Using EPECs to model bilevel games in restructured electricity markets with locational prices

CWPE0619 (EPRG0602) Xinmin Hu and Daniel Ralph (Feb 2006) Using EPECs to model bilevel games in restructured electricity markets with locational prices We study a bilevel noncooperative game-theoretic model of electricity markets with locational marginal prices. Each player faces a bilevel optimization problem that we remodel as a mathematical program with equilibrium constraints, MPEC. This gives an EPEC, equilibrium problem with equilibrium constraints. We establish sufficient conditions for existence of pure strategy Nash equilibria for this class of bilevel games and give some applications. We show by examples the effect of network transmission limits, i.e. congestion, on existence of equilibria. Then we study, for more general EPECs, the weaker pure strategy concepts of local Nash and Nash stationary equilibria. We model the latter via complementarity problems, CPs. Finally, we present numerical examples of methods that attempt to find local Nash or Nash stationary equilibria of randomly generated electricity market games. The CP solver PATH is found to be rather effective in this context

The Limit-Price Dynamics — Uniqueness, Computability and Comparative Dynamics in Competitiive Markets

In this paper, a continuous-time price-quantity trading process is defined for exchange economies with differentiable characteristics. The dynamics is based on boundedly rational agents exchanging limit-price orders to a central clearing house, which rations infinitesimal trades according to Mertens (2003) double auction. Existence of continuous trade and price curves holds under weak conditions and in particular even if there is no long-run competitive equilibrium. Every such curve converges towards a Pareto point, and every Paretian allocation is a locally stable rest-point. Generically, given initial conditions, the trade and price curve is piecewise unique, smooth and computable, hence enables to effectively perform comparative dynamics. Finally, in the 2 x 2 case, the vector field induced by the limit-price dynamics is real-analytic.Non-tatonnement, price-quantity dynamics, limit-price mechanism, myopia, computable general equilibrium.

The Limit-Price Dynamics — Uniqueness, Computability and Comparative Dynamics in Competitiive Markets

URL des Documents de travail :http://ces.univ-paris1.fr/cesdp/CESFramDP2007.htmDocuments de travail du Centre d'Economie de la Sorbonne 2007.20 - ISSN : 1955-611XIn this paper, a continuous-time price-quantity trading process is defined for exchange economies with differentiable characteristics. The dynamics is based on boundedly rational agents exchanging limit-price orders to a central clearing house, which rations infinitesimal trades according to Mertens (2003) double auction. Existence of continuous trade and price curves holds under weak conditions and in particular even if there is no long-run competitive equilibrium. Every such curve converges towards a Pareto point, and every Paretian allocation is a locally stable rest-point. Generically, given initial conditions, the trade and price curve is piecewise unique, smooth and computable, hence enables to effectively perform comparative dynamics. Finally, in the 2 x 2 case, the vector field induced by the limit-price dynamics is real-analytic.On définit un processus d'échanges en prix et en quantités et en temps continue pour des économies différentiables. La dynamique est fondée sur la rationalité limitée d'agents myopes qui adressent des ordres de prix-limites qu'ils adressent à une agence de clearing centrale, laquelle rationne les échanges infinitésimaux en fonction de l'enchère double de Mertens (2003). L'existence de courbes de prix et d'échanges est vérifiée sous de faibles conditions, en particulier en l'absence d'équilibre concurrentiel de long terme. Toute courbe d'échange converge vers un optimum de Pareto et inversement tout optimum est un point stationnaire localement stable de la dynamique. Génériquement, à conditions initiales données, la courbe d'échange et de prix est unique par morceaux, lisse et calculable, ouvrant la possibilité d'une dynamique comparative effective. Enfin, dans le cas 2 x 2, le champ de vecteurs associé à la dynamique est réel-analytique

An Interior-Point Path-Following Method to Compute Stationary Equilibria in Stochastic Games

Subgame perfect equilibrium in stationary strategies (SSPE) is the most important solution concept used in applications of stochastic games, which makes it imperative to develop efficient numerical methods to compute an SSPE. For this purpose, this paper develops an interior-point path-following method (IPM), which remedies a number of issues with the existing method called stochastic linear tracing procedure (SLTP). The homotopy system of IPM is derived from the optimality conditions of an artificial barrier game, whose objective function is a combination of the original payoff function and a logarithmic term. Unlike SLTP, the starting stationary strategy profile can be arbitrarily chosen and IPM does not need switching between different systems of equations. The use of a perturbation term makes IPM applicable to all stochastic games, whereas SLTP only works for a generic stochastic game. A transformation of variables reduces the number of equations and variables of by roughly one half. Numerical results show that our method is more than three times as efficient as SLTP

All solution graphs in multidimensional screening

We study general discrete-types multidimensional screening without any noticeable restrictions on valuations, using instead epsilon-relaxation of the incentive-compatibility constraints. Any active (becoming equality) constraint can be perceived as "envy" arc from one type to another, so the set of active constraints is a digraph. We find that: (1) any solution has an in-rooted acyclic graph ("river"); (2) for any logically feasible river there exists a screening problem resulting in such river. Using these results, any solution is characterized both through its spanning-tree and through its Lagrange multipliers, that can help in finding solutions and their efficiency/distortion properties.incentive compatibility; multidimensional screening; second-degree price discrimination; non-linear pricing; graphs

Numerical Simulation of Nonoptimal Dynamic Equilibrium Models

In this paper we present a recursive method for the computation of dynamic competitive equilibria in models with heterogeneous agents and market frictions. This method is based upon a convergent operator over an expanded set of state variables. The fixed point of this operator defines the set of all Markovian equilibria. We study approximation properties of the operator as well as the convergence of the moments of simulated sample paths. We apply our numerical algorithm to two growth models, an overlapping generations economy with money, and an asset pricing model with financial frictions.Heterogeneous agents, taxes, externalities, financial frictions, competitive equilibrium, computation, simulation