  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 $c_i q_i$, and
then receives reward $\frac{q_i^\alpha}{\sum_j q_j^\alpha}$, for some $\alpha
\geq 1$. When $c_i = c_j$ for all $i,j$, and $\alpha = 1$, 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
$1-\frac{c_j}{c_i}$. 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
($\alpha > 1$), every market participant has a market share of at least
$1-\frac{1}{\alpha}$, implying that the market features at most
$\frac{\alpha}{\alpha - 1}$ 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.
