefore growth – because it is a public good that can be costlessly reused once created. In fact, however, much knowledge in the economy is actually tacit and not easily transmitted –and thus not an obvious source of increasing returns. Several writers have responded to this alarming circumstances by affirming hopefully that knowledge today is increasingly codified, general, and abstract – and increasingly less tacit. This paper disputes such a trend. But all is not lost: for knowledge does not have to be codified to be reused and therefore to generate economic growth.     

A comparison of diffusion approximations and actual limits in birth and death processes of noisy evolution

The stationary distribution of a birth and death process may not be approximated by a diffusion. The general situation is illustrated on the “musical chairs” model by Binmore et al. (1995). This model is shown to generate outcomes which are not captured by the concept of the ultralong run equilibrium selection introduced in Binmore et al. (1995).     

Optimal freeware quality in the presence of network externalities: an evolutionary game theoretical approach

New software products often face difficulty in achieving market penetration. A potential remedy is to offer a freeware version of the software to encourage initial adoption and establish a larger user base for the software, thereby increasing the commercial version's value to adopters in future periods. However, to avoid complete cannibalization of the commercial version, the freeware version's quality must be sufficiently low and the price of the commercial version must not be too high. We model the effect of these two decision variables, price and freeware quality, on the adoption of software using static and evolutionary game theory.     

Where Schumpeter was nearly right – the Swedish model and Capitalism, Socialism and Democracy<--RID="*"-->

In Capitalism, Socialism and Democracy Joseph A. Schumpeter concluded that socialism would eventually displace capitalism in Western democracies. This would come about as a result of the superior performance of capitalism. We extract six “stylized” propositions that are essential elements of Schumpeter's prediction about the fate of capitalism. These propositions are confronted with the development of the Swedish economy. The three main results of the analysis are: (1) The evolution of the Swedish economy closely followed Schumpeter's predictions until about 1980: Large firms became increasingly predominant in production and innovative activity, ownership of firms became more and more concentrated, individual entrepreneurship waned in importance, the general public grew increasingly hostile towards capitalism, and by the late 1970s explicit proposals for a gradual transfer of ownership of firms from private hands were launched. (2) Design of tax and industrial policies fueled a development of the economy along the lines predicted by Schumpeter. In general, the policies discouraged private wealth accumulation. In particular, the policies favored concentration of firms and concentration of private ownership. (3) The turning point away from the path to socialism coincides with real world developments that disclosed two major flaws in Schumpeter's analysis. First, the ever more obvious failure of socialism in Eastern Europe went against Schumpeter's assertion that socialism can work. Second, Schumpeter, who thought that modern technology would make the giant corporation increasingly predominant, did not foresee the revival of entrepreneurship that took place in the Western countries around 1980.     

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.     

Capitalism and democracy in the 21st century: a Kaleckian interpretation of a Schumpeterian problem

The paper reinterprets Schumpeter's views on the dynamic effects of taxation, as originally expressed in Crisis of the Tax State, from a Kaleckian perspective. In light of Schumpeter's rejection of Keynesian and Marshallian approaches to taxation, the paper argues that a recently developed Kaleckian approach provides an appropriate basis from which to analyse the effects on the business cycle of balanced changes in the structure of taxation. It is shown that, under certain shifting assumptions, increases in the taxation of wages or profits will stimulate investment and attenuate the amplitude of the business cycle. Ultimately, the shifting of taxes reduces to a conflict over income shares. The changing distribution of income in the United Kingdom in recent years suggests that investment is likely to remain sluggish unless there is a significant reversal of income shares. This may give rise to increasing economic and political tensions into the 21st century.     

Evolutionary economics and economic geography

This article attempts to explore how key notions from Evolutionary Economics, such as selection, path-dependency, chance and increasing returns, may be applied to two key topics in Economic Geography. The first issue is the problem of how to specify the (potential) impact of the spatial environment on new variety in terms of technological change. Evolutionary thinking may be useful to describe and explain: (1) the process of localized `collective' learning in a regional context, (2) the adjustment problems that regions may be confronted with in a world of increasing variation, and (3) the spatial formation of newly emerging industries as an evolutionary process, in which the spatial connotation of increasing returns (that is, agglomeration economies) may result in a spatial lock-in. The second issue is the problem of how new variety may affect the long-term evolution of the spatial system. We distinguish three approaches that, each in a different way, apply evolutionary notions to the nature of spatial evolution. This is strongly related to the issue whether mechanisms of chance and increasing returns, rather than selection and path-dependency, lay at the root of the spatial evolution of new technology.     

An evolutionary model of the size distribution of firms

An analytical study of the evolution of the distribution of firm size in an industry is presented. A drift-diffusion model is proposed to express the time-evolution of density of firm size within the industry. The model blends the conventional, more or less static, determinants with the kinds of dynamic considerations introduced by stochastic processes of evolutionary dynamics. The steady-state distribution as well as the dynamic behavior of the model are derived. Parameters in the resulting analytical expressions are then fit to a population of firms in the non-manufacturing service sector. The empirical portion of the paper validates the proposed evolutionary model.     

The economic analysis of social customs: the case of pre-marital sex

This paper uses a sample of 18,876 adults aged 16-59 from to estimate logit equations of the determinants of support for the custom of abstaining from pre-marital sex. Three sets of equations are estimated separately for males and females; one set for disapproval of pre-marital sex, one set for disapproval of one night stands and another set for a proxy for involvement in pre-marital sex. These results confirm some expected male-female differences and also show a polarisation amongst the population in their support for, and observance of, customs and practices related to abstaining from sex before marriage.     

Evolution of harvesting strategies: replicator and resource dynamics

Abstract. Economic theories of managing renewable resources, such as fisheries and forestry, traditionally assume that individual harvesters are perfectly rational and thus able to compute the harvesting strategy that maximizes their discounted profits. The current paper presents an alternative approach based on bounded rationality and evolutionary mechanisms. It is assumed that individual harvesters face a choice between two harvesting strategies. The evolution of the distribution of strategies in the population is modeled through a replicator dynamics equation. The latter captures the idea that strategies yielding above average profits are demanded more than strategies yielding below average profits, so that the first type ends up accounting for a larger part in the population. From a mathematical perspective, the combination of resource and evolutionary processes leads to complex dynamics. The paper presents the existence and stability conditions for each steady-state of the system and analyzes dynamic paths to the equilibrium. In addition, effects of changes in prices are analyzed. A main result of the paper is that under certain conditions both strategies can survive in the long-run. Correspondence to: J. Noailly     

Multiple attractors and global bifurcations in a Kaldor-type business cycle model

We consider a Kaldor-type discrete-time nonlinear business cycle model in income and capital, where investment is assumed to depend both on the difference between normal and current levels of capital stock, and on the difference between the current income and its normal level, through a nonlinear S-shaped increasing function. As usual in Kaldor business cycle models, one or three steady states exist, and the standard analysis of the local stability and bifurcations suggests that endogenous oscillations occur in the presence of only one unstable equilibrium, whereas the coexistence of three equilibria is characterized by bi-stability, the central equilibrium being on the boundary which separates the basins of the two stable ones. However, a deeper analysis of the global dynamic properties of the model in the parameter ranges where three steady states exist, reveals the existence of an attracting limit cycle surrounding the three steady states, leading to a situation of multistability, with a rich and complex dynamic structure.     

The learning economy and embodied knowledge flows in Great Britain

This paper examines the `learning economy' from the perspective of occupational characteristics and changes in the British labour market between 1980 and 1992. Following a discussion of the learning and knowledge economy, cross-sectional employment data are analysed to ascertain which occupations can be classified as knowledge-based. Longitudinal career history data are then used to trace the flows of these `knowledge workers' over time. Sectoral shifts are examined, with a particular focus on the knowledge-intensive service sectors. The data come from the Employment in Britain survey: a large-scale employee survey from 1992. The approach used allows us to measure somewhat intangible aspects of economic behaviour such as learning and tacit knowledge and attempt to trace their flows. Shifts in knowledge from the manufacturing to the service sector are shown to be important and related to previous work which demonstrated the importance of knowledge intensive business services for both output and productivity in manufacturing.     

Growth and equilibrium indeterminacy: the role of capital mobility

Summary. The paper presents a human capital driven endogenous growth model which, in general, permits a multiplicity of equilibrium balanced growth paths. It is shown that allowing for perfect capital mobility across countries increases the range of parameter values for which the model permits equilibrium indeterminacy. As opposed to the closed capital markets case, simple restrictions on preferences are no longer sufficient to eliminate the indeterminacy. Intuitively, under perfect capital mobility agents are able to smooth consumption completely. This induces an economy with open capital markets to behave like a closed economy with linear preferences thereby increasing the possibility of equilibrium indeterminacy. Received: 18 November 1998; revised version: 10 August 1999     

The demand for distinction and the evolution of the prestige car

This paper studies the demand for a distinction good – the prestige car – and examines how this demand for distinction influences the evolution of the product. We focus on two very different sorts of distinction car: the Rolls Royce and the Ferrari. These two examples help us to articulate two polar approaches to the demand for distinction: distinction through antiquity and distinction through novelty. The paper sets out a model of the demand for distinction, and uses it to describe two trajectories along which the prestige car – and the demand for it – may evolve.