Non-linear incentive equilibrium strategies for a transboundary pollution differential game

Producción CientíficaIn this paper we apply non-linear incentive strategies to sustain over time an agreement. We illustrate the use of these strategies in a linear-quadratic transboundary pollution differential game. The incentive strategies are constructed in such a way that in the long run the pollution stock (the state variable) is close to the steady state of the pollution stock under the cooperative mode of play. The non-linear incentive functions depend on the emission rates (control variables) of both players and on the current value of the pollution stock. The credibility of the incentive equilibrium strategies is analyzed and the performance of open-loop and feedback incentive strategies is compared in their role of helping to sustain an agreement over time. We present numerical experiments to illustrate the results.This research is partially supported by MINECO under projects MTM2016-78995-P (AEI) and ECO2014-52343-P and ECO2017-82227-P (AEI) and by Junta de Castilla y León VA024P17 and VA105G18 co-financed by FEDER funds (EU

Grad-div stabilization for the time-dependent Boussinesq equations with inf-sup stable finite elements

In this paper we consider inf-sup stable nite element discretizations of the evolutionary Boussinesq equations with a grad-div type stabilization. We prove error bounds for the method with constants independent on the Rayleigh numbersMINECO grant MTM2016-78995-P (AEI)Junta de Castilla y León grant VA024P17Junta de Castilla y León grant VA105G18MINECO grant MTM2015-65608-

Spatial vs. non-spatial transboundary pollution control in a class of cooperative and non-cooperative dynamic games

We analyze a transboundary pollution differential game where, in addition to the standard temporal dimension, a spatial dimension is introduced to capture the geographical relationships among regions. Each region behaves strategically and maximizes its welfare net of environmental damage caused by the pollutant stock. The emission-output ratio is reduced by investment in region specific clean technology which evolves over time. The spatio-temporal dynamics of the pollutant stock is described by a parabolic partial differential equation. Using aggregate variables we study the feedback Nash equilibrium of a discrete- space model which could be seen as a space discretization of the continuous-space model. The discrete- space model presents the three main features of the original formulation: the model is truly dynamic; the agents behave strategically; and the model incorporates spatial aspects. For special functional forms previously used in the literature we analytically characterize the feedback Nash equilibrium and evaluate the impact of the introduction of the spatial dimension in the economic-environmental model. We show that our spatial model is a generalization of the model that disregards the spatial aspects. We analytically show that as the parameter describing how pollution diffuses among regions tends to infinity the equilibrium policies converge to those in the non-spatial setting. In the non-cooperative framework the spatially non-myopic behavior prescribes lower equilibrium emission rates, and consequently a lower global pollution stock. This is compatible with greater long-run welfares. In the cooperative framework, although the strategic interaction among the players does not exist, the only decision-maker still makes spatially strategic decisions.MINECO/AEI Projects MTM2016-78995-P, ECO2014-52343-P, ECO2017-82227-PJunta de Castilla y León, Projects VA024P17 and VA105G1

Does Flexibility Facilitate Sustainability of Cooperation Over Time? A Case Study from Environmental Economics

In this paper we present nonlinear incentive strategies that can be applied to a class of differential games that are frequently used in the literature, in particular, in environmental economics literature. We consider a class of nonlinear incentive functions that depend on the control variables of both players and on the current value of the state variable. The strategies are constructed to allow some flexibility in the sense that, unlike the common literature on the subject, the optimal state path evolves close to the cooperative trajectory. As a consequence of this flexibility, the incentive equilibrium is credible in a larger region than the one associated with the usual linear incentive strategies.This research has been supported by Spanish MINECO, projects ECO2008-01551/ECON, ECO2011-24352 and MTM2010-14919 (cofinanced by FEDER funds)

Fully Discrete Approximations to the Time-Dependent Navier–Stokes Equations with a Projection Method in Time and Grad-Div Stabilization

This paper studies fully discrete approximations to the evolutionary Navier{ Stokes equations by means of inf-sup stable H1-conforming mixed nite elements with a grad-div type stabilization and the Euler incremental projection method in time. We get error bounds where the constants do not depend on negative powers of the viscosity. We get the optimal rate of convergence in time of the projection method. For the spatial error we get a bound O(hk) for the L2 error of the velocity, k being the degree of the polynomials in the velocity approximation. We prove numerically that this bound is sharp for this method.MINECO grant MTM2016-78995-P (AEI)Junta de Castilla y León grant VA024P17Junta de Castilla y León grant VA105G18MINECO grant MTM2015-65608-

Selection of a Markov Perfect Nash Equilibrium in a Class of Differential games

This paper revisits the problem of how to select an equilibrium in a differential game in the case of multiplicity of Nash equilibria. Most of the previous applied dynamic games literature has considered pre-play negotiations between players, implicitly or explicitly, with the aim of reaching an agreement on the selection of the pair of strategies. The main objective of this paper is to determine what would be the equilibrium to be played without pre-play communications. We study the linear and nonlinear Markov perfect Nash equilibria for a class of well-known models in the literature if pre-play communications are eliminated. We analyze both symmetric and nonsymmetric strategies. We show that the nonlinear strategies are not always the optimal strategies implemented when pre-play communications are removed. We conclude that in the presence of multiple equilibria and without pre-play communications the equilibria actually implemented are symmetric piecewise linear Markov perfect Nash equilibria at least for a range of initial values of the state variable

Spatial effects and strategic behavior in a multiregional transboundary pollution dynamic game

We analyze a transboundary pollution differential game where pollution control is spatially distributed among a number of agents with predetermined spatial relationships. The analysis emphasizes, first, the effects of the different geographical relationships among decision makers; and second, the strategic behaviour of the agents. The dynamic game considers a pollution stock (the state variable) distributed among one large region divided in subregions which control their own emissions of pollutants. The emissions are also represented as distributed variables. The dynamics of the pollution stock is defined by a parabolic partial differential equation. We numerically characterize the feedback Nash equilibrium of a discrete-space model that still captures the spatial interactions among agents. We evaluate the impact of the strategic and spatially dynamic behaviour of the agents on the design of equilibrium environmental policiesThis research is partially supported by MINECO under projects MTM2013-42538-P, MTM2016-78995-P (AEI) (first author) and ECO2014-52343-P (second author), co-financed by FEDER funds. The authors thank the support of European Cooperation in Science and Technology through COST Action IS1104, ``The EU in the new complex geography of economic systems: models, tools and policy evaluation"

Optimal Bounds for Numerical Approximations of Infinite Horizon Problems Based on Dynamic Programming Approach

Producción CientíficaIn this paper we get error bounds for fully discrete approximations of infinite horizon problems via the dynamic programming approach. It is well known that considering a time discretization with a positive step size $h$ an error bound of size $h$ can be proved for the difference between the value function (viscosity solution of the Hamilton-Jacobi-Bellman equation corresponding to the infinite horizon) and the value function of the discrete time problem. However, including also a spatial discretization based on elements of size $k$ an error bound of size $O(k/h)$ can be found in the literature for the error between the value functions of the continuous problem and the fully discrete problem. In this paper we revise the error bound of the fully discrete method and prove, under similar assumptions to those of the time discrete case, that the error of the fully discrete case is in fact $O(h+k)$ which gives first order in time and space for the method. This error bound matches the numerical experiments of many papers in the literature in which the behaviour $1/h$ from the bound $O(k/h)$ have not been observed.Spanish MINECO, grant PID2019-104141GB-I00Junta de Castilla y León, grant VA169P20 co-finanzed by FEDER (EU) fund

Error Analysis of Non Inf-sup Stable Discretizations of the time-dependent Navier--Stokes Equations with Local Projection Stabilization

This paper studies non inf-sup stable nite element approximations to the evolutionary Navier{Stokes equations. Several local projection stabilization (LPS) methods corresponding to di erent stabilization terms are analyzed, thereby separately studying the e ects of the di erent stabilization terms. Error estimates are derived in which the constants in the error bounds are independent of inverse powers of the viscosity. For one of the methods, using velocity and pressure nite elements of degree l, it will be proved that the velocity error in L1(0; T;L2( )) decays with rate l + 1=2 in the case that h, with being the dimensionless viscosity and h the mesh width. In the analysis of another method, it was observed that the convective term can be bounded in an optimal way with the LPS stabilization of the pressure gradient. Numerical studies con rm the analytical results.MTM2016-78995-PVA024P1

Equilibrium strategies in a multiregional transboundary pollution differential game with spatially distributed controls

We analyse a differential game with spatially distributed controls to study a multiregional transboundary pollution problem. The dynamics of the state variable (pollution stock) is defined by a two dimensional parabolic partial differential equation. The control variables (emissions) are spatially distributed variables. The model allows for a, possibly large, number of agents with predetermined geographical relationships. For a special functional form previously used in the literature of transboundary pollution dynamic games we analytically characterize the feedback Nash equilibrium. We show that at the equilibrium both the level and the location of emissions of each region depend on the particular geographical relationship among agents. We prove that, even in a simplified model, the geographical considerations can modify the players’ optimal strategies and therefore, the spatial aspects of the model should not be overlooked.Ministerio de Educación (projects PID2019- 104141GB-I00 and ECO2017-82227-P)Junta de Castilla y León - Fondo Europeo de Desarrollo Regional (projects VA169P20 and VA105G18