We study the pay-as-bid auction game, a supply function model with discriminatory pricing and asymmetric firms. In this game, strategies are non-decreasing supply functions relating pric to quantity and the exact choice of the strategy space turns out to be a crucial issue: when it includes all non-decreasing continuous functions, pure-strategy Nash equilibria often fail to exist. To overcome this, we restrict the strategy space to the set of Lipschitz-continuous functions and we prove that Nash equilibria always exist (under standard concavity assumptions) and consist of functions that are affine on their own support and have slope equal to the maximum allowed Lipschitz constant. We further show that the Nash equilibrium is unique up to the market-clearing price when the demand is affine and the asymmetric marginal production costs are homogeneous in zero. For quadratic production costs, we derive a closed-form expression and we compute the limit as the allowed Lipschitz constant grows to infinity. Our results show that in the limit the pay-as-bid auction game achieves perfect competition with efficient allocation and induces a lower market-clearing price compared to supply function models based on uniform price auctions.
Vanelli et al. (Fri,) studied this question.