  We present a new game semantics for Martin-L\"of type theory (MLTT), our aim
is to give a mathematical and intensional explanation of MLTT. Specifically, we
propose a category with families of a novel variant of games, which induces a
surjective and injective (when Id-types are excluded) interpretation of the
intensional variant of MLTT equipped with unit-, empty-, N-, dependent product,
dependent sum and Id-types as well as the cumulative hierarchy of universes for
the first time in the literature (as far as we are aware), though the
surjectivity is accomplished merely by an inductive definition of a certain
class of games and strategies. Our games generalize the existing notion of
games, and achieve an interpretation of dependent types and the hierarchy of
universes in an intuitive yet mathematically precise manner, our strategies can
be seen as algorithms underlying programs (or proofs) in MLTT. A more
fine-grained interpretation of Id-types is left as future work.

Logic in Computer Science;Logic