Equilibria in two-stage electricity markets

Most electricity markets have multiple stages, which include one or more forward markets and the spot market. We consider two stages - a day-ahead market and a real-time market. We study equilibrium outcomes in such markets assuming demand to be deterministic. We show via counterexamples that in such two-stage electricity markets, (i) a Nash equilibrium may not exist, or (ii) there may be multiple inefficient Nash equilibria. We also give two sufficient conditions - a "congestion-free" condition and a "monopoly-free" condition - under which a subgame perfect Nash equilibrium exists and yields efficient outcome.