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.

We study "Nash dynamics" in the context of adversarial deviations in blockchain protocols. We introduce a formal model, within which one can assess whether the Nash dynamics can lead utility-maximizing participants to defect from the "honest" protocol operation, towards variations that exhibit one or more undesirable infractions that affect protocol security, like abstaining from participation and producing conflicting protocol histories. Blockchain protocols that lead to no such infraction states are deemed compliant. Armed with this model, we evaluate the compliance of various Proof-of-Work (PoW) and Proof-of-Stake (PoS) protocol families, under different utility functions and reward schemes, leading to the following results: i) PoW and PoS protocols exhibit different compliance behavior, depending on the lossiness of the network; ii) PoS ledgers can be compliant w.r.t. one realistic infraction (producing conflicting messages) but non-compliant (hence non-equilibria) w.r.t. others (abstaining or an attack we call selfish signing); iii) considering externalities, like exchange rate fluctuations, we quantify the benefit of economic penalties in the context of PoS protocols as mitigation for particular infractions that affect protocol security.

Proof of Work (PoW) blockchains are susceptible to adversarial majority mining attacks in the early stages due to incipient participation and corresponding low net hash power. Bootstrapping ensures safety and liveness during the transient stage by protecting against a majority mining attack, allowing a PoW chain to grow the participation base and corresponding mining hash power. Liveness is especially important since a loss of liveness will lead to loss of honest mining rewards, decreasing honest participation, hence creating an undesired spiral; indeed existing bootstrapping mechanisms offer especially weak liveness guarantees.