Not sure I get this; the weighted average of 1 and 1, for all weights, is 1.

So in the heads world, the average utility is -x. In the tails world, the average utility is 1-x. An unweighted average means that the decision maker goes "and so the utility I evaluate for buying at x is (1-2x)/2". A weighted average means the decision maker goes "and so the utility I evaluate for buying at x is (2-3x)/2".

For an example of a decision maker who takes a weighted average, take the selfish agents in my poorly-modified non-anthropic problem. They multiply the payoff and the probability of the "world coming into existence" (the coin landing heads) to get the payoff to a decider in that world, but weight the average by the frequency with which they're a decider.