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.

1. Ehtamo and Hämäläinen [19, p. 674] already pointed out the interest of considering state-dependent equilibrium strategies instead of decision-dependent equilibrium strategies. Jørgensen and Zaccour [23, p. 818] stated that in the incentive equilibrium the application of state-dependent strategies may be more complicated and may be a promising area for future research.

2. We thank an anonymous reviewer for bringing this point to our attention.

This research has been supported by Spanish MINECO, Projects ECO2008-01551, ECO2011-24352 and MTM2010-14919 (cofinanced by FEDER funds) and by COST Action IS1104, ”The EU in the new economic complex geography: models, tools and policy evaluation”. The authors thank four anonymous reviewers for their valuable comments and suggestions.

Authors and Affiliations

1. IMUVA, Universidad de Valladolid, Valladolid, Spain

   Javier de Frutos & Guiomar Martín-Herrán

Corresponding author

Correspondence to Guiomar Martín-Herrán.

Appendix: The Numerical Method

We discretize problem (5), with the incentive functions \(\psi _i\), \(i=1,2\), defined in (7), by considering a time-discrete problem. To this end, let \(h>0\) be a positive parameter. We introduce the time steps \(t_n=nh\), with \(n\in \mathbb {N}\) a positive integer. We will use the notation \(\bar{u}_{i}\) to represent a sequence of real numbers \(\{u_{i,n}\}_{n=0}^\infty \) with \(u_{i,n}\in \mathbb {U}_i\) for all \(n\in \mathbb {N}\). The set of such sequences is represented by \(\overline{\fancyscript{U}}_i\).

We consider the time discrete, infinite horizon, pair of problems

where \(x_0\) is the initial condition in (5), \(\delta =1-\rho h\) and where superscript \(\xi \) is used to denote \(s\) or \(ns\) depending on the particular realization at hand. For simplicity, we have omitted the time variable in \(\psi _i^\mathrm{{ns}}\). It is assumed that the equilibrium condition

is satisfied. We are using the notation \(\bar{u}_{i}^{*\xi }:=\arg \max _{\bar{u}_{i}}W_{h,i}(\bar{u}_{i},\bar{u}_{j}^{*\xi })\), \(i=1,2\).

Let us observe that the discrete problem (16) corresponds to a discretization of the functional in (5) by means of the rectangle rule with a forward Euler discretization of the dynamics in (5).

The discrete optimal incentive equilibrium trajectory starting in \(x_0^*=x_0\) is computed with the sequence

In the case of the stationary incentive, the (time discrete) value function for problem (16) is defined by the system of Bellman equations

where \(i,j=1,2\), \(i\ne j\). The optimal feedback is defined as

and \(f_i(x,u_i,u_j^{*\mathrm{{s}}})\) and \(g_i^\mathrm{{s}}(x,u_i,u_j^{*\mathrm{{s}}})\) are given by (4) and (10).

The solution of the system of Eq. (18) is approximated using a collocation method based on shape preserving piecewise cubic Hermite interpolation introduced in [32, 33]. More precisely, let us introduce a grid of points \(0=z_0<z_1<\cdots <z_M=X\) for some fixed \(X>0\) that is big enough. We define \(\Delta _k=z_{k+1}-z_k\). The approximation \(V_{h,M,i}^\mathrm{{s}}\) to \(V_{h,i}^\mathrm{{s}}\), can be written for \(x\in [z_k,z_{k+1}]\) as

where \(\Phi (z)=3z^2-2z^3\) and \(\Psi (z)=z^3-z^2\). Note that the coefficients \(F_k\) and \(D_k\), are defined as \(F_k=V_{h,M,i}^\mathrm{{s}}(z_k)\) and \(D_k=\frac{\mathrm {d}}{\mathrm {d}x}V_{h,M,i}^\mathrm{{s}}(z_k)\), \(k=0,1,\ldots ,M\). The values of the slopes \(D_k\) are chosen as in [33]. This choice guarantees that \(V_{h,M,i}^\mathrm{{s}}(x)\) is locally monotone if the data \(F_k\) are locally monotone (see [32, 33]). The interpolant \(V_{h,M,i}^\mathrm{{s}}(x)\) possesses continuous first-order derivatives in \([0,z_M]\). The second derivative is not necessarily continuous.

The piecewise cubic approximation \(V_{h,M,i}^\mathrm{{s}}(x)\) is computed by a fixed-point iteration solving, for \(r\ge 0\),

and

The iteration is initialized with some given \(V_{h,M,i}^{\mathrm{{s}},[0]}(z_k)\) and \(u_{i,k}^{[0]}\), \(i=1,2\), \(k=0,\ldots , M\), and stopped when

where \(\text {TOL}\) is a prescribed tolerance. Once the convergence criterion is satisfied, the functions \(V_{h,M,i}^\mathrm{{s}}:=V_{h,M,i}^{\mathrm{{s}},[r+1]}\) are the desired approximations to the value functions, \(V_i^\mathrm{{s}}\). The approximated optimal policies are defined as the monotone piecewise cubic Hermite interpolant \(u_{M,i}^{*\mathrm{{s}}}\) such that \(u_{M,i}^{*\mathrm{{s}}}(z_k):=u_{i,k}^{[r+1]}\), \(i=1,2\), \(k=0,1,\ldots , M.\) Finally, the approximate optimal trajectory is computed from (17) with \(u_{i,n}^{*\mathrm{{s}}}=u_{M,i}^{*\mathrm{{s}}}(x_n^*)\).

The time-dependent problem (5) with incentive (11) is discretized along the same lines. We introduce a fictitious big enough time horizon \(T=t_N=Nh>0\). The time-discrete, time-dependent value function is defined as the solution of the system of Bellman equations

where functions \(f_i\) and \(g_i^\mathrm{{ns}}\) are given by (4) and (13).

Equation (20) is supplemented with the artificial boundary condition \(V_{h,i}^\mathrm{{ns}}(t_N,x)=V_{h,i}^\mathrm{{s}}(x)\), \(i=1,2\), which is an obvious approximation to (14). The system (20) is numerically solved backward in time by

where, for \(i,j=1,2\), \(i\ne j\),

The notation \(V_{h,M,i}^\mathrm{{ns}}({t_{n}},x)\) represents, as before, the monotone piecewise cubic Hermite interpolant defined by the values \(V_{h,M,i}^\mathrm{{ns}}({t_{n}},x_k)\), \(0\le k\le M\). The backward iteration is initialized using the boundary condition \(V_{h,M,i}^\mathrm{{ns}}({t_N},x_k)=V_{h,M,i}^\mathrm{{s}}(x_k)\), \(0\le k\le M\).

The approximated optimal policy at time \(t_n\), \(1\le n\le N\), is defined as the monotone piecewise cubic Hermite interpolant of the values \(u_{i,n,k}^{*\mathrm{{ns}}}\) and it is denoted by \(u_{M,i}^{*\mathrm{{ns}}}(x)\). Then, the optimal trajectory can be computed from (17) with \(u_{i,n}^{*\mathrm{{ns}}}=u_{M,i}^{*\mathrm{{ns}}}(x_n^*)\). System (21) is solved by a fixed-point iteration similar to that in (19). In this last computation a filtering process is applied to eliminate possible spurious oscillations.

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