Here's the answer. When Amtrak started issuing Incentive Pay to host Railroads for keeping Amtrak ontime whithin the designated segments, SP and yes, believe it or not, UP, yep, UP, would require the Operator at the orriginating station to find out from Amtrak what the total amount of cars were on the train, along with what engines the trains were to have. The Power Desk would calculate the amount of time lost with the consist the way it was. If an additional Host Railroad Unit could save some time for the train, and/or keep it rolling faster, to ensure the train made those segments in the time allocated, to ensure incentive pay, then the host RR would provide an additional engine as it was extremely easy revenue investment for them. This was especially true on the LA to Oakland run, where the Passenger Train Speed did not exceed 70 MPH at that time. On the UP Side, the engines they provided were usually geared for 79 MPH. This started in the early 1980s. This seem to have ended about 1991. You would have never seen this on the AT&SF as they never had any engines equipped with ATS and geared for 90 MPH anyway.
In addition to this, there were times that the Amtrak Power would not be working right and the Host Railroad would provide an engine to Amtrak with the agreement that it be returned by Amtrak to the location from where it was obtained. I remember many a times, UP would add an SD40-2 to train #35 at Salt Lake City. It would stay on the train to LA and Amtrak would service it and return it on the next day's #36 back to the UP at Salt Lake City. This also ended in the early 1990s and the Host Railroad would get their engine back at the destination. There are some rare exceptions.
Hope this answers your questions.
Thanks for listening.