On the continuous equilibria of affiliated-value, all-pay auctions with private budget constraints

•We study the all-pay auction assuming bidders have private budget constraints.•We study the war of attrition assuming bidders have private budget constraints.•We characterize a symmetric equilibrium in continuous strategies.•Budget constraints encourage more aggressive bidding among some participants.

We consider all-pay auctions in the presence of interdependent, affiliated valuations and private budget constraints. For the sealed-bid, all-pay auction we characterize a symmetric equilibrium in continuous strategies for the case of N bidders. Budget constraints encourage more aggressive bidding among participants with large endowments and intermediate valuations. We extend our results to the war of attrition where we show that budget constraints lead to a uniform amplification of equilibrium bids among bidders with sufficient endowments. An example shows that with both interdependent valuations and private budget constraints, a revenue ranking between the two auction formats is generally not possible. Equilibria with discontinuous bidding strategies are discussed.