  This paper uses value functions to characterize the pure-strategy
subgame-perfect equilibria of an arbitrary, possibly infinite-horizon game. It
specifies the game's extensive form as a pentaform (Streufert 2023p,
arXiv:2107.10801v4), which is a set of quintuples formalizing the abstract
relationships between nodes, actions, players, and situations (situations
generalize information sets). Because a pentaform is a set, this paper can
explicitly partition the game form into piece forms, each of which starts at a
(Selten) subroot and contains all subsequent nodes except those that follow a
subsequent subroot. Then the set of subroots becomes the domain of a value
function, and the piece-form partition becomes the framework for a value
recursion which generalizes the Bellman equation from dynamic programming. The
main results connect the value recursion with the subgame-perfect equilibria of
the original game, under the assumptions of upper- and lower-convergence.
Finally, a corollary characterizes subgame perfection as the absence of an
improving one-piece deviation.
