Bertrand’s paradox is a famous problem of probability theory, pointing to a possible inconsistency in Laplace’s principle of insufficient reason. In this article we show that Bertrand’s paradox contains two different problems: an “easy” problem and a “hard” problem. The easy problem can be solved by formulating Bertrand’s question in sufficiently precise terms, so allowing for a non ambiguous modelization of the entity subjected to the randomization. We then show that once the easy problem is settled, also the hard problem becomes solvable, provided Laplace’s principle of insufficient reason is applied not to the outcomes of the experiment, but to the different possible “ways of selecting” an interaction between the entity under investigation and that producing the randomization.

This paper investigates how economic shocks propagate and amplify through the input-output network connecting industrial sectors in developed economies. We study alternative models of diffusion on networks and we calibrate them using input-output data on real-world inter-sectoral dependencies for several European countries before the Great Depression. We show that the impact of economic shocks strongly depends on the nature of the shock and country size. Shocks that impact on final demand without changing production and the technological relationships between sectors have on average a large but very homogeneous impact on the economy.

We present a new type of spin market model, populated by hierarchical agents, represented as configurations of sites and arcs in an evolving network. We describe two analytic techniques for investigating the asymptotic behavior of this model: one based on the spectral theory of Markov chains and another exploiting contingent submartingales to construct a deterministic cellular automaton that approximates the stochastic dynamics. Our study of this system documents a phase transition between a sub-critical and a super-critical regime based on the values of a coupling constant that modulates the tradeoff between local majority and global minority forces. In conclusion, we offer a speculative socioeconomic interpretation of the resulting distributional properties of the system.

An extrapolation of the genetic complexity of organisms to earlier times suggests that life began before the Earth was formed. Life may have started from systems with single heritable elements that are functionally equivalent to a nucleotide. The genetic complexity, roughly measured by the number of non-redundant functional nucleotides, is expected to have grown exponentially due to several positive feedback factors: gene cooperation, duplication of genes with their subsequent specialization, and emergence of novel functional niches associated with existing genes. Linear regression of genetic complexity on a log scale extrapolated back to just one base pair suggests the time of the origin of life 9.7 billion years ago. This cosmic time scale for the evolution of life has important consequences: life took ca. 5 billion years to reach the complexity of bacteria; the environments in which life originated and evolved to the prokaryote stage may have been quite different from those envisaged on Earth; there was no intelligent life in our universe prior to the origin of Earth, thus Earth could not have been deliberately seeded with life by intelligent aliens; Earth was seeded by panspermia; experimental replication of the origin of life from scratch may have to emulate many cumulative rare events; and the Drake equation for guesstimating the number of civilizations in the universe is likely wrong, as intelligent life has just begun appearing in our universe.

This essay discusses the advantages of a probabilistic agent-based approach to questions in theoretical economics, from the nature of economic agents, to the nature of the equilibria supported by their interactions. One idea we propose is that “agents” are meta-individual, hierarchically structured objects, that include as irreducible components groupings of different dimensions. We also explore the effects of non-ergodicity, by constructing a simple stochastic model for the contingent nature of economic interactions.

There are two main approach to probability, one of set-theoretic character where probability is the measure of a set, and another one of linguistic character where probability is the degree of confidence in a proposition. In this work we give an unified algebraic treatment of these approaches through the concept of valued lattice, obtaining as a by-product a translation between them. Then we introduce the concept of partial valuation for DMF-algebras (De Morgan algebras with a single fixed point for negation), giving an algebraic setting for probability of partial events. We introduce the concept of partial probability for propositions, substituting classical logic with Kleene’s logic. In this case too we give a translation between set-theoretic and linguistic probability. Finally, we introduce the concept of conditional partial probability and prove a weak form of Bayes’s Theorem.

Generalized probabilistic theories (GPT) provide a general framework that includes classical and quantum theories. It is described by a cone C and its dual C ∗ . We show that the question whether some one way communication complexity problems can be solved within a GPT is equivalent to the recently introduced cone factorisation of the corresponding communication matrix M . Polytopes and optimising functions over polytopes arise in many areas of discrete mathematics. A conic extension of a polytope is the intersection of a cone C with an affine subspace whose projection onto the original space yields the desired polytope.

We have proposed novel measures based on the Kolmogorov complexity for use in complex system behavior studies and time series analysis. We have considered background of the Kolmogorov complexity and also we have discussed meaning of the physical as well as other complexities. To get better insights into the complexity of complex systems and time series analysis we have introduced the three novel measures based on the Kolmogorov complexity: (i) the Kolmogorov complexity spectrum, (ii) the Kolmogorov complexity spectrum highest value and (iii) the overall Kolmogorov complexity. The characteristics of these measures have been tested using a generalized logistic equation. Finally, the proposed measures have been applied on different time series originating from: the model output (the biochemical substance exchange in a multi-cell system), four different geophysical phenomena (dynamics of: river flow, long term precipitation, indoor 222Rn concentration and UV radiation dose) and economy (stock prices dynamics). Results which are obtained offer deeper insights into complexity of the system dynamics behavior and time series analysis when the proposed complexity measures are applied.

This report undertaken at the IUFM de Cr\’eteil in 1998 under the direction of Evelyne Barbin studies the birth of probability theory. Sources: correspondence between Pascal and Fermat, and Pascal’s “Treatise on Arithmetical Triangle”

The aim of this article is twofold. First, we shall review and analyse the Neo-Kantian justification for the application of probabilistic concepts in physics that was defended by Hans Reichenbach early in his career, notably in his dissertation of 1916. At first sight this Kantian approach seems to contrast sharply with Reichenbach’s later logical positivist, frequentist viewpoint. But, and this is our second goal, we shall attempt to show that there is an underlying continuity in Reichenbach’s thought: typical features of his early Kantian conceptions can still be recognized in his later work.