... agent¹

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 $r_i$ in Equation 8 does not exploit the fact that the sample $S$ contains only a finite set of possibilities for $y_i$, which might make it more robust to inaccuracies in the sampling.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... auction.⁵

For large enough $m$ it is practically the same as the more efficient $m+1$st auction. We use the $m$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 $k$, but expect that the algorithm is not sensitive to it for sufficiently large values of $k$.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...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.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.