Cost of Additional Rooms

Our hotel price predictor described in Section 4 assumes that ATTac-2001's bids do not affect the ultimate closing price (Assumption 3 from Section 2). This assumption holds in a large economy. However in TAC, each hotel auction involved $8$ agents competing for $16$ hotel rooms. Therefore, the actions of each agent had an appreciable effect on the clearing price: the more hotel rooms an agent attempted to purchase, the higher the clearing price would be, all other things being equal. This effect needed to be taken into account when solving the basic allocation problem.

The simplified model used by ATTac-2001 assumed that the $n$th highest bid in a hotel auction was roughly proportional to $c^{-n}$ (over the appropriate range of $n$) for some $c\geq1$. Thus, if the predictor gave a price of $p$, ATTac-2001 only used this for purchasing two hotel rooms (the ``fair'' share of a single agent of the $16$ rooms), and adjusted prices for other quantities of rooms by using $c$.

For example, ATTac-2001 would consider the cost of obtaining $4$ rooms to be $4pc^2$. One or two rooms each cost $p$, but $3$ each cost $pc$, $4$ each cost $pc^2$, 5 each cost $pc^3$, etc. So in total, $2$ rooms cost $2p$, while $4$ cost $4pc^2$. The reasoning behind this procedure is that if ATTac-2001 buys two rooms -- its fair share given that there are 16 rooms and 8 agents, then the 16th highest bid (ATTac-2001's 2 bids in addition to 14 others) sets the price. But if ATTac-2001 bids on an additional unit, the previous 15th highest bid becomes the price-setting bid: the price for all rooms sold goes up from $p$ to $pc$.

The constant $c$ was calculated from the data of several hundred games during the seeding round. In each hotel auction, the ratio of the $14$th and $18$th highest bids (reflecting the most relevant range of $n$) was taken as an estimate of $c^4$, and the (geometric) mean of the resulting estimates was taken to obtain $c = 1.35$.

The LP allocator takes these price estimates into account when computing $G^*$ by assigning higher costs to larger purchase volumes, thus tending to spread out ATTac-2001's demand over the different hotel auctions.

In ATTac-2001, a few heuristics were applied to the above procedure to improve stability and to avoid pathological behavior: prices below $1 were replaced by $1 in estimating $c$; $c=1$ was used for purchasing fewer than two hotel rooms; hotel rooms were divided into early closing and late closing (and cheap and expensive) ones, and the $c$ values from the corresponding subsets of auctions of the seeding rounds were used in each case.