Bidding Reversals in a Multiple-Good Auction with Aggregate Reserve Price Online Appendix

Deviations to other types’ equilibrium strategies. Let π(t̂, t) be the expected payoff of a bidder with type t̂ who makes the equilibrium bid of a bidder with type t. For t ≥ t∗∗, he wins the small item for free when his opponent has type tj > t, and the large object for B(t) otherwise, so π(t̂, t) = (1− t)u(t̂) + tU(t̂)− tE (max{r,X(s)} | s < t) = u(t̂) + tX(t̂)− t∗∗r − ∫ t t∗∗ X(s)ds ↓ ∂ ∂tπ(t̂, t) = X(t̂)−X(t) For t∗ ≤ t < t∗∗, he wins the small object for b(t) when tj > t, and the large object for r otherwise, so π(t̂, t) = (1− t)u(t̂)− (1− t) ( t∗∗−t 1−t E (r −X(s) | s ∈ (t, t ∗∗)) ) + tU(t̂)− tr = u(t̂)− (t∗∗ − t)r + (t∗∗ − t)E (X(s) | s ∈ (t, t∗∗)) + tX(t̂)− tr = u(t̂) + tX(t̂)− t∗∗r + ∫ t∗∗ t X(s)ds ↓ ∂ ∂tπ(t̂, t) = X(t̂)−X(t)