  We study equilibria in two-buyer sequential second-price (or first-price)
auctions for identical goods. Buyers have weakly decreasing incremental values,
and we make a behavioural no-overbidding assumption: the buyers do not bid
above their incremental values. Structurally, we show equilibria are
intrinsically linked to a greedy bidding strategy. We then prove three results.
First, any equilibrium consists of three phases: a competitive phase, a
competition reduction phase and a monopsony phase. In particular, there is a
time after which one buyer exhibits monopsonistic behaviours. Second, the
declining price anomaly holds: prices weakly decrease over time at any
equilibrium in the no-overbidding game, a fact previously known for equilibria
with overbidding. Third, the price of anarchy of the sequential auction is
exactly $1 - 1/e$.
