  Bi-capacities arise as a natural generalization of capacities (or fuzzy
measures) in a context of decision making where underlying scales are bipolar.
They are able to capture a wide variety of decision behaviours, encompassing
models such as Cumulative Prospect Theory (CPT). The aim of this paper in two
parts is to present the machinery behind bi-capacities, and thus remains on a
rather theoretical level, although some parts are firmly rooted in decision
theory, notably cooperative game theory. The present first part is devoted to
the introduction of bi-capacities and the structure on which they are defined.
We define the M\"obius transform of bi-capacities, by just applying the well
known theory of M\" obius functions as established by Rota to the particular
case of bi-capacities. Then, we introduce derivatives of bi-capacities, by
analogy with what was done for pseudo-Boolean functions (another view of
capacities and set functions), and this is the key point to introduce the
Shapley value and the interaction index for bi-capacities. This is done in a
cooperative game theoretic perspective. In summary, all familiar notions used
for fuzzy measures are available in this more general framework.
