Who Would You Back? Really?

So I was sitting at work this week, immersed in some complex n-tier, m-dimensional, z-class np-hard problem, when cooney interjects with a gob-smackingly complexicated puzzler, that spins my mind in a vortex of impossible tangles. Get this:

"Who would you back in a fight? Indiana Jones or Han Solo?"

A riddle wrapped in a puzzle, shipped home in an enigma with pre-paid mystery and a side heaping of conundrums.

After narrowing it down (Sword or Pistols? Stock whip is allowed? Blaster is kosher? Bare knuckle or gloves?) I had the details downpat (bare knuckle, earth, queensbury rules) and took the question home to Mrs Secret Geek to ponder over dinner.

"Both are quick on the draw,"

said mrs secret geek, mostly immersed in her afore-mentioned nintendo ds

"but while Han's a short-sighted scoundrel, Indy has the superior intellect, and less reliance on the sidekick. Clearly, Indy FTW."

I went back to work the next day, happy with this answer, only to be hit with a barrage of further dilemmas:

"Who would you back? Rocky or Rambo?"

And, over lunch, many more in the pattern

"Who would you back? $0 versus $1?"

With $0, $1 represented by various combinations of the following: