The production-distribution system or "beer game" is one of the most well-known system dynamics models. Notorious for the complex dynamics it produces, the beer game has been used for nearly five decades to illustrate how structure generates behavior and to explore human decision making. Here we present a formal bifurcation analysis to analyse the complex dynamics produced by the model. Consistent with the rules of the game, the model constitutes a piecewise-linear map with nonlinearities arising from non-negativity constraints. The bifurcations that occur in piecewise-linear systems are distinctly different from those observed in smooth systems. We show how the model displays abrupt Hopf and period-doubling bifurcations, truncated bifurcation cascades, and various border-collision bifurcations. The latter allow direct transitions from periodic to chaotic dynamics. Bifurcation phenomena in models that use piecewise-linear functions to represent nonlinearities are likely to show similar qualitative differences from the bifurcations known from smooth systems.