- ... agent
^{1} - Top-scoring by one metric, and second place
by another.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... profit.
^{2} - The problem is
computationally difficult in general, but has been solved effectively
in the non-trivial TAC setting [Greenwald BoyanGreenwald Boyan2001,Stone, Littman, Singh, KearnsStone
et al.2001].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... situations.
^{3} - An alternative approach would
be to abstractly calculate the Bayes-Nash
equilibrium [HarsanyiHarsanyi1968] for the game and play the optimal
strategy. We dismissed this approach because of its intractability in
realistically complex situations, including TAC. Furthermore, even if
we were able to approximate the equilibrium strategy, it is reasonable
to assume that our opponents would not play optimal strategies. Thus,
we could gain additional advantage by tuning our
approach to our opponents'
*actual*behavior as observed in the earlier rounds, which is essentially the strategy we adopted.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... value.
^{4} - Note that the strategy
for choosing in Equation 8 does not exploit the fact
that the sample contains only a finite set of possibilities for
, which might make it more robust to inaccuracies in the
sampling.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... auction.
^{5} - For large
enough it is practically the same as the more efficient st
auction. We use the th price model because that is what is used in
TAC's hotel auctions.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... substitutability.
^{6} - Goods are considered
complementary if their value as a package is
greater than the sum of their individual values; goods are considered
substitutable if their value as a package is less than the sum of
their individual values.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ....
^{7} - We did not experiment with
varying , but expect that the algorithm is not sensitive to it for
sufficiently large values of .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...Wellman03rlv.
^{8} - Indeed, in the TAC-03 competition,
ATTac-2001 was entered using the trained models from 2001, and it won
the competition, suggesting further that the failure in 2002 was due
to a problem with the learned models that were used during the finals
in 2002.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... score).
^{9} - We suspect that
were the agents allowed to retrain over the course of the experiments,
*ATTac-2001*would end up improving, as we saw in Phase III of the previous set of experiments. Were this to occur, it is possible that*EarlyBidder*would no longer be able to invade.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .