  Cryptographic Self-Selection is a paradigm employed by modern Proof-of-Stake
consensus protocols to select a block-proposing "leader." Algorand [Chen and
Micali, 2019] proposes a canonical protocol, and Ferreira et al. [2022]
establish bounds $f(\alpha,\beta)$ on the maximum fraction of rounds a
strategic player can lead as a function of their stake $\alpha$ and a network
connectivity parameter $\beta$. While both their lower and upper bounds are
non-trivial, there is a substantial gap between them (for example, they
establish $f(10\%,1) \in [10.08\%, 21.12\%]$), leaving open the question of how
significant of a concern these manipulations are. We develop computational
methods to provably nail $f(\alpha,\beta)$ for any desired $(\alpha,\beta)$ up
to arbitrary precision, and implement our method on a wide range of parameters
(for example, we confirm $f(10\%,1) \in [10.08\%, 10.15\%]$).
  Methodologically, estimating $f(\alpha,\beta)$ can be phrased as estimating
to high precision the value of a Markov Decision Process whose states are
countably-long lists of real numbers. Our methodological contributions involve
(a) reformulating the question instead as computing to high precision the
expected value of a distribution that is a fixed-point of a non-linear sampling
operator, and (b) provably bounding the error induced by various truncations
and sampling estimations of this distribution (which appears intractable to
solve in closed form). One technical challenge, for example, is that natural
sampling-based estimates of the mean of our target distribution are \emph{not}
unbiased estimators, and therefore our methods necessarily go beyond claiming
sufficiently-many samples to be close to the mean.
