I wouldn't sweat it too much - you're basically messing up the notation a bit; my trying to clarify is clearly not helping, so just don't worry about it. It won't cost you more than a mark.
The neatest way (and I mean literally, in the sense of the clearest way of showing what you're doing) of approaching this is probably to write two matrices:
a payoff matrix showing who wins what for each of the 9 possible pairs (a,b),
and a probability matrix showing the probability of each of the 9 possible pairs (a,b).
Then you can just write down the expectation.