Cooperative production and efficiency

We characterize the sharing rule for which a contribution mechanism achieves efficiency in a cooperative production setting when agents are heterogeneous. The sharing rule bears no resemblance to those considered by the previous literature. We also show for a large class of sharing rules that if Nash equilibrium yields efficient allocations, the production function displays constant returns to scale, a case in which cooperation in production is useless.

Peace agreements without commitment

In this paper we present a model of war between two rational and completely informed players. We show that in the absence of binding agreements war can be avoided in many cases by one player transferring money to the other player. In most cases, the "rich" country transfers part of her money to the "poor" country. Only when the military proficiency of the "rich" country is sufficiently great, it could be that the "poor" country can stop the war by transfering part of its resources to the "rich" country.

On the generalized logistic random differential equation: Theoretical analysis and numerical simulations with real-world data

[EN] Based on the previous literature about the random logistic and Gompertz models, the aim of this paper is to extend the investigations to the generalized logistic differential equation in the random setting. First, this is done by rigorously constructing its solution in two different ways, namely, the sample-path approach and the mean-square calculus. Secondly, the probability density function at each time instant is derived in two ways: by applying the random variable transformation technique and by solving the associated Liouville's partial differential equation. It is also proved that both the stochastic solution and its density function converge, under specific conditions, to the corresponding solution and density function of the logistic and Gompertz models, respectively. The investigation finishes showing some examples, where a number of computational techniques are combined to construct reliable approximations of the probability density of the stochastic solution. In particular, we show, step-by-step, how our findings can be applied to a real-world problem. (c) 2022 The Author(s). Published by Elsevier B.V.This work has been supported by the Spanish Agencia Estatal de Investigacion grant PID2020-115270GB-I00. Vicente Bevia acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia.Bevia, V.; Calatayud-Gregori, J.; Cortés, J.; Jornet-Sanz, M. (2023). On the generalized logistic random differential equation: Theoretical analysis and numerical simulations with real-world data. Communications in Nonlinear Science and Numerical Simulation. 116. https://doi.org/10.1016/j.cnsns.2022.10683211

Uncertainty quantification analysis of the biological Gompertz model subject to random fluctuations in all its parameters

[EN] In spite of its simple formulation via a nonlinear differential equation, the Gompertz model has been widely applied to describe the dynamics of biological and biophysical parts of complex systems (growth of living organisms, number of bacteria, volume of infected cells, etc.). Its parameters or coefficients and the initial condition represent biological quantities (usually, rates and number of individual/particles, respectively) whose nature is random rather than deterministic. In this paper, we present a complete uncertainty quantification analysis of the randomized Gomperz model via the computation of an explicit expression to the first probability density function of its solution stochastic process taking advantage of the Liouville-Gibbs theorem for dynamical systems. The stochastic analysis is completed by computing other important probabilistic information of the model like the distribution of the time until the solution reaches an arbitrary value of specific interest and the stationary distribution of the solution. Finally, we apply all our theoretical findings to two examples, the first of numerical nature and the second to model the dynamics of weight of a species using real data.This work has been supported by the Spanish Ministerio de Economia, Industria y Competitividad (MINECO), the Agencia Estatal de Investigacion (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-P.Bevia, V.; Burgos, C.; Cortés, J.; Navarro-Quiles, A.; Villanueva Micó, RJ. (2020). Uncertainty quantification analysis of the biological Gompertz model subject to random fluctuations in all its parameters. Chaos, Solitons and Fractals. 138:1-12. https://doi.org/10.1016/j.chaos.2020.109908S112138Golec, J., & Sathananthan, S. (2003). Stability analysis of a stochastic logistic model. Mathematical and Computer Modelling, 38(5-6), 585-593. doi:10.1016/s0895-7177(03)90029-xCortés, J. C., Jódar, L., & Villafuerte, L. (2009). Random linear-quadratic mathematical models: Computing explicit solutions and applications. Mathematics and Computers in Simulation, 79(7), 2076-2090. doi:10.1016/j.matcom.2008.11.008Dorini, F. A., Cecconello, M. S., & Dorini, L. B. (2016). On the logistic equation subject to uncertainties in the environmental carrying capacity and initial population density. Communications in Nonlinear Science and Numerical Simulation, 33, 160-173. doi:10.1016/j.cnsns.2015.09.009Dorini, F. A., Bobko, N., & Dorini, L. B. (2016). A note on the logistic equation subject to uncertainties in parameters. Computational and Applied Mathematics, 37(2), 1496-1506. doi:10.1007/s40314-016-0409-6Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2019). Analysis of random non-autonomous logistic-type differential equations via the Karhunen–Loève expansion and the Random Variable Transformation technique. Communications in Nonlinear Science and Numerical Simulation, 72, 121-138. doi:10.1016/j.cnsns.2018.12.013Calatayud, J., Cortés, J. C., & Jornet, M. (2019). Improving the approximation of the probability density function of random nonautonomous logistic‐type differential equations. Mathematical Methods in the Applied Sciences, 42(18), 7259-7267. doi:10.1002/mma.5834Casabán, M.-C., Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., Roselló, M.-D., & Villanueva, R.-J. (2016). Probabilistic solution of the homogeneous Riccati differential equation: A case-study by using linearization and transformation techniques. Journal of Computational and Applied Mathematics, 291, 20-35. doi:10.1016/j.cam.2014.11.028Hesam, S., Nazemi, A. R., & Haghbin, A. (2012). Analytical solution for the Fokker–Planck equation by differential transform method. Scientia Iranica, 19(4), 1140-1145. doi:10.1016/j.scient.2012.06.018Lakestani, M., & Dehghan, M. (2009). Numerical solution of Fokker-Planck equation using the cubic B-spline scaling functions. Numerical Methods for Partial Differential Equations, 25(2), 418-429. doi:10.1002/num.20352Mao, X., Yuan, C., & Yin, G. (2005). Numerical method for stationary distribution of stochastic differential equations with Markovian switching. Journal of Computational and Applied Mathematics, 174(1), 1-27. doi:10.1016/j.cam.2004.03.016Casabán, M.-C., Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., Roselló, M.-D., & Villanueva, R.-J. (2017). Computing probabilistic solutions of the Bernoulli random differential equation. Journal of Computational and Applied Mathematics, 309, 396-407. doi:10.1016/j.cam.2016.02.034Kegan, B., & West, R. W. (2005). Modeling the simple epidemic with deterministic differential equations and random initial conditions. Mathematical Biosciences, 195(2), 179-193. doi:10.1016/j.mbs.2005.02.004Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2017). Full solution of random autonomous first-order linear systems of difference equations. Application to construct random phase portrait for planar systems. Applied Mathematics Letters, 68, 150-156. doi:10.1016/j.aml.2016.12.015Cortés, J. C., Navarro‐Quiles, A., Romero, J., & Roselló, M. (2019). (CMMSE2018 paper) Solving the random Pielou logistic equation with the random variable transformation technique: Theory and applications. Mathematical Methods in the Applied Sciences, 42(17), 5708-5717. doi:10.1002/mma.5440Dorini, F. A., & Cunha, M. C. C. (2011). On the linear advection equation subject to random velocity fields. Mathematics and Computers in Simulation, 82(4), 679-690. doi:10.1016/j.matcom.2011.10.008Slama, H., El-Bedwhey, N. A., El-Depsy, A., & Selim, M. M. (2017). Solution of the finite Milne problem in stochastic media with RVT Technique. The European Physical Journal Plus, 132(12). doi:10.1140/epjp/i2017-11763-6Hussein, A., & Selim, M. M. (2013). A general analytical solution for the stochastic Milne problem using Karhunen–Loeve (K–L) expansion. Journal of Quantitative Spectroscopy and Radiative Transfer, 125, 84-92. doi:10.1016/j.jqsrt.2013.03.018Hussein, A., & Selim, M. M. (2019). A complete probabilistic solution for a stochastic Milne problem of radiative transfer using KLE-RVT technique. Journal of Quantitative Spectroscopy and Radiative Transfer, 232, 54-65. doi:10.1016/j.jqsrt.2019.04.034Cortés, J.-C., Jódar, L., Camacho, F., & Villafuerte, L. (2010). Random Airy type differential equations: Mean square exact and numerical solutions. Computers & Mathematics with Applications, 60(5), 1237-1244. doi:10.1016/j.camwa.2010.05.046Bekiryazici, Z., Merdan, M., & Kesemen, T. (2020). Modification of the random differential transformation method and its applications to compartmental models. Communications in Statistics - Theory and Methods, 50(18), 4271-4292. doi:10.1080/03610926.2020.1713372Calatayud, J., Cortés, J.-C., Díaz, J. A., & Jornet, M. (2020). Constructing reliable approximations of the probability density function to the random heat PDE via a finite difference scheme. Applied Numerical Mathematics, 151, 413-424. doi:10.1016/j.apnum.2020.01.012Laird, A. K. (1965). Dynamics of Tumour Growth: Comparison of Growth Rates and Extrapolation of Growth Curve to One Cell. British Journal of Cancer, 19(2), 278-291. doi:10.1038/bjc.1965.32Nahashon, S. N., Aggrey, S. E., Adefope, N. A., Amenyenu, A., & Wright, D. (2006). Growth Characteristics of Pearl Gray Guinea Fowl as Predicted by the Richards, Gompertz, and Logistic Models. Poultry Science, 85(2), 359-363. doi:10.1093/ps/85.2.35

Probabilistic analysis of a general class of nonlinear random differential equations with state-dependent impulsive terms via probability density functions

[EN] In this contribution, we rigorously construct a pathwise solution to a general scalar random differential equation with state-dependent Dirac-delta impulse terms at a finite number of time instants. Furthermore, we obtain the first probability density function of the solution by combining two main results, firstly, the Liouville-Gibbs equation between the impulse instants, and secondly, the Random Variable Transformation technique at the impulse times. Finally, all theoretical findings are illustrated on two stochastic models, widely used in mathematical modeling, carrying on computational simulations.(c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).This work has been partially supported by the Spanish Ministerio de Economia, Industria y Competitividad (MINECO), the Agencia Estatal de Investigacion (AEI), Spain, and Fondo Europeo de Desarrollo Regional (FEDER UE) grant PID2020-115270GB-I00, the Generalitat Valenciana, Spain (grant AICO/2021/302) and by el Fondo Social Europeo y la Iniciativa de Empleo Juvenil EDGJID/2021/185. Vicente Bevia acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia, Spain.Bevia, VJ.; Cortés, J.; Jornet-Sanz, M.; Villanueva Micó, RJ. (2023). Probabilistic analysis of a general class of nonlinear random differential equations with state-dependent impulsive terms via probability density functions. Communications in Nonlinear Science and Numerical Simulation. 119. https://doi.org/10.1016/j.cnsns.2023.10709711

Uncertainty Quantification of Random Microbial Growth in a Competitive Environment via Probability Density Functions

[EN] The Baranyi-Roberts model describes the dynamics of the volumetric densities of two interacting cell populations. We randomize this model by considering that the initial conditions are random variables whose distributions are determined by using sample data and the principle of maximum entropy. Subsequenly, we obtain the Liouville-Gibbs partial differential equation for the probability density function of the two-dimensional solution stochastic process. Because the exact solution of this equation is unaffordable, we use a finite volume scheme to numerically approximate the aforementioned probability density function. From this key information, we design an optimization procedure in order to determine the best growth rates of the Baranyi-Roberts model, so that the expectation of the numerical solution is as close as possible to the sample data. The results evidence good fitting that allows for performing reliable predictions.This work has been supported by the Spanish Ministerio de Economia, Industria y Competitividad (MINECO), the Agencia Estatal de Investigacion (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-P.Bevia-Escrig, V.; Burgos-Simon, C.; Cortés, J.; Villanueva Micó, RJ. (2021). Uncertainty Quantification of Random Microbial Growth in a Competitive Environment via Probability Density Functions. Fractal and Fractional. 5(2):1-18. https://doi.org/10.3390/fractalfract5020026S1185

Optimism and commitment: An elementary theory of bargaining and war

We propose an elementary theory of wars fought by fully rational contenders. Two parties play a Markov game that combines stages of bargaining with stages where one side has the ability to impose surrender on the other. Under uncertainty and incomplete information, in the unique equilibrium of the game, long confrontations occur: war arises when reality disappoints initial (rational) optimism, and it persist longer when both agents are optimists but reality proves both wrong. Bargaining proposals that are rejected initially might eventually be accepted after several periods of confrontation. We provide an explicit computation of the equilibrium, evaluating the probability of war, and its expected losses as a function of i) the costs of confrontation, ii) the asymmetry of the split imposed under surrender, and iii) the strengths of contenders at attack and defense. Changes in these parameters display non-monotonic effects

Forward uncertainty quantification in random differential equation systems with delta-impulsive terms: Theoretical study and applications

[EN] This contribution aims at studying a general class of random differential equations with Dirac-delta impulse terms at a finite number of time instants. Our approach directly addresses calculating the so-called first probability density function, from which all the relevant statistical information about the solution, a stochastic process, can be extracted. We combine the Liouville partial differential equation and the random variable transformation method to conduct our study. Finally, all our theoretical findings are illustrated on two stochastic models, widely used in mathematical modeling, for which numerical simulations are carried out.Spanish Ministerio de Economia, Industria y Competitividad (MINECO); Agencia Estatal de Investigacion (AEI); Fondo Europeo de Desarrollo Regional (FEDER UE), Grant/Award Number: PID2020-115270GB-I00; Generalitat Valenciana, Grant/Award Number: AICO/2021/302; el Fondo Social Europeo y la Iniciativa de Empleo Juvenil, Grant/Award Number: EDGJID/2021/185; Ayuda a Primeros Proyectos de Investigacion, Grant/Award Number: PAID-06-22; Vicerrectorado de Investigacion de la Universitat Politecnica de Valencia (UPV).Bevia-Escrig, V.; Cortés, J.; Villanueva Micó, RJ. (2023). Forward uncertainty quantification in random differential equation systems with delta-impulsive terms: Theoretical study and applications. Mathematical Methods in the Applied Sciences. 1-21. https://doi.org/10.1002/mma.922612

Rational Sabotage in Cooperative Production with Heterogeneous Agents

We present a model of cooperative production in which rational agents might carry out sabotage activities that decrease output. We provide necessary and sufficient conditions for the existence of a Nash equilibrium without sabotage. It is shown that the absence of sabotage in equilibrium depends on the interplay between technology, relative productivity of agents and the degree of meritocracy. In particular we show that, ceteris paribus, meritocratic systems give more incentives to sabotage than egalitarian systems.

On the generic imposibility of truthful behavior A simple approach

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