Assessing the predictive effectiveness of the variety seeking and reinforcement models

Prediction of customer choice behaviour has been a big challenge for marketing researchers. They have adopted various models to represent customers purchase patterns. Some researchers considered simple zero–order models. Others proposed higher–order models to represent explicitly customers tendency to seek [variety] or [reinforcement] as they make repetitive choices. Nevertheless, the question [Which model has the highest probability of representing some future data?] still prevails. The objective of this paper is to address this question. We assess the predictive effectiveness of the well–known customer choice models. In particular, we compare the predictive ability of the [dynamic attribute satiation] (DAS) model due to McAlister (Journal of Consumer Research, 91, pp. 141–150, 1982) with that of the well–known stochastic variety seeking and reinforcement behaviour models. We found that the stochastic [beta binomial] model has the best predictive effectiveness on both simulated and real purchase data. Using simulations, we also assessed the effectiveness of the stochastic models in representing various complex choice processes generated by the DAS. The beta binomial model mimicked the DAS processes the best. In this research we also propose, for the first time, a stochastic choice rule for the DAS model.     

Monitoring cooperative equilibria in a stochastic differential game

This paper deals with a class of equilibria which are based on the use of memory strategies in the context of continuous-time stochastic differential games. In order to get interpretable results, we will focus the study on a stochastic differential game model of the exploitation of one species of fish by two competing fisheries. We explore the possibility of defining a so-called cooperative equilibrium, which will implement a fishing agreement. In order to obtain that equilibrium, one defines a monitoring variable and an associated retaliation scheme. Depending on the value of the monitoring variable, which provides some evidence of a deviation from the agreement, the probability increases that the mode of a game will change from a cooperative to a punitive one. Both the monitoring variable and the parameters of this jump process are design elements of the cooperative equilibrium. A cooperative equilibrium designed in this way is a solution concept for a switching diffusion game. We solve that game using the sufficient conditions for a feedback equilibrium which are given by a set of coupled HJB equations. A numerical analysis, approximating the solution of the HJB equations through an associated Markov game, enables us to show that there exist cooperative equilibria which dominate the classical feedback Nash equilibrium of the original diffusion game model.This research was supported by FNRS-Switzerland, NSERC-Canada, FCAR-Quebec.     

A statistical mechanics approach to Granovetter theory

In this paper we try to bridge breakthroughs in quantitative sociology/econometrics, pioneered during the last decades by Mac Fadden, Brock–Durlauf, Granovetter and Watts–Strogatz, by introducing a minimal model able to reproduce essentially all the features of social behavior highlighted by these authors.     

Diffusion in flashing periodic potentials

The one-dimensional overdamped Brownian motion in a symmetric periodic potential modulated by external time-reversible noise is analyzed. The calculation of the effective diffusion coefficient is reduced to the mean first passage time problem. We derive general equations to calculate the effective diffusion coefficient of Brownian particles moving in arbitrary supersymmetric potential modulated by: (i) an external white Gaussian noise and (ii) a Markovian dichotomous noise. For both cases the exact expressions for the effective diffusion coefficient are derived. We obtain acceleration of diffusion in comparison with the free diffusion case for fast fluctuating potentials with arbitrary profile and for sawtooth potential in case (ii). In this case the parameter region where this effect can be observed is given. We obtain also a finite net diffusion in the absence of thermal noise. For rectangular potential the diffusion slows down, for all parameters of noise and of potential, in comparison with the case when particles diffuse freely.     

Renormalization of branching models of triggered seismicity from total to observable seismicity

Several recent works point out that the crowd of small unobservable earthquakes (with magnitudes below the detection threshold m_d) may play a significant and perhaps dominant role in triggering future seismicity. Using the ETAS branching model of triggered seismicity, we apply the formalism of generating probability functions to investigate how the statistical properties of observable earthquakes differ from the statistics of all events. The ETAS (epidemic-type aftershock sequence) model assumes that each earthquake can trigger other earthquakes (“aftershocks”). An aftershock sequence results in this model from the cascade of aftershocks of each past earthquake. The triggering efficiency of earthquakes is assumed to vanish below a lower magnitude limit m₀, in order to ensure the convergence of the theory and may reflect the physics of state-and-velocity frictional rupture. We show that, to a good approximation, the statistical distribution of seismic rates of events with magnitudes above m_d generated by an ETAS model with branching ratio n is the same as that of events generated by another ETAS model with effective parameter n(m_d). Our present analysis thus confirms, for the full statistical (time-independent or large time-window approximation) properties, the results obtained previously by one of us and Werner, based solely on the average seismic rates (the first-order moment of the statistics). Our analysis also demonstrates that this correspondence is not exact, as there are small corrections which can be systematically calculated, in terms of additional contributions that can be mapped onto a different branching model. We also show that this approximate correspondence of the ETAS model onto itself obtained by changing m₀ into m_d, and n into n(m_d) holds only with respect to its statistical properties and not for all its space-time properties.     

Triangular arbitrage as an interaction among foreign exchange rates

We first show that there are in fact triangular arbitrage opportunities in the spot foreign exchange markets, analyzing the time dependence of the yen–dollar rate, the dollar–euro rate and the yen–euro rate. Next, we propose a model of foreign exchange rates with an interaction. The model includes effects of triangular arbitrage transactions as an interaction among three rates. The model explains the actual data of the multiple foreign exchange rates well.     

Dynamics of bound vector solitons induced by stochastic perturbations: Soliton breakup and soliton switching

We respectively investigate breakup and switching of the Manakov-typed bound vector solitons (BVSs) induced by two types of stochastic perturbations: the homogenous and nonhomogenous. Symmetry-recovering is discovered for the asymmetrical homogenous case, while soliton switching is found to relate with the perturbation amplitude and soliton coherence. Simulations show that soliton switching in the circularly-polarized light system is much weaker than that in the Manakov and linearly-polarized systems. In addition, the homogenous perturbations can enhance the soliton switching in both of the Manakov and non-integrable (linearly- and circularly-polarized) systems. Our results might be helpful in interpreting dynamics of the BVSs with stochastic noises in nonlinear optics or with stochastic quantum fluctuations in Bose–Einstein condensates.     

Analysis of slip activity and heterogeneous deformation in tension and tension-creep of Ti–5Al–2.5Sn (wt %) using in-situ SEM experiments

The deformation behavior of a Ti–5Al–2.5Sn (wt %) near-α alloy was investigated during in-situ deformation inside a scanning electron microscope. Tensile experiments were performed at 296?K and 728?K (≈0.4?T _m), while tensile-creep experiments were performed at 728?K and 763?K. Active deformation systems were identified using electron backscattered diffraction-based slip trace analysis. Both basal and prismatic slip systems were active during the tensile experiments. Basal slip was observed for grains clustered around high Schmid factor orientations, while prismatic slip exhibited less dependence on the crystallographic orientation. The tension-creep experiments revealed less slip but more development of grain boundary ledges than in the higher strain rate tensile experiments. Some of the grain boundary ledges evolved into grain boundary cracks, and grain boundaries oriented nearly perpendicular to the tensile axis formed ledges earlier in the deformation process. Grain boundaries with high misorientations also tended to form ledges earlier than those with lower misorientations. Most of the grain boundary cracks formed in association with grains displaying hard orientations, where the c-axis was nearly perpendicular to the tensile direction. For the tension-creep experiments, pronounced basal slip was observed in the lower-stress creep regime and the activity of prismatic slip increased with increasing creep stress and temperature.     

Comparison of the deformation behaviour of commercially pure titanium and Ti–5Al–2.5Sn(wt.%) at 296 and 728 K

The tension and tensile-creep deformation behaviours of a fully-α phase commercially pure (CP) Ti and a near-α Ti–5Al–2.5Sn(wt.%) alloy deformed in situ inside a scanning electron microscope were compared. Tensile tests were performed at 296 and 728?K, while tensile-creep tests were performed at 728?K. The yield stress of CP Ti decreased dramatically with increasing temperature. In contrast, temperature had much smaller effect on the yield stress of Ti–5Al–2.5Sn(wt.%). Electron backscattered diffraction was performed both before and after the deformation, and slip trace analysis was used to determine the active slip and twinning systems, as well as the associated global stress state Schmid factors. In tension tests of CP Ti, prismatic slip was the most likely slip system to be activated when the Schmid factor exceeded 0.4. Prismatic slip was observed over the largest Schmid factor range, indicating that the local stress tensor varies significantly from the global stress state of uniaxial tension. The basal slip activity in Ti–5Al–2.5Sn(wt.%) was observed in a larger faction of grains than in CP Ti. Pyramidal ?c?+?a? slip was more prevalent in CP Ti. Although twinning was an active deformation mode in tension tests of the CP Ti, it was rare in Ti–5Al–2.5Sn(wt.%). During creep, dislocation slip was the primary apparent deformation mechanism in CP Ti, while evidence for dislocation slip was much less apparent in Ti–5Al–2.5Sn(wt.%), where grain boundary sliding was dominant. A robust statistical analysis was carried out to assess the significance of the comparative activity of the different slip systems under the variety of experimental conditions examined.     

Non-equilibrium stochastic model for stock exchange market

We study the effect of the topology of industrial relationship (IR) between the companies in a stock exchange market on the universal features in the market. For this we propose a stochastic model for stock exchange markets based on the behavior of technical traders. From the numerical simulations we measure the return distribution, P(R)$P\left(R\right)$, and the autocorrelation function of the volatility, C(T)$C\left(T\right)$, and find that the observed universal features in real financial markets are originated from the heterogeneity of IR network topology. Moreover, the heterogeneous IR topology can also explain Zipf–Pareto’s law for the distribution of market value of equity in the real stock exchange markets.