Two questions:
In a symmetric game, TDT agents choose symmetric strategies. Without a source of randomness, this entails playing symmetrically as well.
I'm not sure why you're talking about shared random input. If both agents get the same input, they can both be expected to treat it in the same way and make the same decision, regardless of the input's source. Each agent needs an independent source of randomness in order to play the mixed equilibrium; if my strategy is to play C 30% of the time, I need to know whether this iteration is part of that 30%, which I can't do deterministically because my opponent is simulating me.
A monthly thread for posting rationality-related quotes you've seen recently (or had stored in your quotesfile for ages).