Consider a first-price, sealed-bid auction with interdependent valuations and private budget constraints. Private budget constraints introduce subtle strategic tradeoffs with first-order consequences for equilibrium bidding. In a pure-strategy, symmetric equilibrium, agents may adopt discontinuous bidding strategies resulting in a stratification of competition along the budget dimension. In an asymmetric setting, equilibria in “nondecreasing” strategies exist, albeit in a qualified sense. Private budgets introduce significant confounds for the interpretation of bidding data due to their interaction with risk preferences and their countervailing strategic implications.
Kotowski, Maciej. "First-Price Auctions with Budget Constraints." HKS Faculty Research Working Paper Series RWP19-021, June 2019.