Central to many aspects of economic analysis (and even political theory) is the concept of equilibrium. Similarly as in mechanics, formal characterizations of this concept typically come as complementarity problems or variational inequalities.
KeywordsNash Equilibrium Variational Inequality Economic Application Positive Definiteness Equilibrium Constraint
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- The Cournot equilibrium was thoroughly studied in Murphy et al., 1982; Harker, 1984; Cach, 1996 in connection with algorithms for its computation. Lemmas 12.1,12.2 come from Murphy et al., 1982 and Cach, 1996, respectively. The Stackelberg problem (12.8) was used in Harker and Choi, 1987 to test a special penalty method for the numerical solution of MPECs. The results of Section 12.1 originate from Outrata and Zowe, 1995b, but the satisfaction of (A3) was only conjectured there. Here we were able to verify (A3) on the basis of Lemma 12.2.Google Scholar
- The GNE was investigated in Debreu, 1952; Ichiishi, 1983; Harker, 1991; Flåm and Kummer, 1992 both from the theoretical and the numerical point of view. The idea to compute these equilibria via a suitable MPEC was proposed and briefly sketched in Outrata and Zowe, 1995b. Here we analyzed it in more detail. In Outrata and Zowe, 1995a and Kočvara and Outrata, 1995a an effective numerical approach, based on the nonsmooth Newton’s method from Chapter 3, was proposed for the solution of a class of quasivariational inequalities. This approach, however, cannot be applied to the computation of GNE, since the essential regularity assumption is violated at all equilibria at which the coupling constraints are active.Google Scholar