  In the present paper, based on the previous work (Part I), we present a game
semantics for the intensional variant of intuitionistic type theory that
refutes the principle of uniqueness of identity proofs and validates the
univalence axiom, though we do not interpret non-trivial higher propositional
equalities. Specifically, following the historic groupoid interpretation by
Hofmann and Streicher, we equip predicative games in Part I with a groupoid
structure, which gives rise to the notion of (predicative) gamoids. Roughly,
gamoids are "games with (computational) equalities specified", which interpret
subtleties in Id-types. We then formulate a category with families of
predicative gamoids, equipped with dependent product, dependent sum, and
Id-types as well as universes, which forms a concrete instance of the groupoid
model. We believe that this work is an important stepping-stone towards a
complete interpretation of homotopy type theory.
