See my other example. Most sets of non-transitive dice do not have the same expected value, and the expected value doesn't tell you which one will win. To use an extreme example, take two dice, (2 2 2 2 2 2) and (1 1 1 1 1 999). The average value of the first die is much smaller than that of the second, yet it wins 5/6ths of the time.