I don't know that they aren't allowed to trade on the screen these days, but they were forbidden in the 90's to take any speculative positions.  BP was the only real "player" among the majors.  The small guys, Koch, Valero Conoco, Neste had some aggressive traders though.  Neste's boys lost about $40 meg in a few months and that shop closed down about 1995.  Ditto Chevron for the most part.  But it goes in cycles.

Big oil only wishes they have control.  I'd guess Morgan and J. Aron have many times the influence of Big Oil