A Complementarity Framework for Forward Contracting Under Uncertainty

We consider a particular instance of a stochastic multi-leader multi-follower equilibrium problem in which players compete in the forward and spot markets in successive periods. Proving the existence of such equilibria has proved difficult, as has the construction of globally convergent algorithms for obtaining such points. By conjecturing a relationship between forward and spot decisions, we consider a variant of the original game and relate the equilibria of this game to a related simultaneous stochastic Nash game where forward and spot decisions are made simultaneously. We characterize the complementarity problem corresponding to the simultaneous Nash game and prove that it is indeed solvable. Moreover, we show that an equilibrium to this Nash game is a local Nash equilibrium of the conjectured variant of the multi-leader multi-follower game of interest. Numerical tests reveal that the difference between equilibrium profits between the original and constrained games are small. Under uncertainty, the equilibrium point of interest is obtainable as the solution to a stochastic mixed-complementarity problem. Based on matrix-splitting methods, a globally convergent decomposition method is suggested for such a class of problems. Computational tests show that the effort grows linearly with the number of scenarios. Further tests show that the method can address larger networks as well. Finally, some policy-based insights are drawn from utilizing the framework to model a two-settlement six-node electricity market.