The Full Approach

Leveraging from the preceding analysis, we define our sampling-based
approach to profit prediction in general simultaneous, multi-unit
auctions for interacting goods. In this scenario, let there be
simultaneous, multi-unit auctions for interacting goods
. The auctions might close at different times and
these times are not, in general, known in advance to the bidders.
When an auction closes, let us assume that the units available are
distributed irrevocably to the highest bidders, who each need to
pay the price bid by the th highest bidder. This scenario
corresponds to an th price ascending auction.^{5} Note that the same bidder may place multiple
bids in an
auction, and thereby potentially win multiple units. We assume that
after the auction closes, the bidders will no longer have any
opportunity to acquire additional copies of the goods sold in that
auction (*i.e.*, there is no aftermarket).

Our approach is based upon five assumptions. For
, let
represent the
value derived by the agent if it owns units of the commodity
being sold in auction . Note that is independent of the
*costs* of the commodities. Note further that this
representation allows for interacting goods of all kinds, including
complementarity and substitutability.^{6}
The assumptions of our approach are as follows:

- Closing prices are somewhat, but only
somewhat, predictable. That is, given a set of input features ,
for each auction , there exists a sampling rule that outputs a
closing price according to a probability distribution of
predicted closing prices for .
- Given a vector of holdings
where represents the quantity of the commodity being
sold in auction that are already owned by the agent, and given a
vector of fixed closing prices
, there exists a
tractable procedure
to determine the optimal set of
purchases
where
represents the number of goods to be purchased in auction such
that

for all . This procedure corresponds to the optimization problem in Equation 3. - An individual agent's bids do not have an
appreciable effect on the economy (large population assumption).
- The agent is free to change existing bids in
auctions that have not yet closed.
- Future decisions are made in the presence of complete price information. This assumption corresponds to the operator reordering approximation from the previous section.

By Assumption 3, the price predictor can generate predicted prices prior to considering one's bids. Thus, we can sample from these distributions to produce complete sets of closing prices of all goods.

For each good under consideration, we assume that it is the next one
to close. If a different auction closes first, we can then revise our
bids later (Assumption 4). Thus, we would like to bid
exactly the good's *expected marginal utility* to us. That is, we
bid the difference between the expected utilities attainable with and
without the good. To compute these expectations, we simply average
the utilities of having and not having the good under different price
samples as in Equation 8. This strategy rests on
Assumption 5 in that we assume that bidding the good's
current expected marginal utility cannot adversely affect our future
actions, for instance by impacting our future space of possible bids.
Note that as time proceeds, the price distributions change in response
to the observed price trajectories, thus causing the agent to
continually revise its bids.

Table 1 shows pseudo-code for the entire algorithm. A fully detailed description of an instantiation of this approach is given in Section 5.