  We study game-theoretic models for capturing participation in blockchain
systems. Permissionless blockchains can be naturally viewed as games, where a
set of potentially interested users is faced with the dilemma of whether to
engage with the protocol or not. Engagement here implies that the user will be
asked to complete certain tasks, whenever they are selected to contribute
(typically according to some stochastic process) and be rewarded if they choose
to do so. Apart from the basic dilemma of engaging or not, even more strategic
considerations arise in settings where users may be able to declare
participation and then retract before completing their tasks (but are still
able to receive rewards) or are rewarded independently of whether they
contribute. Such variations occur naturally in the blockchain setting due to
the complexity of tracking ``on-chain'' the behavior of the participants.
  We capture these participation considerations offering a series of models
that enable us to reason about the basic dilemma, the case where retraction
effects influence the outcome and the case when payments are given universally
irrespective of the stochastic process. In all cases we provide
characterization results or necessary conditions on the structure of Nash
equilibria. Our findings reveal that appropriate reward mechanisms can be used
to stimulate participation and avoid negative effects of free riding, results
that are in line but also can inform real world blockchain system deployments.
