  In this paper, we propose a scheme for the problem of cache-aided multi-user
private information retrieval with small caches, in which $K$ users are
connected to $S$ non-colluding databases via shared links. Each database
contains a set of $N$ files, and each user has a dedicated cache of size
equivalent to the size of $M$ files. All the users want to retrieve a file
without revealing their demands to the databases. During off-peak hours, all
the users will fill their caches, and when required, users will demand their
desired files by cooperatively generating query sets for each database. After
receiving the transmissions from databases, all the users should get their
desired files using transmitted data and their cache contents. This problem has
been studied in [X. Zhang, K. Wan, H. Sun, M. Ji and G. Caire, \tqt{Fundamental
limits of cache-aided multiuser private information retrieval}, IEEE Trans.
Commun., 2021], in which authors proposed a product design scheme. In this
paper, we propose a scheme that gives a better rate for a particular value of
$M$ than the product design scheme. We consider a slightly different approach
for the placement phase. Instead of a database filling the caches of all users
directly, a database will broadcast cache content for all users on a shared
link, and then the users will decide unitedly which part of the broadcasted
content will be stored in the cache of each user. This variation facilitates
maintaining the privacy constraint at a reduced rate.
