Virtual Bidding: Equilibrium, Learning, and the Wisdom of Crowds*

We present a theoretical analysis of virtual bidding in a stylized model of a single bus, two-settlement electricity market. North-American ISOs typically take a conservative approach to uncertainty, scheduling supply myopically in day-ahead (DA) markets to meet expected demand, neglecting the subsequent cost of recourse required to correct imbalances in the realtime (RT) market. This can result in generation costs that far exceed the minimum expected cost of supply. We explore the idea that virtual bidding can mitigate this excess cost incurred by myopic scheduling on the part of the ISO. Adopting a game-theoretic model of virtual bidding, we show that as the number of virtual bidders increases, the equilibrium market outcome tends to the socially optimal DA schedule, and prices converge between the DA and RT markets. We additionally analyze the effects of virtual bidding on social welfare and the variance of the price spread. Finally, we establish a repeated game formulation of virtual bidding, and investigate simple learning strategies for virtual bidders that guarantee convergence to the Nash equilibrium.