77 research outputs found

    Approximate Efficient Solutions of the Vector Optimization Problem on Hadamard Manifolds via Vector Variational Inequalities

    Get PDF
    This article has two objectives. Firstly, we use the vector variational-like inequalities problems to achieve local approximate (weakly) efficient solutions of the vector optimization problem within the novel field of the Hadamard manifolds. Previously, we introduced the concepts of generalized approximate geodesic convex functions and illustrated them with examples. We see the minimum requirements under which critical points, solutions of Stampacchia, and Minty weak variational-like inequalities and local approximate weakly efficient solutions can be identified, extending previous results from the literature for linear Euclidean spaces. Secondly, we show an economical application, again using solutions of the variational problems to identify Stackelberg equilibrium points on Hadamard manifolds and under geodesic convexity assumptions

    Moral hazard and private monitoring

    Get PDF
    game theory;moral hazard;information

    Characterization of transport optimizers via graphs and applications to Stackelberg-Cournot-Nash equilibria

    Full text link
    We introduce graphs associated to transport problems between discrete marginals, that allow to characterize the set of all optimizers given one primal optimizer. In particular, we establish that connectivity of those graphs is a necessary and sufficient condition for uniqueness of the dual optimizers. Moreover, we provide an algorithm that can efficiently compute the dual optimizer that is the limit, as the regularization parameter goes to zero, of the dual entropic optimizers. Our results find an application in a Stackelberg-Cournot-Nash game, for which we obtain existence and characterization of the equilibria

    Economics of Electric Vehicle Charging: A Game Theoretic Approach

    Full text link
    In this paper, the problem of grid-to-vehicle energy exchange between a smart grid and plug-in electric vehicle groups (PEVGs) is studied using a noncooperative Stackelberg game. In this game, on the one hand, the smart grid that acts as a leader, needs to decide on its price so as to optimize its revenue while ensuring the PEVGs' participation. On the other hand, the PEVGs, which act as followers, need to decide on their charging strategies so as to optimize a tradeoff between the benefit from battery charging and the associated cost. Using variational inequalities, it is shown that the proposed game possesses a socially optimal Stackelberg equilibrium in which the grid optimizes its price while the PEVGs choose their equilibrium strategies. A distributed algorithm that enables the PEVGs and the smart grid to reach this equilibrium is proposed and assessed by extensive simulations. Further, the model is extended to a time-varying case that can incorporate and handle slowly varying environments

    A Model for the Global Crude Oil Market Using a Multi-Pool MCP Approach

    Get PDF
    This paper proposes a partial equilibrium model to describe the global crude oil market. Pricing on the global crude oil market is strongly influenced by price indices such as WTI (USA) and Brent (Northwest Europe). Adapting an approach for pool-based electricity markets, the model captures the particularities of these benchmark price indices and their influence on the market of physical oil. This approach is compared to a model with bilateral trade relations as is traditionally used in models of energy markets. With these two model approaches, we compute the equilibrium solutions for several market power scenarios to investigate whether the multi-pool approach may be better suited than the bilateral trade model to describe the crude oil market. The pool-based approach yields, in general, results closer to observed quantities and prices, with the best fit obtained by the scenario of an OPEC oligopoly. We conclude that the price indices indeed are important on the global crude market in determining the prices and flows, and that OPEC effectively exerts market power, but in a non-cooperative way.crude oil, market structure, cartel, pool market, simulation model

    Price competition and market concentration: An experimental study

    Get PDF
    price competition;industrial concentration

    Modelling and Analysis of Global Coal Markets

    Get PDF
    The thesis comprises four interrelated essays featuring modelling and analysis of coal markets. Each of the four essays has a dedicated chapter in this thesis. Chapters 2 to 4 have, from a topical perspective, a backward-looking focus and deal with explaining recent market outcomes in the international coal trade. The findings of those essays may serve as guidance for assessing current coal market outcomes as well as expected market outcomes in the near to medium-term future. Chapter 5 has a forward-looking focus and builds a bridge between explaining recent market outcomes and projecting long-term market equilibria. Chapter 2, Strategic Behaviour in International Metallurgical Coal Markets, deals with market conduct of large exporters in the market of coals used in steel-making in the period 2008 to 2010. In this essay I analyse whether prices and trade-flows in the international market for metallurgical coals were subject to non-competitive conduct in the period 2008 to 2010. To do so, I develop mathematical programming models - a Stackelberg model, two varieties of a Cournot model, and a perfect competition model - for computing spatial equilibria in international resource markets. Results are analysed with various statistical measures to assess the prediction accuracy of the models. The results show that real market equilibria cannot be reproduced with a competitive model. However, real market outcomes can be accurately simulated with the non-competitive models, suggesting that market equilibria in the international metallurgical coal trade were subject to the strategic behaviour of coal exporters. Chapter 3 and chapter 4 deal with market power issues in the steam coal trade in the period 2006 to 2008. Steam coals are typically used to produce steam either for electricity generation or for heating purposes. In Chapter 3 we analyse market behaviour of key exporting countries in the steam coal trade. This chapter features the essay Market Structure Scenarios in International Steam Coal Trade. In this paper, we analyse steam coal market equilibria in the years 2006 and 2008 by testing for two possible market structure scenarios: perfect competition and an oligopoly setup with major exporters competing in quantities. The assumed oligopoly scenario cannot explain market equilibria for any year. While we find that the competitive model simulates market equilibria well in 2006, the competitive model is not able to reproduce real market outcomes in 2008. The analysis shows that not all available supply capacity was utilised in 2008. We conclude that either unknown capacity bottlenecks or more sophisticated non-competitive strategies were the cause for the high prices in 2008. Chapter 4 builds upon the findings of the analysis in chapter 3 and adds a more detailed representation of domestic markets. The corresponding essay is titled Nations as Strategic Players in Global Commodity Markets: Evidence from World Coal Trade. In this chapter we explore the hypothesis that export policies and trade patterns of national players in the steam coal market are consistent with non-competitive market behaviour. We test this hypothesis by developing a static equilibrium model which is able to model coal producing nations as strategic players. We explicitly account for integrated seaborne trade and domestic markets. The global steam coal market is simulated under several imperfect market structure setups. We find that trade and prices of a China - Indonesia duopoly fits the real market outcome best and that real Chinese export quotas in 2008 were consistent with simulated exports under a Cournot-Nash strategy. Chapter 5 looks at the long-term effect of Chinese energy system planning decisions. The time horizon is 2006 to 2030. The analysis in this chapter combines a dynamic equilibrium model with the scenario analysis technique. The corresponding essay is titled Coal Lumps vs. Electrons: How Do Chinese Bulk Energy Transport Decisions Affect the Global Steam Coal Market? The essay demonstrates the ways in which different Chinese bulk energy transport strategies affect the future steam coal market in China and in the rest of the world. Increasing Chinese energy demand will require additional energy to be transported from the supply to the demand regions. If domestic transport costs escalate, Chinese coal consumers could increasingly import coal. We analyse two settings: one in which coal is increasingly transported by rail and one in which coal energy is transported as electricity. A key finding is that if coal were converted into electricity early in the supply chain, worldwide marginal costs off coal supply would be lower than if coal were hauled by train. Furthermore, China's dependence on imports is significantly reduced in this context. Allocation of welfare changes particularly in favour of Chinese consumers while rents of international producers decrease

    Multi-player games in the era of machine learning

    Full text link
    Parmi tous les jeux de société joués par les humains au cours de l’histoire, le jeu de go était considéré comme l’un des plus difficiles à maîtriser par un programme informatique [Van Den Herik et al., 2002]; Jusqu’à ce que ce ne soit plus le cas [Silveret al., 2016]. Cette percée révolutionnaire [Müller, 2002, Van Den Herik et al., 2002] fût le fruit d’une combinaison sophistiquée de Recherche arborescente Monte-Carlo et de techniques d’apprentissage automatique pour évaluer les positions du jeu, mettant en lumière le grand potentiel de l’apprentissage automatique pour résoudre des jeux. L’apprentissage antagoniste, un cas particulier de l’optimisation multiobjective, est un outil de plus en plus utile dans l’apprentissage automatique. Par exemple, les jeux à deux joueurs et à somme nulle sont importants dans le domain des réseaux génératifs antagonistes [Goodfellow et al., 2014] ainsi que pour maîtriser des jeux comme le Go ou le Poker en s’entraînant contre lui-même [Silver et al., 2017, Brown andSandholm, 2017]. Un résultat classique de la théorie des jeux indique que les jeux convexes-concaves ont toujours un équilibre [Neumann, 1928]. Étonnamment, les praticiens en apprentissage automatique entrainent avec succès une seule paire de réseaux de neurones dont l’objectif est un problème de minimax non-convexe et non-concave alors que pour une telle fonction de gain, l’existence d’un équilibre de Nash n’est pas garantie en général. Ce travail est une tentative d'établir une solide base théorique pour l’apprentissage dans les jeux. La première contribution explore le théorème minimax pour une classe particulière de jeux non-convexes et non-concaves qui englobe les réseaux génératifs antagonistes. Cette classe correspond à un ensemble de jeux à deux joueurs et a somme nulle joués avec des réseaux de neurones. Les deuxième et troisième contributions étudient l’optimisation des problèmes minimax, et plus généralement, les inégalités variationnelles dans le cadre de l’apprentissage automatique. Bien que la méthode standard de descente de gradient ne parvienne pas à converger vers l’équilibre de Nash de jeux convexes-concaves simples, il existe des moyens d’utiliser des gradients pour obtenir des méthodes qui convergent. Nous étudierons plusieurs techniques telles que l’extrapolation, la moyenne et la quantité de mouvement à paramètre négatif. La quatrième contribution fournit une étude empirique du comportement pratique des réseaux génératifs antagonistes. Dans les deuxième et troisième contributions, nous diagnostiquons que la méthode du gradient échoue lorsque le champ de vecteur du jeu est fortement rotatif. Cependant, une telle situation peut décrire un pire des cas qui ne se produit pas dans la pratique. Nous fournissons de nouveaux outils de visualisation afin d’évaluer si nous pouvons détecter des rotations dans comportement pratique des réseaux génératifs antagonistes.Among all the historical board games played by humans, the game of go was considered one of the most difficult to master by a computer program [Van Den Heriket al., 2002]; Until it was not [Silver et al., 2016]. This odds-breaking break-through [Müller, 2002, Van Den Herik et al., 2002] came from a sophisticated combination of Monte Carlo tree search and machine learning techniques to evaluate positions, shedding light upon the high potential of machine learning to solve games. Adversarial training, a special case of multiobjective optimization, is an increasingly useful tool in machine learning. For example, two-player zero-sum games are important for generative modeling (GANs) [Goodfellow et al., 2014] and mastering games like Go or Poker via self-play [Silver et al., 2017, Brown and Sandholm,2017]. A classic result in Game Theory states that convex-concave games always have an equilibrium [Neumann, 1928]. Surprisingly, machine learning practitioners successfully train a single pair of neural networks whose objective is a nonconvex-nonconcave minimax problem while for such a payoff function, the existence of a Nash equilibrium is not guaranteed in general. This work is an attempt to put learning in games on a firm theoretical foundation. The first contribution explores minimax theorems for a particular class of nonconvex-nonconcave games that encompasses generative adversarial networks. The proposed result is an approximate minimax theorem for two-player zero-sum games played with neural networks, including WGAN, StarCrat II, and Blotto game. Our findings rely on the fact that despite being nonconcave-nonconvex with respect to the neural networks parameters, the payoff of these games are concave-convex with respect to the actual functions (or distributions) parametrized by these neural networks. The second and third contributions study the optimization of minimax problems, and more generally, variational inequalities in the context of machine learning. While the standard gradient descent-ascent method fails to converge to the Nash equilibrium of simple convex-concave games, there exist ways to use gradients to obtain methods that converge. We investigate several techniques such as extrapolation, averaging and negative momentum. We explore these techniques experimentally by proposing a state-of-the-art (at the time of publication) optimizer for GANs called ExtraAdam. We also prove new convergence results for Extrapolation from the past, originally proposed by Popov [1980], as well as for gradient method with negative momentum. The fourth contribution provides an empirical study of the practical landscape of GANs. In the second and third contributions, we diagnose that the gradient method breaks when the game’s vector field is highly rotational. However, such a situation may describe a worst-case that does not occur in practice. We provide new visualization tools in order to exhibit rotations in practical GAN landscapes. In this contribution, we show empirically that the training of GANs exhibits significant rotations around Local Stable Stationary Points (LSSP), and we provide empirical evidence that GAN training converges to a stable stationary point, which is a saddle point for the generator loss, not a minimum, while still achieving excellent performance
    • …
    corecore