Re-entry Effect in War of Attrition and Pay-to-bid Auctions

We study the effect of entry and re-entry option in the pay-to-bid auctions in which risk-loving bidders pay a constant fee each time they bid for an object whose monetary value is common knowledge among the bidders, and bidding fees are the only source of benefit for the seller. We show that for the proposed model there exists a unique Symmetric Subgame Perfect Equilibrium (SSPE). The characterized SSPE is stationary when re-entry in the auction is allowed, and it is Markov perfect when re-entry is forbidden. Furthermore, we fully characterize the expected revenue of the seller. Generally, with or without re-entry, it is more beneficial for the seller to choose v (value of the object), s (sale price), and c (bidding fee) such that v−s c becomes sufficiently large. In particular, when re-entry is permitted: the expected revenue of the seller is independent of the number of bidders, decreasing in the sale price, increasing in the value of the object, and decreasing in the bidding fee; Moreover, the seller’s revenue is equal to the value of the object when players are risk neutral, and it is strictly greater than the value of the object when bidders are risk-loving. Finally, and most importantly, we show that allowing re-entry can be important in practice. Because, if the seller were to run such an auction without allowing re-entry, the auction would last a long time, and for almost all of its duration have only two remaining players. Thus, the seller’s revenue relies on those two players being willing to participate, without any breaks, in an auction that might last for thousands of rounds.