On the Properties of Equilibria in Private Value Divisible Good Auctions with Constrained Bidding

I analyze a model of a private value divisible good auction with different payment rules, standard rationing rule pro-rata on-the-margin and both with and without a restriction on the number of bids (steps) bidders can submit. I provide characterization of equilibrium bidding strategies in a model with restricted strategy sets and I show that these equilibria converge to some equilibrium of the model with unrestricted strategy sets as the restrictions are relaxed. In particular, the equilibrium conditions require that the Euler condition characterizing equilibrium strategies in the unrestricted model holds “on average” over the intervals defined by the length of each step of the restricted strategy, where the average is taken with respect to the endogenous distribution of the market clearing price. I also prove that the foregone surplus of bidders from using K steps rather than the optimal continuous bids is proportional to 1 K2 . Sufficient conditions for equilibrium existence are also provided.