  Despite the popularity and practical applicability of blockchains, there is
very limited work on the theoretical foundation of blockchains: The lack of
rigorous theory and analysis behind the curtain of blockchains has severely
staggered its broader applications. This paper attempts to lay out a
theoretical foundation for a specific type of blockchains---the ones requiring
basic authenticity from the participants, also called \textit{permissioned
blockchain}. We formulate permissioned blockchain systems and operations into a
game-theoretical problem by incorporating constraints implied by the wisdom
from distributed computing and Byzantine systems. We show that in a
noncooperative blockchain game (NBG), a Nash equilibrium can be efficiently
found in a closed-form even though the game involves more than two players.
Somewhat surprisingly, the simulation results of the Nash equilibrium implies
that the game can reach a stable status regardless of the number of Byzantine
nodes and trustworthy players. We then study a harder problem where players are
allowed to form coalitions: the coalitional blockchain game (CBG). We show that
although the Shapley value for a CBG can be expressed in a more succinct form,
its core is empty.
