Price Competition, Fluctuations, and Welfare Guarantees

In various markets where sellers compete in price, price oscillations are observed rather than convergence to equilibrium. Such fluctuations have been empirically observed in the retail market for gasoline, in airline pricing and in the online sale of consumer goods. Motivated by this, we study a model of price competition in which an equilibrium rarely exists. We seek to analyze the welfare, despite the nonexistence of an equilibrium, and present welfare guarantees as a function of the market power of the sellers. We first study best response dynamics in markets with sellers that provide a homogeneous good, and show that except for a modest number of initial rounds, the welfare is guaranteed to be high. We consider two variations: in the first the sellers have full information about the valuation of the buyer. Here we show that if there are $n$ items available across all sellers and $n_{\max}$ is the maximum number of items controlled by any given seller, the ratio of the optimal welfare to the achieved welfare will be at most $\log(\frac{n}{n-n_{\max}+1})+1$. As the market power of the largest seller diminishes, the welfare becomes closer to optimal. In the second variation we consider an extended model where sellers have uncertainty about the buyer's valuation. Here we similarly show that the welfare improves as the market power of the largest seller decreases, yet with a worse ratio of $\frac{n}{n-n_{\max}+1}$. The exponential gap in welfare between the two variations quantifies the value of accurately learning the buyer valuation. Finally, we show that extending our results to heterogeneous goods in general is not possible. Even for the simple class of $k$-additive valuations, there exists a setting where the welfare approximates the optimal welfare within any non-zero factor only for $O(1/s)$ fraction of the time, where $s$ is the number of sellers

Layered Cellular Automata

Layered Cellular Automata (LCA) extends the concept of traditional cellular automata (CA) to model complex systems and phenomena. In LCA, each cell's next state is determined by the interaction of two layers of computation, allowing for more dynamic and realistic simulations. This thesis explores the design, dynamics, and applications of LCA, with a focus on its potential in pattern recognition and classification. The research begins by introducing the limitations of traditional CA in capturing the complexity of real-world systems. It then presents the concept of LCA, where layer 0 corresponds to a predefined model, and layer 1 represents the proposed model with additional influence. The interlayer rules, denoted as f and g, enable interactions not only from adjacent neighboring cells but also from some far-away neighboring cells, capturing long-range dependencies. The thesis explores various LCA models, including those based on averaging, maximization, minimization, and modified ECA neighborhoods. Additionally, the implementation of LCA on the 2-D cellular automaton Game of Life is discussed, showcasing intriguing patterns and behaviors. Through extensive experiments, the dynamics of different LCA models are analyzed, revealing their sensitivity to rule changes and block size variations. Convergent LCAs, which converge to fixed points from any initial configuration, are identified and used to design a two-class pattern classifier. Comparative evaluations demonstrate the competitive performance of the LCA-based classifier against existing algorithms. Theoretical analysis of LCA properties contributes to a deeper understanding of its computational capabilities and behaviors. The research also suggests potential future directions, such as exploring advanced LCA models, higher-dimensional simulations, and hybrid approaches integrating LCA with other computational models.Comment: This thesis represents the culmination of my M.Tech research, conducted under the guidance of Dr. Sukanta Das, Associate Professor at the Department of Information Technology, Indian Institute of Engineering Science and Technology, Shibpur, West Bengal, India. arXiv admin note: substantial text overlap with arXiv:2210.13971 by other author

Advances in Bayesian inference and stable optimization for large-scale machine learning problems

A core task in machine learning, and the topic of this thesis, is developing faster and more accurate methods of posterior inference in probabilistic models. The thesis has two components. The first explores using deterministic methods to improve the efficiency of Markov Chain Monte Carlo (MCMC) algorithms. We propose new MCMC algorithms that can use deterministic methods as a “prior” to bias MCMC proposals to be in areas of high posterior density, leading to highly efficient sampling. In Chapter 2 we develop such methods for continuous distributions, and in Chapter 3 for binary distributions. The resulting methods consistently outperform existing state-of-the-art sampling techniques, sometimes by several orders of magnitude. Chapter 4 uses similar ideas as in Chapters 2 and 3, but in the context of modeling the performance of left-handed players in one-on-one interactive sports. The second part of this thesis explores the use of stable stochastic gradient descent (SGD) methods for computing a maximum a posteriori (MAP) estimate in large-scale machine learning problems. In Chapter 5 we propose two such methods for softmax regression. The first is an implementation of Implicit SGD (ISGD), a stable but difficult to implement SGD method, and the second is a new SGD method specifically designed for optimizing a double-sum formulation of the softmax. Both methods comprehensively outperform the previous state-of-the-art on seven real world datasets. Inspired by the success of ISGD on the softmax, we investigate its application to neural networks in Chapter 6. In this chapter we present a novel layer-wise approximation of ISGD that has efficiently computable updates. Experiments show that the resulting method is more robust to high learning rates and generally outperforms standard backpropagation on a variety of tasks

Tools for the management and conservation of the European eel (Anguilla anguilla): an application to Santo André lagoon

The critical status of the European eel (Anguilla anguilla) population is an ongoing concern for fisheries and environmental managers. The European Eel Regulation approved in 2007 established the framework for the eel recovery, although incomplete knowledge about the stock hampers management that ensure the species’ sustainability. The present research investigated the dynamics of the European eel in a coastal lagoon, contributing to increase the knowledge about the species and to the assessment of the status of the stock in Portuguese inland waters. The information gathered on eel recruitment, and on the species exploitation and life history traits in a Portuguese brackish system, the Santo André lagoon, provided relevant data to develop stock assessment tools, and to analyse and understand the impact of management decisions on the local eel stock. Data on glass eel from the rivers Minho and Lis provide insights on recruitment trends in Portugal in recent decades and on the drivers of glass eel ingress into Portuguese coastal systems and were used as a proxy to the recruitment dynamics in Santo André lagoon. Results suggest that eel recruitment shows no evidence of a dramatic decline in Portugal, compared to trends reported for other European rivers. The dynamics of the eel population in continental waters was studied from a socio-economic and biological perspective in Santo André lagoon, where the eel yield is significant. The fishery statistics since the 1980’s and the field work conducted in 2011/12 and in 2015/16 have made possible to gather relevant information on the exploitation and biology and have provided the necessary data to characterize the local eel population. Data collected were used to calibrate a Bayesian state-space Integral Projection Model (SSIPM) that described the eel dynamics in the lagoon and estimated relevant eel biomass indicators. The eel population in the lagoon is male-skewed and characterized by a relatively young population (average age is 2.3 years), with only 13.7 % of individuals above 350 mm (13.7%). Eel growth in the lagoon is among the fastest reported for the species and explains the early onset of maturity in males, which metamorphose into silver eels, on average, at three years of age. The male dominance in the lagoon is most likely related to the high eel density estimates obtained with the SSIPM, ranging between 6 and 139 kg ha-1 in the years studied (2008-2017). The high productivity of the system supports one of the highest eel fishing yields in the species’ range, which represents a substantial share (50%) of local fishermen’s income. The increase in the catch per unit effort between 2006 and 2017 obtained from fisheries statistical data, was hypothesized to be a consequence of fishing effort reduction under the eel regulation, combined with an increase in natural recruitment. These results point out to the positive signs of the adaptive approach adopted in local fisheries management, although the eel regulation targets (silver eel biomass escapement) may be compromised. The closed nature of the lagoon translates into a late escapement of silver eels, which occurs only in the spring when the connection to the sea is artificially re-established, making them very vulnerable to capture by the fishery. Despite being an illegal activity, silver eel fishing occurs, showing that the management system must be improved. There are conditions to implement fisheries co-management in this socio-ecological system, and the estimated fishing yield and silver eel escapement under different management scenarios suggest that there are opportunities to improve the sustainability of the local European eel population. The SSIPM developed proved to be a useful tool to describe the European eel population in Santo André lagoon and can be used to support the evaluation of the Portuguese eel management plan, particularly in coastal brackish systems. The framework can be adapted to other eel habitats where eel abundance time-series are available, ideally incorporating information on sex ratio, growth, and length at silvering of the local population, since those life history traits are environmentally dependent

Tools for the management and conservation of the European eel (Anguilla anguilla): an application to Santo André lagoon

The critical status of the European eel (Anguilla anguilla) population is an ongoing concern for fisheries and environmental managers. The European Eel Regulation approved in 2007 established the framework for the eel recovery, although incomplete knowledge about the stock hampers management that ensure the species’ sustainability. The present research investigated the dynamics of the European eel in a coastal lagoon, contributing to increase the knowledge about the species and to the assessment of the status of the stock in Portuguese inland waters. The information gathered on eel recruitment, and on the species exploitation and life history traits in a Portuguese brackish system, the Santo André lagoon, provided relevant data to develop stock assessment tools, and to analyse and understand the impact of management decisions on the local eel stock. Data on glass eel from the rivers Minho and Lis provide insights on recruitment trends in Portugal in recent decades and on the drivers of glass eel ingress into Portuguese coastal systems and were used as a proxy to the recruitment dynamics in Santo André lagoon. Results suggest that eel recruitment shows no evidence of a dramatic decline in Portugal, compared to trends reported for other European rivers. The dynamics of the eel population in continental waters was studied from a socio-economic and biological perspective in Santo André lagoon, where the eel yield is significant. The fishery statistics since the 1980’s and the field work conducted in 2011/12 and in 2015/16 have made possible to gather relevant information on the exploitation and biology and have provided the necessary data to characterize the local eel population. Data collected were used to calibrate a Bayesian state-space Integral Projection Model (SSIPM) that described the eel dynamics in the lagoon and estimated relevant eel biomass indicators. The eel population in the lagoon is male-skewed and characterized by a relatively young population (average age is 2.3 years), with only 13.7 % of individuals above 350 mm (13.7%). Eel growth in the lagoon is among the fastest reported for the species and explains the early onset of maturity in males, which metamorphose into silver eels, on average, at three years of age. The male dominance in the lagoon is most likely related to the high eel density estimates obtained with the SSIPM, ranging between 6 and 139 kg ha-1 in the years studied (2008-2017). The high productivity of the system supports one of the highest eel fishing yields in the species’ range, which represents a substantial share (50%) of local fishermen’s income. The increase in the catch per unit effort between 2006 and 2017 obtained from fisheries statistical data, was hypothesized to be a consequence of fishing effort reduction under the eel regulation, combined with an increase in natural recruitment. These results point out to the positive signs of the adaptive approach adopted in local fisheries management, although the eel regulation targets (silver eel biomass escapement) may be compromised. The closed nature of the lagoon translates into a late escapement of silver eels, which occurs only in the spring when the connection to the sea is artificially re-established, making them very vulnerable to capture by the fishery. Despite being an illegal activity, silver eel fishing occurs, showing that the management system must be improved. There are conditions to implement fisheries co-management in this socio-ecological system, and the estimated fishing yield and silver eel escapement under different management scenarios suggest that there are opportunities to improve the sustainability of the local European eel population. The SSIPM developed proved to be a useful tool to describe the European eel population in Santo André lagoon and can be used to support the evaluation of the Portuguese eel management plan, particularly in coastal brackish systems. The framework can be adapted to other eel habitats where eel abundance time-series are available, ideally incorporating information on sex ratio, growth, and length at silvering of the local population, since those life history traits are environmentally dependent

Advances in scalable learning and sampling of unnormalised models

We study probabilistic models that are known incompletely, up to an intractable normalising constant. To reap the full benefit of such models, two tasks must be solved: learning and sampling. These two tasks have been subject to decades of research, and yet significant challenges still persist. Traditional approaches often suffer from poor scalability with respect to dimensionality and model-complexity, generally rendering them inapplicable to models parameterised by deep neural networks. In this thesis, we contribute a new set of methods for addressing this scalability problem. We first explore the problem of learning unnormalised models. Our investigation begins with a well-known learning principle, Noise-contrastive Estimation, whose underlying mechanism is that of density-ratio estimation. By examining why existing density-ratio estimators scale poorly, we identify a new framework, telescoping density-ratio estimation (TRE), that can learn ratios between highly dissimilar densities in high-dimensional spaces. Our experiments demonstrate that TRE not only yields substantial improvements for the learning of deep unnormalised models, but can do the same for a broader set of tasks including mutual information estimation and representation learning. Subsequently, we explore the problem of sampling unnormalised models. A large literature on Markov chain Monte Carlo (MCMC) can be leveraged here, and in continuous domains, gradient-based samplers such as Metropolis-adjusted Langevin algorithm (MALA) and Hamiltonian Monte Carlo are excellent options. However, there has been substantially less progress in MCMC for discrete domains. To advance this subfield, we introduce several discrete Metropolis-Hastings samplers that are conceptually inspired by MALA, and demonstrate their strong empirical performance across a range of challenging sampling tasks

From control to constraint: a study of reproduction in the eusocial honeybee and the solitary red mason bee

The major evolutionary transition from solitary to eusocial living is hallmarked by the reproductive division of labour. I investigated mechanisms underlying reproductive control in a solitary bee (Osmia bicornis), with the aim of informing how and why such mechanisms were co-opted into reproductively constraining workers in a eusocial species (Apis mellifera, mrca 95 mya; Peters et al., 2017). I start out by introducing the problems of reproductive constraint and the evolution of eusociality (Chapter 1). In order to test functional links and perform manipulation, it is imperative to establish a reference species within the laboratory. To address the lack of a temperate European solitary model species, I attempted to establish O. bicornis in a laboratory environment (Chapter 3). Preliminary erratic successes of nesting and egg-laying behaviour were achieved, and future recommendations were laid out. To further facilitate O. bicornis as a model species; microsatellite markers were mined, designed, tested and validated in collaboration with the NERC Biomolecular analysis facility in Sheffield (Chapter 4). The broad applicability of these markers is discussed. The capricious nature of laboratory egg-laying necessitated appraising reproductive control directly. To enable assaying oogenesis, I performed the first microstructural study of the O. bicornis ovary (Chapter 5). Since mating plays an important role in the ovary activation of eusocial queens and other insect species, I concurrently examined the effect of mating status on the ovary of O. bicornis (Chapter 5) — with special reference to the potential role of mating status in reproductive constraint. Finally, I investigated how a known mechanism of reproductive constraint (Duncan et al., 2016) operates in the related solitary bee, to ascertain its ancestral role (Chapter 6).The mechanism was found to be reversed in O. bicornis. Chapter 7 places the overall findings within their wider context, and outlines future avenues of research