We consider matching markets, where an auctioneer is selling  a set of
indivisible items E to a set of bidders  with  private valuations v_j.
In an ascending auction, the auctioneer decides on an assignment of
items to bidders, together with prices p_i for all items i in E, based
on the demand indicated by the bidders throughout the auction.

In the first part of the talk, we consider computability and
monotonicity of buyer-optimal Walrasian equilibria in ascending
auctions. A price vector p is called "Walrasian", if  there exists an
assignment under which every bidder j receives a bundle S of maximum
payoff v_j(S)-p(S), and, additionally, all items of positive price are
sold. Walrasian prices are called "buyer-optimal" if they minimise the
total payment of the bidders among all Walrasian prices. It is
well-known that buyer-optimal Walrasian prices exist and can be computed
efficiently if the valuations v_j are gross-substitute.
Moreover, it is known that an ascending auction, which iteratively
raises the prices uniformly on items in a so-called "inclusionswise
minimal maximal overdemanded set" terminates with buyer-optimal
Walrasian prices, and  that these  overdemanded sets can be computed
efficiently with tools from Discrete Convexity.
We will see how to compute the desired overdemanded set in each
iteration in a simple combinatorial way with the matroid partition
algorithm. Moreover, we will use the  obtained structural insights to
show that the buyer optimal Walrasian prices are monotone in supply and
demand.

In the second part of the talk, we consider ascending auctions where the
auctioneer is restricted to sell a basis of a matroid to the bidders. It
is known that a very natural and simple ascending auction is
welfare-maximising and truthful in the sense that the prices coincide
with the Vickrey payments. We will provide a simple proof of this
result, combined with certain sensitivity results.

(Joint work with Katharina Eickhoff, Niklas Rieken, and Laura Vargas Koch).