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Although Bitcoin was intended to be a decentralized digital currency, in practice, mining power is quite concentrated. This fact is a persistent source of concern for the Bitcoin community. We provide an explanation using a simple model to capture miners' incentives to invest in equipment. In our model, n miners compete for a prize of fixed size. Each miner chooses an investment q_i, incurring cost c_iq_i, and then receives reward q^{\alpha}∑_j q_j^{\alpha}, for some \alpha≥1. When c_i = c+j for all i,j, and α=1, there is a unique equilibrium where all miners invest equally. However, we prove that under seemingly mild deviations from this model, equilibrium outcomes become drastically more centralized. In particular, (a) When costs are asymmetric, if miner i chooses to invest, then miner j has market share at least 1−c_j/c_i. That is, if miner j has costs that are (e.g.) 20% lower than those of miner i, then miner j must control at least 20% of the \emph{total} mining power. (b) In the presence of economies of scale (α>1), every market participant has a market share of at least 1−1/α, implying that the market features at most α/(α−1) miners in total. We discuss the implications of our results for the future design of cryptocurrencies. In particular, our work further motivates the study of protocols that minimize "orphaned" blocks, proof-of-stake protocols, and incentive compatible protocols. 

We consider the problem of a single seller repeatedly selling a single item to a single buyer (specifically, the buyer has a value drawn fresh from known distribution $$D$$ in every round). Prior work assumes that the buyer is fully rational and will perfectly reason about how their bids today affect the seller's decisions tomorrow. In this work we initiate a different direction: the buyer simply runs a no-regret learning algorithm over possible bids. We provide a fairly complete characterization of optimal auctions for the seller in this domain. Specifically: 1) If the buyer bids according to EXP3 (or any ``mean-based'' learning algorithm), then the seller can extract expected revenue arbitrarily close to the expected welfare. This auction is independent of the buyer's valuation $$D$$, but somewhat unnatural as it is sometimes in the buyer's interest to overbid. 2) There exists a learning algorithm $$\mathcal{A}$$ such that if the buyer bids according to $$\mathcal{A}$$ then the optimal strategy for the seller is simply to post the Myerson reserve for $$D$$ every round. 3) If the buyer bids according to EXP3 (or any ``mean-based'' learning algorithm), but the seller is restricted to ``natural'' auction formats where overbidding is dominated (e.g. Generalized First-Price or Generalized Second-Price), then the optimal strategy for the seller is a pay-your-bid format with decreasing reserves over time. Moreover, the seller's optimal achievable revenue is characterized by a linear program, and can be unboundedly better than the best truthful auction yet simultaneously unboundedly worse than the expected welfare.