We view a conic optimization problem that has a unique solution as a map from its data to its solution. If sufficient regularity conditions hold at a solution point, namely that the implicit function theorem applies to the normalized residual function of [Busseti et al., 2018], the problem solution map is differentiable. We obtain the derivative, in the form of an abstract linear operator. This applies to any convex optimization problem in conic form, while a previous result [Amos et al., 2016] studied strictly convex quadratic programs. Such differentiable problems can be used, for example, in machine learning, control, and related areas, as a layer in an end-to-end learning and control procedure, for backpropagation. We accompany this note with a lightweight Python implementation which can handle problems with the cone constraints commonly used in practice.

The main focus of researchers in energy markets is typically placed on the analysis of the supply side. The demand side, despite its critical importance, still requires a more profound academic investigation. In particular, the number of studies on the demand elasticity in a wholesale market is limited to merely several pieces. In this paper we extend this field of study and propose a new method for determining the demand elasticity. More specifically, we decompose the data observed in the wholesale market into individual supply and demand schedules of the market participants. These schedules are then used to construct a fundamental model of an electricity market. The main feature of our fundamental model is that our demand curve is not perfectly inelastic and is a better approximation of the true demand curve. Hence, our approach allows us to precisely compute the elasticity of the demand curve in an electricity market. Moreover, we obtain a deeper understanding of the bidding behavior of the market participants and hence can make more thorough inferences about the functioning of an electricity market or produce more accurate forecasts.

In this paper we propose an efficient method to compute the price of American basket options, based on Machine Learning and Monte Carlo simulations. Specifically, the options we consider are written on a basket of assets, each of them following a Black-Scholes dynamics. The method we propose is a backward dynamic programming algorithm which considers a finite number of uniformly distributed exercise dates. On these dates, the value of the option is computed as the maximum between the exercise value and the continuation value, which is approximated via Gaussian Process Regression. Specifically, we consider a finite number of points, each of them representing the values reached by the underlying at a certain time. First of all, we compute the continuation value only for these points by means of Monte Carlo simulations and then we employ Gaussian Process Regression to approximate the whole continuation value function. Numerical tests show that the algorithm is fast and reliable and it can handle also American options on very large baskets of assets, overcoming the problem of the curse of dimensionality.

A new approach to obtaining market--directional information, based on a non--stationary solution to the dynamic equation "future price tends to the value that maximizes the number of shares traded per unit time" [1] is presented. In our previous work[2], we established that it is the share execution flow ($I=dV/dt$) and not the share trading volume ($V$) that is the driving force of the market, and that asset prices are much more sensitive to the execution flow $I$ (the dynamic impact) than to the traded volume $V$ (the regular impact). In this paper, an important advancement is achieved: we define the "scalp--price" ${\cal P}$ as the sum of only those price moves that are relevant to market dynamics; the criterion of relevance is a high $I$. Thus, only "follow the market" (and not "little bounce") events are included in ${\cal P}$. Changes in the scalp--price defined this way indicate a market trend change --- not a bear market rally or a bull market sell--off. The software calculating the scalp--price given market observations triples (time, execution price, shares traded) is available from the authors.

When multiple firms are simultaneously running experiments on a platform, the treatment effects for one firm may depend on the experimentation policies of others. This paper presents a set of causal estimands that are relevant to such an environment. We also present an experimental design that is suitable for facilitating experimentation across multiple competitors in such an environment. Together, these can be used by a platform to run experiments "as a service," on behalf of its participating firms. We show that the causal estimands we develop are identified nonparametrically by the variation induced by the design, and present two scalable estimators that help measure them in typical high-dimensional situations. We implement the design on the advertising platform of JD.com, an eCommerce company, which is also a publisher of digital ads in China. We discuss how the design is engineered within the platform's auction-driven ad-allocation system, which is typical of modern, digital advertising marketplaces. Finally, we present results from a parallel experiment involving 16 advertisers and millions of JD.com users. These results showcase the importance of accommodating a role for interactions across experimenters and demonstrates the viability of the framework.

Increased data gathering capacity, together with the spreading of data analytics techniques, has allowed an unprecedented concentration of information related to the individuals' preferences in the hands of a few gatekeepers. In such context, the traditional economic literature has been attempting to frame all the data-driven economy features. Such features, although being able to bring about a more efficient matching of people and relevant purchase opportunities, also result into distortions and disequilibria, up to market failures. Data-economy market disequilibria can be decrypted by leveraging on some of the known network properties, thus obtaining general results suitable for building a new theoretical framework for economic phenomena. Starting from the hypothesis that a digital company can always benefit from an underlying network of consumers or items related to its market, their representation can indeed provide significant competitive advantages, also enhancing the platforms' capacity to implement discriminatory practices by means of an increased ability to estimate individuals' preferences. In the present paper, we propose a measure called Information Patrimony, considering the amount of information available within the system and we look into how platforms may exploit data stemming from connected profiles within a network, with a view to obtaining competitive advantages. Such information flow may eventually allow to envisage the emergence of a new hybrid price discrimination pattern, through which platforms may influence and steer individuals' purchase choices, as well as to apply different prices to different customers.

Anthropologists have long appreciated that single-layer networks are insufficient descriptions of human interactions---individuals are embedded in complex networks with dependencies. One debate explicitly about this surrounds food sharing. Some argue that failing to find reciprocal food sharing means that some process other than reciprocity must be occurring, whereas others argue for models that allow reciprocity to span domains. The analysis of multi-dimensional social networks has recently garnered the attention of the mathematics and physics communities. Multilayer networks are ubiquitous and have consequences, so processes giving rise to them are important social phenomena. Recent models of these processes show how ignoring layer interdependencies can lead one to miss why a layer formed the way it did, and/or draw erroneous conclusions. Understanding the structuring processes that underlie multiplex networks will help understand increasingly rich datasets, which give better, richer, and more accurate pictures of social interactions.