Does protobuf-net have built-in compression for serialization?

No it does not; there is no "compression" as such specified in the protobuf spec; however, it does (by default) use "varint encoding" - a variable-length encoding for integer data that means small values use less space; so 0-127 take 1 byte plus the header. Note that varint by itself goes pretty loopy for negative numbers, so "zigzag" encoding is also supported which allows small magnitude numbers to be small (basically, it interleaves positive and negative pairs).

Actually, in your case for Scores you should also look at "packed" encoding, which requires either [ProtoMember(4, IsPacked = true)] or the equivalent via TypeModel in v2 (v2 supports either approach). This avoids the overhead of a header per value, by writing a single header and the combined length. "Packed" can be used with varint/zigzag. There are also fixed-length encodings for scenarios where you know the values are likely large and unpredictable.

Note also: but if your data has lots of text you may benefit from additionally running it through gzip or deflate; if it doesn't, then both gzip and deflate could cause it to get bigger.

An overview of the wire format is here; it isn't very tricky to understand, and may help you plan how best to further optimize.