Homotopy Methods to Compute Equilibria in Game Theory

This paper presents a complete survey of the use of homotopy methods in game theory.Homotopies allow for a robust computation of game-theoretic equilibria and their refinements. Homotopies are also suitable to compute equilibria that are selected by variousselection theories. We present all relevant techniques underlying homotopy algorithms.We give detailed expositions of the Lemke-Howson algorithm and the Van den Elzen-Talman algorithm to compute Nash equilibria in 2-person games, and the Herings-Vanden Elzen, Herings-Peeters, and McKelvey-Palfrey algorithms to compute Nash equilibriain general n-person games.operations research and management science;

A Differentiable Homotopy to Compute Nash Equilibria of n-Person Games

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 an unique feasible path. The numerical performance of the algorithm is illustrated by means of several examples.microeconomics ;

Lubricant sensitivity in function of paddle movement in the forced feeder of a high-speed tablet press

Context: The negative impact of magnesium stearate (MgSt) on the hardness of tablets is a well-known phenomenon, but the influence of paddle movement in the forced feeder on the lubricant effect during tablet compression is often neglected. Objective: The purpose of this research was to investigate the influence of paddle speed in the forced feeder on tablet tensile strength (TS).Materials and methods: Mixtures of microcrystalline cellulose (MCC) and MgSt (0.5%) were blended using different methods (low & high shear). After blending, the formulations were compressed into tablets. All parameters of the tableting cycle were kept constant except the speed of the paddles in the forced feeder. Results and discussion: The blending technique affected the sensitivity of the formulation to the paddle speed. The TS of pure MCC tablets did not change in function of paddle speed, while tablets prepared by low shear mixing became softer at higher paddle speed. The TS of tablets manufactured using the high-shear mixed blend was low and did not vary in function of paddle speed, suggesting that overlubrication already occurred during the initial blending step. Furthermore, analysis of the machine parameters allowed evaluation of the influence of the paddles on the flowability, initial packing, and compactability of the powder mixtures. Conclusion: The results elucidated that during manufacturing of tablets using MgSt-containing blends care should not only be taken during the blending step prior to tableting, but also during the tableting process itself, as paddle speed can affect tablet TS, a critical quality attribute

Reduction of tablet weight variability by optimizing paddle speed in the forced feeder of a high-speed rotary tablet press

Context: Tableting is a complex process due to the large number of process parameters that can be varied. Knowledge and understanding of the influence of these parameters on the final product quality is of great importance for the industry, allowing economic efficiency and parametric release. Objective: The aim of this study was to investigate the influence of paddle speeds and fill depth at different tableting speeds on the weight and weight variability of tablets. Materials and methods: Two excipients possessing different flow behavior, microcrystalline cellulose (MCC) and dibasic calcium phosphate dihydrate (DCP), were selected as model powders. Tablets were manufactured via a high-speed rotary tablet press using design of experiments (DoE). During each experiment also the volume of powder in the forced feeder was measured. Results and discussion: Analysis of the DoE revealed that paddle speeds are of minor importance for tablet weight but significantly affect volume of powder inside the feeder in case of powders with excellent flowability (DCP). The opposite effect of paddle speed was observed for fairly flowing powders (MCC). Tableting speed played a role in weight and weight variability, whereas changing fill depth exclusively influenced tablet weight. Conclusion: The DoE approach allowed predicting the optimum combination of process parameters leading to minimum tablet weight variability. Monte Carlo simulations allowed assessing the probability to exceed the acceptable response limits if factor settings were varied around their optimum. This multi-dimensional combination and interaction of input variables leading to response criteria with acceptable probability reflected the design space

Competition against peer-to-peer networks

In this paper, we consider the competition of providers of information products against P2P networks that offer illegal versions of the information products. Depending on the generic cost factor of downloading—incorporating factors including, among other things, the degree of legal enforcement of intellectual property rights—we find that the firm may employ pricing strategies to either deter the entry of a network or to accommodate it. In the latter case, we find that the equilibrium price moves in the opposite direction of the generic cost factor of downloading. This counter-intuitive result corresponds to a very subtle form of platform competition between the firm and the network. Furthermore, profits for the firm ambiguously decrease when the generic cost factor of downloading declines, whereas total welfare unambiguously increases. This implies that it may well be welfare enhancing to relax the legal enforcements of intellectual property rights.Strategy;

Residential equilibrium in a multifractal metropolitan area

A residential location model derived from urban economics is combined with the geometry of a multifractal Sierpinski carpet to represent and model a metropolitan area. This area is made up of a system of built-up patches hierarchically organised around a city centre, and green areas arranged in an inverse hierarchical order (large open-spaces in the periphery). An analytical solution is obtained by using a specific geographic coding system for computing distances. The values of the parameters used in the model are based on the French medium sized metropolitan areas; a realistic benchmark is proposed and comparative-statics simulations are performed. The results show that the French peri-urbanisation process (which took place from 1970 onward) can be explained by an increase in income and a reduction in transport costs. Nevertheless, changes in household preferences, in particular an increased taste for open spaces, can also contribute to urban sprawl by making the gradient of land rents less steep and by making peripheral household locations more desirableperi-urban, residential localisation, fractal geometry, amenities