  Seymour conjectured that every oriented simple graph contains a vertex whose
second neighborhood is at least as large as its first. Seymour's conjecture has
been verified in several special cases, most notably for tournaments by Fisher.
  One extension of the conjecture that has been used by several researchers is
to consider vertex-weighted digraphs. In this paper we introduce a version of
the conjecture for arc-weighted digraphs. We prove the conjecture in the
special case of arc-weighted tournaments, strengthening Fisher's theorem. Our
proof does not rely on Fisher's result, and thus can be seen as an alternate
proof of said theorem.
