Deciding whether a pure Bayes-Nash equilibrium exists in a discrete first-price auction with correlated values is NP-hard, while approximate monotone equilibria for affiliated values admit polynomial-time algorithms.
Monotone comparative statics under uncertainty
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Equilibrium Computation in First-Price Auctions with Correlated Priors
Deciding whether a pure Bayes-Nash equilibrium exists in a discrete first-price auction with correlated values is NP-hard, while approximate monotone equilibria for affiliated values admit polynomial-time algorithms.