An approximation scheme for a class of risk-averse stochastic equilibrium problems

Abstract

We consider two models for stochastic equilibrium: one based on the variational equilibrium of a generalized Nash game, and the other on the mixed complementarity formulation. Each agent in the market solves a single-stage risk-averse optimization problem with both here-and-now (investment) variables and (production) wait-and-see variables. A shared constraint couples almost surely the wait-and-see decisions of all the agents. An important characteristic of our approach is that the agents hedge risk in the objective functions (on costs or profits) of their optimization problems, which has a clear economic interpretation. This feature is obviously desirable, but in the risk-averse case it leads to variational inequalities with set-valued operators -a class of problems for which no established software is currently available. To overcome this difficulty, we define a sequence of approximating differentiable variational inequalities based on smoothing the nonsmooth risk measure in the agents' problems, such as average or conditional value-at-risk. The smoothed variational inequalities can be tackled by the PATH solver, for example. The approximation scheme is shown to converge, including the case when smoothed problems are solved approximately. An interesting by-product of our proposal is that the smoothing approach allows us to show existence of an equilibrium for the original problem. To assess the proposed technique, numerical results are presented. The first set of experiments is on randomly generated equilibrium problems, for which we compare the proposed methodology with the standard smooth reformulation of average value-at-risk minimization (using additional variables to rewrite the associated max-function). The second set of experiments deals with a part of the real-

Discrete approximation of two-stage stochastic and distributionally robust linear complementarity problems

“…This is a closed loop two-stage stochastic Nash-Cournot game where each player (firm) needs to make a decision on capacity before realization of uncertainty anticipating competition in future (second stage). At this point, we refer readers to Wongrin et al [31] for a deterministic model with application in electricity markets, and a more sophisticated two-stage stochastic model by Luna, Sagastizábal and Solodov [20] where each player is risk-averse and all players share an identical constraint in the second stage. Similar models can also be found in Ralph and Smeers [22] for stochastic endogenous equilibrium in asset pricing.…”

Risk aversion in imperfect natural gas markets

“…This allows us to solve the problem directly, thus avoiding smoothed auxiliary variants of the genuine problem (cf. Luna et al (2016)). Focusing on shale gas, and the uncertainties and risks connected to it in a realistic representation of the north and central European gas market, we show how risk aversion affects reserve exploration and production capacity expansion in conventional and shale gas by the suppliers.…”

“…The Average Value-at-Risk and the risk functional R are convex, and thus, sub-differentiable, although not differentiable in the strict sense. Luna et al (2016) employ the identity Eq. ( 12) and replace the kink-function…”

“…Chin and Siddiqui (2014) develop a two-stage electricity market model with generation capacity expansions, and forward and spot markets; they analyze a monopoly, duopoly, and perfectly competitive setting with risk-averse suppliers. To the best of our knowledge, Luna et al (2016) is the only reference considering risk aversion in a natural gas market in a two-stage setting, however for one type of natural gas resources only. They develop an approximation scheme to solve two model variants.…”

“…They develop an approximation scheme to solve two model variants. We implement the same risk measure as Luna et al (2016), but show that an approximation-decomposition approach is not needed to solve the resulting model.…”