We were house-hunting in London last year, and yes, it's not strange at all to offer less. In fact, our agents showed us places above our range with the idea that we would offer our actual target price. We were given serious consideration for one flat at much below the asking price, but pulled out of negotiations when we found another place that was more suitable. (And that one was priced attractively, so we offered the full rent as we thought it was fine. We are now up for renewal, and our rent will not be increased. Not sure what will happen in the third year, though.)