  We present an explicit treatment of the two-particle-irreducible (2PI)
effective action for a zero-dimensional quantum field theory. The advantage of
this simple playground is that we are required to deal only with functions
rather than functionals, making complete analytic approximations accessible and
full numerical evaluation of the exact result possible. Moreover, it permits us
to plot intuitive graphical representations of the behaviour of the effective
action, as well as the objects out of which it is built. We illustrate the
subtleties of the behaviour of the sources and their convex-conjugate
variables, and their relation to the various saddle points of the path
integral. With this understood, we describe the convexity of the 2PI effective
action and provide a comprehensive explanation of how the Maxwell construction
arises in the case of multiple, classically stable saddle points, finding
results that are consistent with previous studies of the
one-particle-irreducible (1PI) effective action.
