Perfect Competition

In his 1987 entry on ‘Perfect Competition’ in The New Palgrave, the author reviewed the question of the perfectness of perfect competition, and gave four alternative formalisations rooted in the so-called Arrow-Debreu-Mckenzie model. That entry is now updated for the second edition to include work done on the subject during the last twenty years. A fresh assessment of this literature is offered, one that emphasises the independence assumption whereby individual agents are not related except through the price system. And it highlights a ‘linguistic turn’ whereby Hayek’s two fundamental papers on ‘division of knowledge’ are seen to have devastating consequences for this research programme.Allocation of Resources; Perfect Competition; Exchange Economy

Perfect Competition

In his 1987 entry on ‘Perfect Competition’ in The New Palgrave, the author reviewed the question of the perfectness of perfect competition, and gave four alternative formalisations rooted in the so-called Arrow-Debreu-Mckenzie model. That entry is now updated for the second edition to include work done on the subject during the last twenty years. A fresh assessment of this literature is offered, one that emphasises the independence assumption whereby individual agents are not related except through the price system. And it highlights a ‘linguistic turn’ whereby Hayek’s two fundamental papers on ‘division of knowledge’ are seen to have devastating consequences for this research programmeAllocation of Resources, Perfect Competition, Exchange Economy

Unifying a Geometric Framework of Evolutionary Algorithms and Elementary Landscapes Theory

Evolutionary algorithms (EAs) are randomised general-purpose strategies, inspired by natural evolution, often used for finding (near) optimal solutions to problems in combinatorial optimisation. Over the last 50 years, many theoretical approaches in evolutionary computation have been developed to analyse the performance of EAs, design EAs or measure problem difficulty via fitness landscape analysis. An open challenge is to formally explain why a general class of EAs perform better, or worse, than others on a class of combinatorial problems across representations. However, the lack of a general unified theory of EAs and fitness landscapes, across problems and representations, makes it harder to characterise pairs of general classes of EAs and combinatorial problems where good performance can be guaranteed provably. This thesis explores a unification between a geometric framework of EAs and elementary landscapes theory, not tied to a specific representation nor problem, with complementary strengths in the analysis of population-based EAs and combinatorial landscapes. This unification organises around three essential aspects: search space structure induced by crossovers, search behaviour of population-based EAs and structure of fitness landscapes. First, this thesis builds a crossover classification to systematically compare crossovers in the geometric framework and elementary landscapes theory, revealing a shared general subclass of crossovers: geometric recombination P-structures, which covers well-known crossovers. The crossover classification is then extended to a general framework for axiomatically analysing the population behaviour induced by crossover classes on associated EAs. This shows the shared general class of all EAs using geometric recombination P-structures, but no mutation, always do the same abstract form of convex evolutionary search. Finally, this thesis characterises a class of globally convex combinatorial landscapes shared by the geometric framework and elementary landscapes theory: abstract convex elementary landscapes. It is formally explained why geometric recombination P-structure EAs expectedly can outperform random search on abstract convex elementary landscapes related to low-order graph Laplacian eigenvalues. Altogether, this thesis paves a way towards a general unified theory of EAs and combinatorial fitness landscapes

Complexity and algorithms related to two classes of graph problems

This thesis addresses the problems associated with conversions on graphs and editing by removing a matching. We study the f-reversible processes, which are those associated with a threshold value for each vertex, and whose dynamics depends on the number of neighbors with different state for each vertex. We set a tight upper bound for the period and transient lengths, characterize all trees that reach the maximum transient length for 2-reversible processes, and we show that determining the size of a minimum conversion set is NP-hard. We show that the AND-OR model defines a convexity on graphs. We show results of NP-completeness and efficient algorithms for certain convexity parameters for this new one, as well as approximate algorithms. We introduce the concept of generalized threshold processes, where the results are NP-completeness and efficient algorithms for both non relaxed and relaxed versions. We study the problem of deciding whether a given graph admits a removal of a matching in order to destroy all cycles. We show that this problem is NP-hard even for subcubic graphs, but admits efficient solution for several graph classes. We study the problem of deciding whether a given graph admits a removal of a matching in order to destroy all odd cycles. We show that this problem is NP-hard even for planar graphs with bounded degree, but admits efficient solution for some graph classes. We also show parameterized results.Esta tese aborda problemas associados a conversões em grafos e de edição pela remoção de um emparelhamento. Estudamos processos f-reversíveis, que são aqueles associados a um valor de limiar para cada vértice e cuja dinâmica depende da quantidade de vizinhos com estado contrário para cada vértice. Estabelecemos um limite superior justo para o tamanho do período e transiente, caracterizamos todas as árvores que alcançam o transiente máximo em processos 2-reversíveis e mostramos que determinar o tamanho de um conjunto conversor mínimo é NP-difícil. Mostramos que o modelo AND-OR define uma convexidade sobre grafos. Mostramos resultados de NP-completude e algoritmos eficientes para certos parâmetros de convexidade para esta nova, assim como algoritmos aproximativos. Introduzimos o conceito de processos de limiar generalizados, onde mostramos resultados de NP-completude e algoritmos eficientes para ambas as versões não relaxada e relaxada. Estudamos o problema de decidir se um dado grafo admite uma remoção de um emparelhamento de modo a remover todos os ciclos. Mostramos que este problema é NP-difícil mesmo para grafos subcúbicos, mas admite solução eficiente para várias classes de grafos. Estudamos o problema de decidir se um dado grafo admite uma remoção de um emparelhamento de modo a remover todos os ciclos ímpares. Mostramos que este problema é NP-difícil mesmo para grafos planares com grau limitado, mas admite solução eficiente para algumas classes de grafos. Mostramos também resultados parametrizados

EDGE-EFFECT EXTERNALITIES: THEORETICAL AND EMPIRICAL IMPLICATIONS OF SPATIAL HETEROGENEITY

This dissertation examines the impacts of distance-dependent spatial externalities on patterns of economic activity in a free-market setting. This class of externalities, which include such examples as smog dispersal, pesticide drift, and habitat degradation from roads, are referred to as ``edge-effect externalities''. Under edge-effect externalities, economic optimality will require not only the correct allocation of land to different uses, but also the correct arrangement of land uses. However, an unregulated free market will potentially fail to achieve an efficient arrangement of land uses. Chapter 2 develops a spatially continuous one-dimensional model of edge-effect externalities. The model demonstrates that, while the externality creates an incentive for a recipient to distance himself from the generator, this distance is too small from a social standpoint. The model also demonstrates the potential for positive externalities between those impacted by the edge-effect externality. Chapter 3 formally demonstrates the potential for edge-effect externalities to create non-convexities in the production possibilities frontier. Further, it demonstrates that conflicting border per unit area is a summary measure of landscape efficiency under edge-effect externalities, but this ratio will vary with the number, shape, and geographic concentration of parcels in the externality-receiving use. Chapter 4 develops a two dimensional agent-based cellular automaton model of free-market land use in an economy impacted by edge-effect externalities. It demonstrates that in an unregulated free-market without bargaining, both Pareto-efficient and inefficient equilibrium landscape patterns are possible. Initial configurations of firms, permanent geographic features, and transportation costs will impact final outcomes. Chapter 5 tests the hypothesis that production patterns for California Certified Organic Farms reflect possible avoidance of negative spatial spillovers from surrounding conventional farms. Differences in parcel size, shape, and surroundings between C.C.O.F. and non-C.C.O.F. parcels are demonstrated. While inherently more vulnerable to losses from mandatory buffer zones, C.C.O.F. parcels are shown to potentially lose a much lower proportion of their land to buffers than non-C.C.O.F. parcels. However, very few C.C.O.F. farms border C.C.O.F. farms under separate management, indicating that growers have not managed to coordinate to capture potential positive externalities.Resource /Energy Economics and Policy,