Forward-Looking Bidding in Online Auctions. - The University of ...

game. Finally, it is notable that bidders only submit positive bids on products of their desiredtype – a result that will lead to identification of personal preferences in the empirical test.The bidding function is fully characterized by the expected surplus function S, which is inturn characterized by the steady-state distribution of the future competition c 1 conditional on thecurrent competition c 0 and all the state variables involved in the relationship between them:( | , , , , , )G c c ϕ ϕ ϕ ω ω . In equilibrium, the surplus function must reflect the actual expected1 0 0 1 2 1 2surplus given that everyone uses the optimal bidding strategy (2). Therefore, the equilibriumexpected surplus function must satisfy the Bellman equation (3). Such an S function existsbecause of the continuity of f, compactness of its support, and the fact (shown in the proof ofProposition 1) that the slope of S in any of its arguments is uniformly bounded. However,equilibrium S cannot be expressed in closed form even for a simple distribution F and small N.Despite the lack of a closed-form solution, some general comparative statics of the biddingfunction can be derived from an analysis of the Bellman equation (see Appendix for a proof):Proposition 2: For all F, the equilibriumb( ϕ , ϕ , ω , v)0 1 1has the following properties:( )1) b 1, ϕ , ω , v increases in ω2)1 1( ) ( )1( )b 0, ϕ , ω , v = 0< b 1,1, ω , v < b 1,0, ω , v < v for all v>0.1 1 1 1( )3) b 1, ϕ , ω , v decreases in ρ1 1The first property shows that bids decrease when the future gets closer in the sense that ω 1decreases. In the empirical section, a generalization of this result will be tested, namely theprediction that bids decrease as the number of auctions in the next hour increases. The secondproperty contains several important results. The first inequality shows that all bidders withpositive valuations submit positive bids on objects. This both guarantees trade and allows an13