The notion that you should start by offering half is right in some places, but not in others. For starters, most vendors know this so-called rule.
I remember my son asking a vendor in the Silk Market in Beijing what a tie cost -- the vendor turned over the tie and on the name tag it read "US$250."
My kid offered US$1. Settled for US$5 and he still probably didn't come close to the local price.