Liquidity Risk in Sequential Trading Networks: Online Appendix

The pure common-value auction with private budget constraints that we study admits multiple equilibria, which replicate the bid distribution from the monotone, pure strategy equilibrium defined in Section 3. As explained in the main text, a bidder is indifferent among a range of bids. Thus, he has multiple best reply bids and may randomize among them. Provided that the resulting aggregate distribution of bids (averaged over all types of bidders) replicates the distribution of bids from the monotone, pure strategy equilibrium another equilibrium obtains. To formalize this observation, we can define a family of equilibria in distributional strategies. Following Milgrom and Weber (1985), a distributional strategy ηri(b, w) for bidder i in row r as a joint probability measure over bids and types. Let ν∗ r−1 and Ur(w) be defined as in Theorem 1 and define w∗ r as in Corollary 1. It is straightforward to verify that any family of ηri(w, b) satisfying the following conditions is an equilibrium of the trading game.