Mixed Strategies in Discriminatory Divisible-Good Auctions

Abstract

Using the concept of market-distribution functions, we derive general optimality conditions for discriminatory divisible-good auctions, which are also applicable to Bertrand games and non-linear pricing. We introduce the concept of offer distribution function to analyze randomized offer curves, and characterize mixed-strategy Nash equilibria for pay-as-bid auctions where demand is uncertain and costs are common knowledge; a setting for which pure-strategy supply function equilibria typically do not exist. We generalize previous results on mixtures over horizontal offers as in Bertrand-Edgeworth games, but more importantly we characterize novel mixtures over partly increasing supply functions.

Electricity Markets: Designing Auctions Where Suppliers Have Uncertain Costs

“…This concurs with the competitive outcomes for non-pivotal producers in the Bertrand game, in von der Fehr and Harbord (1993) and in Fabra et al (2006). If signals are independent and producers pivotal, it follows from Harsanyi's (1973) puri…cation theorem that in the limit when costs are almost surely common knowledge, our Bayesian Nash equilibria correspond to the mixed-strategy NE that Anderson et al (2013), Anwar (2006), Fabra et al (2006), Genc (2009), Son et al (2004) and von der Fehr and Harbord (1993 derive for uniform-price and discriminatory auctions.…”

“…In practice, a producer is normally allowed to make more than the single bid that is considered here and in von der Fehr and Harbord (1993). However, as shown by Genc (2009) and Anderson et al (2013), there are many circumstances in discriminatory auctions where a producer …nds it optimal to o¤er its whole production capacity at one price, and then the equilibrium is not in ‡uenced by the single-bid restriction. Similar to the extension of a singlebid model to a multi-bid model by Fabra et al (2006), we could also argue that it is enough to solve for the marginal bids of the two producers as long as other bids never clear the market, i.e.…”

Supply Function Equilibria in Networks with Transport Constraints

“…For example, consider networks with discriminatory pricing as in the electricity market of Britain. Anderson et al [3] show that the optimality condition of a …rm's o¤er in such an auction is given by 19 …”

“…Firms can in ‡uence the market integration factor with their o¤er curves, so it is endogenous. We use our optimality conditions to solve for symmetric equilibria in two-node and star networks 3 with multi-dimensional uniformly distributed demand shocks. In this case, the market integration factor is found to be constant for a given network with given transmission capacities and total production capacities.…”

“…Thus with the slope of residual demand we here mean the slope of residual demand conditional on a …xed output. 3 There are no strategic producers in the center node of the star network. Thus the network is symmetric from the producers'perspective.…”