Bitcoin, a cryptocurrency built on the blockchain data structure, has generated significant academic and commercial interest. Contrary to prior expectations, recent research has shown that participants of the protocol (the so-called “miners”) are not always incentivized to follow the protocol. We study the game induced by one such attack – the pool block withholding attack – in which mining pools (groups of miners) attack other mining pools. We focus on the case of two pools attacking each other, with potentially other mining power in the system.We show that this game always admits a pure Nash equilibrium, and its pure price of anarchy, which intuitively measures how much computational power can be wasted due to attacks in an equilibrium, is at most 3. We conjecture, and prove in special cases, that it is in fact at most 2. Our simulations provide compelling evidence for this conjecture, and show that players can quickly converge to the equilibrium by following best response strategies.

Popular word embedding algorithms exhibit stereotypical biases, such as gender bias. The widespread use of these algorithms in machine learning systems can amplify stereotypes in important contexts. Although some methods have been developed to mitigate this problem, how word embedding biases arise during training is poorly understood. In this work we develop a technique to address this question. Given a word embedding, our method reveals how perturbing the training corpus would affect the resulting embedding bias. By tracing the origins of word embedding bias back to the original training documents, one can identify subsets of documents whose removal would most reduce bias. We demonstrate our methodology on Wikipedia and New York Times corpora, and find it to be very accurate.

Combinatorial auctions are multiple-item auctions in which bidders may place bids on any package (subset) of goods. This additional expressibility produces benefits that have led to combinatorial auctions becoming extremely important both in practice and in theory. In the computer science community, auction design has focused primarily on computational practicality and incentive compatibility. The latter concerns mechanisms that are resistant to bidders misrepresenting themselves via a single false identity; however, with modern forms of bid submission, such as electronic bidding, other types of cheating have become feasible. Prominent amongst them is false-name bidding; that is, bidding under pseudonyms. For example, the ubiquitous Vickrey-Clarke-Groves (VCG) mechanism is incentive compatible and produces optimal allocations, but it is not false-name-proof–bidders can increase their utility by submitting bids under multiple identifiers. Thus, there has recently been much interest in the design and analysis of false-name-proof auction mechanisms. These false-name-proof mechanisms, however, have polynomially small efficiency guarantees: they can produce allocations with very low economic efficiency/social welfare. In contrast, we show that, provided the degree to which different goods are complementary is bounded (as is the case in many important, practical auctions), the VCG mechanism gives a constant efficiency guarantee. Constant efficiency guarantees hold even at equilibria where the agents bid in a manner that is not individually rational. Thus, while an individual bidder may personally benefit greatly from making false-name bids, this will have only a small detrimental effect on the objective of the auctioneer: maximizing economic efficiency. So, from the auctioneer's viewpoint the VCG mechanism remains preferable to false-name-proof mechanisms.