Feedback Equilibria for a General Class of Non-linear Differential Games with Application to Environmental and Resource Management

As it is well known in differential games, the open-loop Nash equilibrium (OLNE) concept with an infinite period of commitment is weakly time-consistent but not strongly time-consistent. On the other hand, the feedback Nash equilibrium (FBNE) is Markov perfect by construction and thus a more satisfactory solution concept, but solutions are usually very difficult to derive. Explicit solutions can be obtained for the so-called linearquadratic differential game. However, the transition equations in realistic differential game models for environmental and resource management contain non-linear feedbacks, so that the linear-quadratic structure of the game is lost. Thus the attempt to make the natural system more realistic complicates the use of the FBNE concept, since the non-linear structure of the differential game does not allow the standard determination of feedback equilibrium strategies. The purpose of this paper is to develop an algorithm to solve a non-linear-quadratic differential game and to explicitly determine the non-linear feedback equilibrium strategies. In particular we consider a class of non-linear differential games which are often encountered in environmental and resource management problems. At the first stage we analyze the cooperative solution and the OLNE of the underlying differential game. These solutions can be regarded as benchmark cases that expose the existence of multiple equilibria and of “good” and “bad” basins of attraction. At the second stage we analyze the FBNE of the non-linear differential game and we construct a procedure that determines the feedback equilibrium strategies numerically. Since the non-linearity of the problem induces multiple equilibria, our procedure determines the ∗The Beijer International Institute of Ecological Economics, The Royal Swedish Academy of Sciences, [email protected] †Department of Economics, University of Crete, University Campus, 74 100 Rethymno, Crete, GREECE, tel +30 831 0 77861, fax:+30 831 0 77860 [email protected] ‡Department of Economics and CentER, Tilburg University, [email protected] Feedback Equilibria 2 feedback equilibrium strategies locally in the basin of attraction of the corresponding open-loop Nash equilibrium.