Nils Bertschinger

• Goethe University Frankfurt

We study a game-theoretic variant of the maximum circulation problem. In a flow allocation game, we are given a directed flow network. Each node is a rational agent and can strategically allocate any incoming flow to the outgoing edges. Given the strategy choices of all agents, a maximal circulation that adheres to the chosen allocation strategies...

The complex interactions between algorithmic trading agents can have a severe influence on the functioning of our economy, as witnessed by recent banking crises and trading anomalies. A common phenomenon in these situations are fire sales, a contagious process of asset sales that trigger further sales. We study the existence and structure of equili...

We demonstrate how a systematic theory of complexity emerges from information theoretical concepts. The complexity of a structure may refer to the difficulty of its description, the encoding of its regularities or the relations between its elements, components or parts. All such measures can be and usually are quantified with the help of informatio...

In this paper, we examine the interlinkages among firms through a financial network where cross-holdings on both equity and debt are allowed. We relate mathematically the correlation among equities with the unconditional correlation of the assets, the values of their business assets and the sensitivity of the network, particularly the $\Delta$-Gree...

The statistical description and modeling of volatility plays a prominent role in econometrics, risk management and finance. GARCH and stochastic volatility models have been extensively studied and are routinely fitted to market data, albeit providing a phenomenological description only. In contrast, agent-based modeling starts from the premise that...

Volatility clustering and fat tails are prominently observed in financial markets. Here, we analyze the underlying mechanisms of three agent-based models explaining these stylized facts in terms of market instabilities and compare them on empirical grounds. To this end, we first develop a general framework for detecting tail events in stock markets...

Due to increased awareness of data protection and corresponding laws many data, especially involving sensitive personal information, are not publicly accessible. Accordingly, many data collecting agencies only release aggregated data, e.g. providing the mean and selected quantiles of population distributions. Yet, research and scientific understand...

We consider biological individuality in terms of information theoretic and graphical principles. Our purpose is to extract through an algorithmic decomposition system-environment boundaries supporting individuality. We infer or detect evolved individuals rather than assume that they exist. Given a set of consistent measurements over time, we discov...

Estimating covariances between financial assets plays an important role in risk management. In practice, when the sample size is small compared to the number of variables, the empirical estimate is known to be very unstable. Here, we propose a novel covariance estimator based on the Gaussian Process Latent Variable Model (GP-LVM). Our estimator can...

In their seminal work on systemic risk in financial markets, Eisenberg and Noe [13] proposed and studied a model with n firms embedded into a network of debt relations. We analyze this model from a game-theoretic point of view. Every firm is a rational agent in a directed graph that has an incentive to allocate payments in order to clear as much of...

The latest financial crisis has painfully revealed the dangers arising from a globally interconnected financial system. Conventional approaches based on the notion of the existence of equilibrium and those which rely on statistical forecasting have seen to be inadequate to describe financial systems in any reasonable way. A more natural approach is...

The latest financial crisis has painfully revealed the dangers arising from a globally interconnected financial system. Conventional approaches based on the notion of the existence of equilibrium and those which rely on statistical forecasting are seen to be inadequate to describe financial systems in any reasonable way. A more natural approach is...

In their seminal work on systemic risk in financial markets, Eisenberg and Noe proposed and studied a model with $n$ firms embedded into a network of debt relations. We analyze this model from a game-theoretic point of view. Every firm is a rational agent in a directed graph that has an incentive to allocate payments in order to clear as much of it...

We consider probabilistic PCA and related factor models from a Bayesian perspective. These models are in general not identifiable as the likelihood has a rotational symmetry. This gives rise to complicated posterior distributions with continuous subspaces of equal density and thus hinders efficiency of inference as well as interpretation of obtaine...

Stochastic volatility models describe asset prices [Formula: see text] as driven by an unobserved process capturing the random dynamics of volatility [Formula: see text]. We quantify how much information about [Formula: see text] can be inferred from asset prices [Formula: see text] in terms of Shannon’s mutual information in a twofold way: theoret...

Since the latest financial crisis, the idea of systemic risk has received considerable interest. In particular, contagion effects arising from cross-holdings between interconnected financial firms have been studied extensively. Drawing inspiration from the field of complex networks, these attempts are largely unaware of models and theories for cred...

Estimating covariances between financial assets plays an important role in risk management and optimal portfolio allocation. In practice, when the sample size is small compared to the number of variables, i.e. when considering a wide universe of assets over just a few years, this poses considerable challenges and the empirical estimate is known to...

The formulation of the Partial Information Decomposition (PID) framework by Williams and Beer in 2010 attracted a significant amount of attention to the problem of defining redundant (or shared), unique and synergistic (or complementary) components of mutual information that a set of source variables provides about a target. This attention resulted...

Volatility is a widely recognized measure of market risk. As volatility is not observed it has to be estimated from market prices, i.e., as the implied volatility from option prices. The volatility index VIX making volatility a tradeable asset in its own right is computed from near- and next-term put and call options on the S&P 500 with more than 2...

When the strict rationality underlying the Nash equilibria in game theory is relaxed, one arrives at the quantal response equilibria introduced by McKelvey and Palfrey. Here, the players are assigned parameters measuring their degree of rationality, and the resulting equilibria are Gibbs type distribution. This brings us into the realm of the expon...

We consider the problem of quantifying the information shared by a pair of random variables $X_{1},X_{2}$ about another variable $S$. We propose a new measure of shared information, called extractable shared information that is left monotonic; that is, the information shared about $S$ is bounded from below by the information shared about $f(S)$ for...

Suppose we have a pair of information channels, $\kappa_{1},\kappa_{2}$ with a common input. The Blackwell order is a partial order over channels that compares $\kappa_{1}$ and $\kappa_{2}$ by the maximal expected utility an agent can obtain when decisions are based on the outputs of $\kappa_{1}$ and $\kappa_{2}$. Equivalently, $\kappa_{1}$ is said...

Suppose we have a pair of information channels, $\kappa_{1},\kappa_{2}$, with a common input. The Blackwell order is a partial order over channels that compares $\kappa_{1}$ and $\kappa_{2}$ by the maximal expected utility an agent can obtain when decisions are based on the channel outputs. Equivalently, $\kappa_{1}$ is said to be Blackwell-inferio...

We consider the problem of quantifying the information shared by a pair of random variables $X_{1},X_{2}$ about another variable $S$. We propose a new measure of shared information, called extractable shared information, that is left monotonic; that is, the information shared about $S$ is bounded from below by the information shared about $f(S)$ fo...

We address the information content of European option prices about volatility in terms of the Fisher information matrix. We assume that observed option prices are centred on the theoretical price provided by Heston's model disturbed by additive Gaussian noise. We fit the likelihood function on the components of the VIX, i.e., near- and next-term pu...

Stochastic volatility models describe stock returns rt as driven by an unobserved process capturing the random dynamics of volatility vt. The present paper quantifies how much information about volatility vt and future stock returns can be inferred from past returns in stochastic volatility models in terms of Shannon's mutual information. In partic...

Stochastic volatility models describe stock returns $r_t$ as driven by an unobserved process capturing the random dynamics of volatility $v_t$. The present paper quantifies how much information about volatility $v_t$ and future stock returns can be inferred from past returns in stochastic volatility models in terms of Shannon's mutual information.

Stochastic volatility models describe asset prices $S_t$ as driven by an unobserved process capturing the random dynamics of volatility $\sigma_t$. Here, we quantify how much information about $\sigma_t$ can be inferred from asset prices $S_t$ in terms of Shannon's mutual information $I(S_t : \sigma_t)$. This motivates a careful numerical and analy...

Recently, a series of papers addressed the problem of decomposing the information of two random variables into shared information, unique information and synergistic information. Several measures were proposed, although still no consensus has been reached. Here, we compare these proposals with an older approach to define synergistic information bas...

The information that two random variables $Y$, $Z$ contain about a third random variable $X$ can have aspects of shared information (contained in both $Y$ and $Z$), of complementary information (only available from $(Y,Z)$ together) and of unique information (contained exclusively in either $Y$ or $Z$). Here, we study measures $\widetilde{SI}$ of s...

We quantify the relationship between the dynamics of a time-discrete dynamical system, the tent map T and its iterations T(m), and the induced dynamics at a symbolical level in information theoretical terms. The symbol dynamics, given by a binary string s of length m, is obtained by choosing a partition point [Formula: see text] and lumping togethe...

Levels of a complex system are characterized by the fact that they admit a closed functional description in terms of concepts and quantities intrinsic to that level. Several ideas have come up so far in order to make the notion of a closed description precise. In this paper, we present four of these approaches and investigate their mutual relations...

Blackwell's theorem shows the equivalence of two preorders on the set of information channels. Here, we restate, and slightly generalize, his result in terms of random variables. Furthermore, we prove that the corresponding partial order is not a lattice; that is, least upper bounds and greatest lower bounds do not exist.

In order to further a classifier construction, feature selection algorithms reduce the input dimensionality to a subset of the most informative features. Usually, such subset is fixed and chosen on the preprocessing step before the actual classification. However, when it is difficult to find a small number of features sufficient for classification...

What is the "value of information" in non-cooperative games with imperfect information? To answer this question, we propose to quantify information using concepts from Shannon's information theory. We then relate quantitative changes to the information structure of a game to changes in the expected utility of the players. Our approach is based on t...

We propose new measures of shared information, unique information and synergistic information that can be used to decompose the multi-information of a pair of random variables $(Y,Z)$ with a third random variable $X$. Our measures are motivated by an operational idea of unique information which suggests that shared information and unique informatio...

How can the information that a set {X 1,…,X n } of random variables contains about another random variable S be decomposed? To what extent do different subgroups provide the same, i.e. shared or redundant, information, carry unique information or interact for the emergence of synergistic information? Recently Williams and Beer proposed such a decom...

Feature selection helps to focus resources on relevant dimensions of input data. Usually, reducing the input dimensionality to the most informative features also simplifies subsequent tasks, such as classification. This is, for instance, important for systems operating in online mode under time constraints. However, when the training data is of lim...

We adapt the method used by Jaynes to derive the equilibria of statistical physics to instead derive equilibria of bounded rational game theory. We analyze the dependence of these equilibria on the parameters of the underlying game, focusing on hysteresis effects. In particular, we show that by gradually imposing individual-specific tax rates on th...

We develop a geometric approach to complexity based on the principle that complexity requires interactions at different scales of description. Complex systems are more than the sum of their parts of any size and not just more than the sum of their elements. Using information geometry, we therefore analyze the decomposition of a system in terms of a...

We analyze the time honored subject of emergence with modern tools from theoretical computer science, dynamical systems and complex systems theory. The features we identify are regularities that allow for a reduced description at a higher scale, coordination between the parts that enable a system to explore new regions of its state space, and the c...

We adapt the method used by Jaynes to derive the equilibria of statistical physics to instead derive equilibria of bounded rational game theory. We analyze the dependence of these equilibria on the parameters of the underlying game, focusing on hysteresis effects. In particular, we show that by gradually imposing individual-specific tax rates on th...

We investigate exponential families of random graph distributions as a framework for systematic quantification of structure in networks. In this paper we restrict ourselves to undirected unlabeled graphs. For these graphs, the counts of subgraphs with no more than k links are a sufficient statistics for the exponential families of graphs with inter...

The response to a knockout of a node is a characteristic feature of a networked dynamical system. Knockout resilience in the dynamics of the remaining nodes is a sign of robustness. Here we study the effect of knockouts for binary state sequences and their implementations in terms of Boolean threshold networks. Besides random sequences with biologi...

. We study how statistical complexity depends on the system size and how the complexity of the whole system relates to the complexity of its subsystems. We study this size dependence for two well-known complexity measures, the excess entropy of Grassberger and the neural complexity introduced by Tononi, Sporns and Edelman. We compare these results...

Measures of complexity are of immediate interest for the field of autonomous robots both as a means to classify the behavior and as an objective function for the autonomous development of robot behavior. In the present paper we consider predictive information in sensor space as a measure for the behavioral complexity of a two-wheel embodied robot...

We present a tentative proposal for a quantitative measure of autonomy. This is something that, surprisingly, is rarely found in the literature, even though autonomy is considered to be a basic concept in many disciplines, including artificial life. We work in an information theoretic setting for which the distinction between system and environment...

We develop an analysis of complex systems in terms of statistical correlations between the dynamics of its subsystems as a formal framework within which to understand processes of system differentiation.

The notion of closure plays a prominent role in systems theory where it is used to identify or define the system in distinction from its environment and to explain the autonomy of the system. Here, we present a quantitative measure, as opposed to the already existing qualitative notions, of closure. We shall elaborate upon the observation that cogn...

We develop a unifying approach for complexity measures, based on the principle that complexity requires interactions at different scales of description. Complex systems are more than the sum of their parts of any size, and not just more than the sum of their elements. We therefore analyze the decomposition of a system in terms of an interaction hie...

The aim of this study is to produce travelling waves in a planar net of artificial spiking neurons. Provided that the parameters of the waves--frequency, wavelength and orientation--can be sufficiently controlled, such a network can serve as a model of the spinal pattern generator for swimming and terrestrial quadruped locomotion. A previous implem...

Depending on the connectivity, recurrent networks of simple computational units can show very different types of dynamics, ranging from totally ordered to chaotic. We analyze how the type of dynamics (ordered or chaotic) exhibited by randomly connected networks of threshold gates driven by a time-varying input signal depends on the parameters descr...

In this paper we analyze the relationship between the computational ca- pabilities of randomly connected networks of threshold gates in the time- series domain and their dynamical properties. In particular we propose a complexity measure which we find to assume its highest values near the edge of chaos, i.e. the transition from ordered to chaotic d...

\ What makes a neural microcircuit computationally powerful? Or more precisely, which measurable quantities could explain why one microcircuit is better suited for a particular family of computational tasks than another microcircuit ? We propose in this article quantitative measures for evaluating the computational power and generalization capabil...

We develop an analysis of complex systems in terms of statistical cor- relations between the dynamics of its subsystems as a formal framework within which to understand processes of system dierentiation.