Bidding process in online auctions and winning strategy: Rate equation approach

Online auctions have expanded rapidly over the last decade and have become a fascinating new type of business or commercial transaction in this digital era. Here we introduce a master equation for the bidding process that takes place in online auctions. We find that the number of distinct bidders who bid $k$ times up to the $t$th bidding progresses, called the $k$-frequent bidder, seems to scale as $n_k\left(t\right)\sim t{k}^{-2.4}$. The successfully transmitted bidding rate by the $k$-frequent bidder is likely to scale as $q_k\left(t\right)\sim k^{-1.4}$, independent of $t$ for large $t$. This theoretical prediction is close to empirical data. These results imply that bidding at the last moment is a rational and effective strategy to win in an eBay auction.

Figure 1

Plot of $N\left(T\right)$ vs $T$ for the eBay (a) and the Korean auction (b), where $N\left(T\right)$ is the number of distinct bidders who successfully bid at least once up to terminal time $T$. The dotted lines have slopes of 1 for (a) and 0.22 for (b).

Figure 2

Plot of $P_f\left(T\right)$ vs $T$ in (a) and (c), and $P_n\left(N\right)$ vs $N$ in (b) and (d), where $P_f\left(T\right)$ is the distribution of total number of bids, and $P_n\left(N\right)$ is the distribution of total number of participating bidders on each items, for the eBay (a) and (b) and the Korean auction (c) and (d) in semilogarithmic scale. The dotted lines have slopes of $2.5$ in (a), $5.6$ in (b), $7.4$ in (c), and $10.8$ in (d).

Figure 3

Plot of the transition rate $⟨dk∕dt⟩$ averaged over different items vs $k∕t$ for the eBay (a) and for the Korean auction (b). The dotted lines, obtained by the least square fit in the range [0.1:1] for (a) and [0.01:1] for (b), respectively, fit to the formula, $\approx 0.7k∕t$ for both data.

Figure 4

Plot of $⟨N\left(t\right)⟩$ vs $t$, averaged over different items for the eBay data. The straight line has a slope of $0.7$ obtained from the least square fit.

Figure 5

Plot of $n_k\left(t\right)∕n_1\left(t\right)$ vs $k$ for the eBay auction (a) and for the Korean auction (b) for various terminal times $T$. Different terminal times are represented by different symbols shown in each panel. The solid lines have a slope of $-2.4$ drawn for guidance.

Figure 6

Plot of the relative winning probability $q_k\left(T\right)∕q_1\left(T\right)$ of the $k$-frequent bidder to that of the one-frequent bidder at the last moment vs frequency $k$. The dotted line has a slope of $-1.4$ drawn for guidance.