  For any two complete discrete valued fields $K_1$ and $K_2$ of mixed
characteristic with perfect residue fields, we show that if the $n$-th valued
hyperfields of $K_1$ and $K_2$ are isomorphic over $p$ for each $n\ge1$, then
$K_1$ and $K_2$ are isomorphic. More generally, for $n_1,n_2\ge 1$, if $n_2$ is
large enough, then any homomorphism, which is over $p$, from the $n_1$-th
valued hyperfield of $K_1$ to the $n_2$-th valued hyperfield of $K_2$ can be
lifted to a homomorphism from $K_1$ to $K_2$. We compute such $n_2$
effectively, which depends only on the ramification indices of $K_1$ and $K_2$.
Moreover, if $K_1$ is tamely ramified, then any homomorphism over $p$ between
the first valued hyperfields is induced from a unique homomorphism of valued
fields. Using this lifting result, we deduce a relative completeness theorem of
AKE-style in terms of valued hyperfields.
  We also study some relationships between valued hyperfields, truncated
discrete valuation rings, and complete discrete valued fields of mixed
characteristic. For a prime number $p$ and a positive integer $e$ and for large
enough $n$, we show that a certain category of valued hyperfields is equivalent
to the category of truncated discrete valuation rings of length $n$ and the
ramification indices $e$ having perfect residue fields of characteristic $p$.
Furthermore, in the tamely ramified case, we show that a subcategory of this
category of valued hyperfields is equivalent to the category of complete
discrete valued rings of mixed characteristic $(0,p)$ having perfect residue
fields.
