  We analyze bribing attacks in Proof-of-Stake distributed ledgers from a game
theoretic perspective. In bribing attacks, an adversary offers participants a
reward in exchange for instructing them how to behave, with the goal of
attacking the protocol's properties. Specifically, our work focuses on
adversaries that target blockchain safety. We consider two types of bribing,
depending on how the bribes are awarded: i) guided bribing, where the bribe is
given as long as the bribed party behaves as instructed; ii) effective bribing,
where bribes are conditional on the attack's success, w.r.t. well-defined
metrics. We analyze each type of attack in a game theoretic setting and
identify relevant equilibria. In guided bribing, we show that the protocol is
not an equilibrium and then describe good equilibria, where the attack is
unsuccessful, and a negative one, where all parties are bribed such that the
attack succeeds. In effective bribing, we show that both the protocol and the
"all bribed" setting are equilibria. Using the identified equilibria, we then
compute bounds on the Prices of Stability and Anarchy. Our results indicate
that additional mitigations are needed for guided bribing, so our analysis
concludes with incentive-based mitigation techniques, namely slashing and
dilution. Here, we present two positive results, that both render the protocol
an equilibrium and achieve maximal welfare for all parties, and a negative
result, wherein an attack becomes more plausible if it severely affects the
ledger's token's market price.
