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On the contrary, marginal values for AON auctions do not contain similarly valuable
information. By assuming divisibility with AON auctions, the marginal value provided
is neither accurate for the marginal nor a bound on the price for additional units. In
this case, the quote provides a very loose approximation of value. Similarly, if we took
into consideration indivisibility constraints when quoting marginal values with AON
auctions, the result would be such very conservative. It is often undefined for the
BID
(we need an agent or combination of them intending to buy a single unit), and the
ASK
can often be excessively high (when the auction matches bids with sell quantities much
larger than the marginal). Moreover, this quote provides no information about the unit
price beyond the marginal, showing only that the total payment for more units will be
at least the price for a single unit.
The effects described above were confirmed in part by measurements applied to our
simulation results. We define a trade as
desired
(A) with respect to agent
i
, if, once ex-
ecuted, it increases or maintains the profit (i.e., income minus costs plus cashflow) of
i
assuming that no further trades occur. We identified three possible reasons for agent
i
to engage in undesired trading:
outdated information
(B1),
misleading non-anonymous
quotes
(B2), or
misleading anonymous quotes
(B3). A bid can contain outdated infor-
mation because its points were calculated incrementally or due to the asynchronous
nature of the bidding process. Outdated information (B1) thus refers to the case in
which
i
engaged in a trade that it would have rejected had it reevaluated its bid using
the most up-to-date information (i.e., current holdings and quotes). Misleading non-
anonymous (B2) or anonymous (B3) quotes are those that made
i
believe that it could
buy or sell goods at the quote, when that was actually not possible. Non-anonymous
quotes are those calculated by excluding from the order topic the bid sent by agent
i
.
Finally, every transaction that decreased utility and cannot be otherwise explained must
have occurred because of a
dependency on other auctions
(C). Such dependencies exist
because agents construct their bids in an auction assuming they could trade in other auc-
tions at the quote. Since communication is asynchronous and auctions clear at different
times, some intermediate decreases in utility are normal and expected.
Suppose we had
T
transactions, and transaction
t
occurred in auction
g
for quantity
q
t
and payment
p
t
. We perform two different optimizations for each agent
i
:
-
R
∗
(
H
) is the profit achieved by
i
when calculating its optimal production plan based
on holdings
H
, assuming that
i
cannot trade further.
-
P
∗
(
H
,
Q
) is the highest payment that
i
is willing to offer to trade quantity
q
t
in
auction
g
, assuming it holds goods
H
and that it could freely trade goods in auctions
other than
g
at the prices given by quotes
Q
.
We further define
Q
i
and
Q
as the most up-to-date non-anonymous and anonymous
quotes, respectively. Holdings
H
t
are the goods held by
i
in all auctions right after
t
occurred; holdings
H
0
are initial endowments; holdings
H
t
i
and
H
t
are the goods held
right after
t
and an hypothetical clear of all auctions other than
g
occurred, assuming
that
i
bid to achieve optimal holdings as calculated for
P
∗
(
H
t
−
1
,
Q
i
) and
P
∗
(
H
t
−
1
,
Q
),
respectively; and
p
t
is the lowest hypothetical payment that
i
could have bid in order to
trade
q
t
in
g
. Negative coefficients for payments and quantities are used for sell offers.
Given these definitions, we can classify transaction
t
for agent
i
as follows.
