The Cost of Pricing Rules in Electricity Auctions: A Neural Network Approach to Counterfactual Bayesian Nash Equilibria

This paper studies counterfactual equilibrium bidding in centralized electricity procurement auctions, where strategic behavior depends on the payment rule. We compare pay-as-clear (PAC) and pay-as-bid (PAB) mechanisms. PAC is often valued for its simplicity and for inducing bids closer to marginal cost, but it can generate substantial infra-marginal rents because the market-clearing price is paid on all accepted units. PAB, by contrast, pays each accepted bidder its own offer, compressing infra-marginal rents but inducing upward bid shading. Procurement expenditures therefore reflect a trade-off between infra-marginal rents under PAC and bid shading under PAB, making the ranking between the two formats theoretically ambiguous. To address this question empirically,we adopt a two-stage framework. First, we recover the distribution of firms’ marginal costs using a residual-demand bootstrap estimator that conditions on common demand and supply shocks (Balat et al. (2022)). Second, we compute counterfactual Bayesian Nash equilibria by approximating bidding policies with neural networks trained through an evolution-strategies procedure inspired by the Neural Pseudogradient Ascent (NPGA, Bichler et al. (2021)) algorithm. We extend this procedure to accommodate both partially and fully asymmetric environments across firms. We apply this framework to the Colombian electricity market, estimate counterfactual equilibria under PAB, and compare the resulting procurement expenditures with a model-consistent PAC benchmark. Our empirical results indicate that, under the maintained thermal-plant simple-bidding model, PAC reduces average daily procurement expenditures in the spot market for thermal plants by approximately COP 2.4 billion relative to the PAB counterfactual under constant marginal costs, and by COP 3.5 billion under variable and ramping costs, over the sample period. More broadly, the results suggest that the markup distortion under PAB can outweigh its rent-compression effect, making PAC less costly in similar market environments.