An upper bound on the number of zeros of a piecewise polinomial function

A precise tie between a univariate spline's knots and its zeros abundance and dissemination is formulated. As an application, a conjecture formulated by De Concini and Procesi is shown to be true in the special univariate, unimodular case. As a supplement, the same conjecture is shown, through computing a counterexample, to be false when unimodularity hypothesis is dropped.Comment: 1 figur

Variety, Consumption and Growth

This paper attempts at fixing some guide-posts on the relation between variety, consumption and growth, while abstracting from the well known effect that variety may exert on productivity, through specialization. A mechanism is first described, through which preference for variety expressed by intertemporally-optimizing consumers perfectly predicting the endogenously growing future consumption opportunities can cause faster steady-state growth. The mechanism amounts to a substitution of future for present consumption causing a higher steady-state savings ratio and is most naturally, but not exclusively, embedded in the intertemporal-equilibrium approach to growth modeling. The paper shows that this growth enhancing effect of preference for variety may not be unambiguous, if the creation of new goods is endogenous and costly. Some of the results obtained in this part of the paper hinge upon the assumption that there are constant returns to the endogenous factor, all factors are producible and that each type of variety can be used both as a consumption good and as a intermediate good in the production of capital by competitive firms. Dissatisfaction with the approach to preference for variety and innovation within the mechanism above is then motivated. The approach is oblivious of endogenous preference formation and the relation between innovation, consumption knowledge and consumption activities. Some research indications concerning long-term growth analysis in a world of endogenous preference formation are then drawn.

Budget Imbalance Criteria for Auctions: A Formalized Theorem

We present an original theorem in auction theory: it specifies general conditions under which the sum of the payments of all bidders is necessarily not identically zero, and more generally not constant. Moreover, it explicitly supplies a construction for a finite minimal set of possible bids on which such a sum is not constant. In particular, this theorem applies to the important case of a second-price Vickrey auction, where it reduces to a basic result of which a novel proof is given. To enhance the confidence in this new theorem, it has been formalized in Isabelle/HOL: the main results and definitions of the formal proof are re- produced here in common mathematical language, and are accompanied by an informal discussion about the underlying ideas.Comment: 6th Podlasie Conference on Mathematics 2014, 11 page

Drift and equilibrium selection with human and computer players

The theory of drift (Binmore and Samuelson 1999) concerns equilibrium selection in which second-order disturbances may have first-order effects in the emergence of one equilibrium over the other. We provided experimental evidence with human players supporting the model in Caminati, Innocenti and Ricciuti (2006). In this paper we test it with conditioning by computer players. When computers are removed and humans are matched against each other, the comparative static properties of the model are confirmed.evolutionary games

Drift and Equilibrium Selection with Human and Computer Players

The theory of drift (Binmore and Samuelson 1999) concerns equilibrium selection in which second order disturbances may have first-order effects in the emergence of one equilibrium over the other. We provided experimental evidence with human players supporting the model in Caminati, Innocenti and Ricciuti (2006). In this paper we test it with conditioning by computer players. When computers are removed and humans are matched against each other, the comparative static properties of the model are confirmed.drift, equilibrium selection, evolutionary games, experiments.

Harrodian Instability and Learning

It is a standard result in macroeconomic models populated by boundedly-rational agents that the stability of equilibria is left undetermined by the deep structural parameters, because it depends upon the nature of the prevailing beliefs. Apparently, a mean of avoiding this sort of indeterminacy is making belief formation endogenous, via methods and results borrowed from the learning literature. The paper attempts at clarifying the difficulties encountered by this line of argument when predictions of a variable are based on expectation functions defined on a space of realized values of the same variable and learning is based on gradient descent procedures. Since the analysis is carried out with reference to a simple macroeconomic model of Harrodian inspiration, the paper gives also some local results on the theme of Harrodian instabiliity

Drift effect and timing without observability: experimental evidence

We provide experimental evidence to Binmore and Samuelson’s (1999) insights for modeling the learning process through which equilibrium is selected. They proposed the concept of drift to describe the effect of perturbations on the dynamic process leading to equilibrium in evolutionary games with boundedly rational agents. We test within a random matched population two different versions of the Dalek game where the forward induction equilibrium weakly iterately dominates the other Nash equilibrium in pure strategies. We also assume that the first mover makes her decision first (“timing”) but the second mover is not informed of the first mover's choice (“lack of observability”). Both players are informed of their position in the sequence and of the fact that the second player will decide without knowing the decision of the first player. If the actual observed choices are only those made by other players in previous interactions, the role played by forward induction is replaced with the learning process taking place within the population. Our results support Binmore and Samuelson’s model because the frequency of the forward induction outcome is payoff-sensitive: it strongly increases when we impose a slight change in the payoffs that does not change equilibrium predictions. This evidence reinforces the evolutionary nature of the drift effect.evolutionary games, experiments, drift, forward induction, order of play. J.E.L. Classification: C72, C91

Patterns of Discovery

From a given directed weighted network of knowledge links between technology fields, the paper develops a multisector dynamic model of incremental innovation and R&D activity in these fields. The model is focused on the equilibrium share distribution of these variables, which is proved to be locally stable, with reference to a simple low dimensional case. Simulation methods suggest that local, and also global, stability extend to any model dimension. It is also shown how different network structures map to different asymptotic share distributions. Using the NBER patents and patent citation data files, the analytical framework is then used to analyse some general features of the pattern of knowledge creation and transfer in the period 1975-1999. From a descriptive viewpoint, the changes in the share distribution of innovation activity predicted by the model match reasonably well the actual changes in the perioddirected weighted network, knowledge spillovers, share distribution, incremental innovation and R&D dynamics, local stability, simulation, patents and patent citations

A knowledge based approach to collaboration in basic research

This paper suggests a knowledge based approach to the formation of collaboration networks in basic research. Though mainly focused on foundations, it provides the example of a set of knowledge distributions supporting effort allocations that are pairwise equilibria of the collaboration game. These equilibrium outcomes produce a collaboration network consisting of connected quasi stars