For real numbers, the continued fraction expansion arises from the dynamics of the Gauss map and encodes precise information on Diophantine approximation. Ergodic properties of the Gauss map therefore translate into deep arithmetic consequences.

Multidimensional continued fractions (MCFs) aim to extend this correspondence to the approximation of vectors in R^d by rationals with a common denominator. Many algorithms have been introduced for this purpose, but no single model has emerged as canonical.

Several frameworks have been developed to study MCF algorithms from a unified perspective. In particular, simplex-splitting algorithms, initiated by Lagarias, describe the dynamics through locally projective linear maps acting on simplices. On the other hand, ideas originating in the study of measured foliations and interval exchange transformations led Kerckhoff and later Nogueira--Chaika to generalized Rauzy–Veech type inductions, where the dynamics act on a simplex together with the vertices of a graph through elementary binary splittings.

In this talk, I will introduce a common formalism encompassing both approaches. To every labeled graph we associate a deterministic dynamical system, called a win–lose induction. I will explain how classical multidimensional continued fraction algorithms fit naturally into this framework.

I will then present recent results giving a graph-theoretic criterion for ergodicity of win–lose inductions, a condition satisfied by the classical examples. These results also yield exponential tail estimates and spectral gap properties, leading in particular to uniqueness of the measure of maximal entropy and central limit theorems for multidimensional continued fraction algorithms.