Georgios Zervas — Page 2

arXiv2010

Heapable Sequences and Subsequences

Let us call a sequence of numbers heapable if they can be sequentially inserted to form a binary tree with the heap property, where each insertion subsequent to the first occurs at a leaf of the tree, i.e. below a previously placed number. In this paper we consider a variety of problems related to heapable sequences and subsequences that do not appear to have been studied previously. Our motivation for introducing these concepts is two-fold. First, such problems correspond to natural extensions of the well-known secretary problem for hiring an organization with a hierarchical structure. Second, from a purely combinatorial perspective, our problems are interesting variations on similar longest increasing subsequence problems, a problem paradigm that has led to many deep mathematical connections. We provide several basic results. We obtain an efficient algorithm for determining the heapability of a sequence, and also prove that the question of whether a sequence can be arranged in a complete binary heap is NP-hard. Regarding subsequences we show that, with high probability, the longest heapable subsequence of a random permutation of n numbers has length (1 - o(1)) n, and a subsequence of length (1 - o(1)) n can in fact be found online with high probability. We similarly show that for a random permutation a subsequence that yields a complete heap of size αn for a constant αcan be found with high probability. Our work highlights the interesting structure underlying this class of subsequence problems, and we leave many further interesting variations open for future work.

cs.DS↗

arXiv2010

Information Asymmetries in Pay-Per-Bid Auctions: How Swoopo Makes Bank

Innovative auction methods can be exploited to increase profits, with Shubik's famous "dollar auction" perhaps being the most widely known example. Recently, some mainstream e-commerce web sites have apparently achieved the same end on a much broader scale, by using "pay-per-bid" auctions to sell items, from video games to bars of gold. In these auctions, bidders incur a cost for placing each bid in addition to (or sometimes in lieu of) the winner's final purchase cost. Thus even when a winner's purchase cost is a small fraction of the item's intrinsic value, the auctioneer can still profit handsomely from the bid fees. Our work provides novel analyses for these auctions, based on both modeling and datasets derived from auctions at Swoopo.com, the leading pay-per-bid auction site. While previous modeling work predicts profit-free equilibria, we analyze the impact of information asymmetry broadly, as well as Swoopo features such as bidpacks and the Swoop It Now option specifically, to quantify the effects of imperfect information in these auctions. We find that even small asymmetries across players (cheaper bids, better estimates of other players' intent, different valuations of items, committed players willing to play "chicken") can increase the auction duration well beyond that predicted by previous work and thus skew the auctioneer's profit disproportionately. Finally, we discuss our findings in the context of a dataset of thousands of live auctions we observed on Swoopo, which enables us also to examine behavioral factors, such as the power of aggressive bidding. Ultimately, our findings show that even with fully rational players, if players overlook or are unaware any of these factors, the result is outsized profits for pay-per-bid auctioneers.

cs.GT↗