Mean-square stability and error analysis of implicit time-stepping schemes for linear parabolic SPDEs with multiplicative Wiener noise in the first derivative

In this article, we extend a Milstein finite difference scheme introduced in [Giles & Reisinger(2011)] for a certain linear stochastic partial differential equation (SPDE), to semi- and fully implicit timestepping as introduced by [Szpruch(2010)] for SDEs. We combine standard finite difference Fourier analysis for PDEs with the linear stability analysis in [Buckwar & Sickenberger(2011)] for SDEs, to analyse the stability and accuracy. The results show that Crank-Nicolson timestepping for the principal part of the drift with a partially implicit but negatively weighted double It\^o integral gives unconditional stability over all parameter values, and converges with the expected order in the mean-square sense. This opens up the possibility of local mesh refinement in the spatial domain, and we show experimentally that this can be beneficial in the presence of reduced regularity at boundaries

Unique Equilibrium in Two-Part Tariff Competition between Two-Sided Platforms

Two-sided market models in which platforms compete via two-part tariffs, i.e. a subscription and a per-transaction fee, are often plagued by a continuum of equilibria. This paper augments existing models by allowing for heterogeneous rading behavior of agents on both sides. We show that this simple method yields a unique equilibrium even in the limit as the heterogeneity vanishes. In case of competitive bottlenecks we find that in this equilibrium platforms benefit from the possibility to price discriminate if per-transaction costs are relatively large. This is the case because two-part tariffs allow platforms to better distribute these costs among the two sides. Under two-sided single-homing price discrimination hurts platforms if per-transaction fees can be negative

The non-locality of Markov chain approximations to two-dimensional diffusions

In this short paper, we consider discrete-time Markov chains on lattices as approximations to continuous-time diffusion processes. The approximations can be interpreted as finite difference schemes for the generator of the process. We derive conditions on the diffusion coefficients which permit transition probabilities to match locally first and second moments. We derive a novel formula which expresses how the matching becomes more difficult for larger (absolute) correlations and strongly anisotropic processes, such that instantaneous moves to more distant neighbours on the lattice have to be allowed. Roughly speaking, for non-zero correlations, the distance covered in one timestep is proportional to the ratio of volatilities in the two directions. We discuss the implications to Markov decision processes and the convergence analysis of approximations to Hamilton-Jacobi-Bellman equations in the Barles-Souganidis framework.Comment: Corrected two errata from previous and journal version: definition of R in (5) and summations in (7

The Effects of Product Bundling in Duopoly

This paper studies the incentives for multiproduct duopolists to sell their products as a bundle. It is shown that contrary to the monopoly case bundling may reduce profits and increase consumer rent. This is the case if consumers' reservation values are negatively correlated. The reason is that bundling reduces consumer heterogeneity and makes price competition more aggressive. This effect can dominate the sorting effect that is well known for the monopoly case. Firms are in a prisoner's dilemma situation because they would be better off without bundling. Despite the lower prices a welfare loss occurs because some consumers do not buy their prefered product which results in distributive inefficiency. If firms can influence the correlation by choosing their location in the product range they try to avoid negative correlation and choose minimal differentiation in one good

Exclusive vs Overlapping Viewers in Media Markets

This paper investigates competition for advertisers in media markets when viewers can subscribe to multiple channels. A central feature of the model is that channels are monopolists in selling advertising opportunities toward their exclusive viewers, but they can only obtain a competitive price for advertising opportunities to multi-homing viewers. Strategic incentives of firms in this setting are different than those in former models of media markets. If viewers can only watch one channel, then firms compete for marginal consumers by reducing the amount of advertising on their channels. In our model, channels have an incentive to increase levels of advertising, in order to reduce the overlap in viewership. We take an account of the differences between the predictions of the two types of models and find that our model is more consistent with recent developments in broadcasting markets. We also show that if channels can charge subscription fees on viewers, then symmetric firms can end up in an asymmetric equilibrium in which one collects all or most of its revenues from advertisers, while the other channel collects most of its revenues via viewer fees

Two-Sided Markets with Negative Externalities

This paper analyses a two-sided market in which two platforms compete against each other. One side, the advertisers, exerts a negative externality on the ther side, the users. It is shown that if platforms can charge advertisers only, a higher degree of competition for users can lead to higher profits because competition on the advertisers' side is reduced. If platforms can charge users as well, profits might increase or decrease, the latter because of increased competition through the additional instrument of the user fee. Nevertheless the equilibrium with user fee is more efficient

The Effects of Product Bundling in Duopoly

This paper studies the incentives for multiproduct duopolists to sell their products as a bundle. It is shown that contrary to the monopoly case bundling may reduce profits and increase consumer rent. This is the case if consumers' reservation values are negatively correlated. The reason is that bundling reduces consumer heterogeneity and makes price competition more aggressive. This effect can dominate the sorting effect that is well known for the monopoly case. Firms are in a prisoner's dilemma situation because they would be better off without bundling. Despite the lower prices a welfare loss occurs because some consumers do not buy their prefered product which results in distributive inefficiency. If firms can influence the correlation by choosing their location in the product range they try to avoid negative correlation and choose minimal differentiation in one good.Product Bundling ; Price Competition ; Price Discrimination ; Product Differentiation

Two-Sided Markets with Negative Externalities

This paper analyses a two-sided market in which two platforms compete against each other. One side, the advertisers, exerts a negative externality on the ther side, the users. It is shown that if platforms can charge advertisers only, a higher degree of competition for users can lead to higher profits because competition on the advertisers' side is reduced. If platforms can charge users as well, profits might increase or decrease, the latter because of increased competition through the additional instrument of the user fee. Nevertheless the equilibrium with user fee is more efficient.Negative Externalities ; Price Competition ; Two-Sided Markets