The male, who wore a brown leather jacket, is said to have had a slight build, while the females, one with blonde hair and the other with red hair that was possibly dyed, wore a pink hooded top and a blue top respectively.
Shortly after 5.30pm, a 3O-year-old woman reported an attempted theft of her handbag on Duncan Street. The description of the suspects was similar to that of the previous two incidents, police said.
Lothian and Borders Police said they are now investigating whether all three crimes are linked.
A spokesman for the force said: "All three victims were left shaken as a result of these incidents and we are conducting local enquiries to identify the suspects. Anyone who can assist with our investigations is asked to contact police immediately.
"Members of the public are encouraged to add tracking software to the mobile telephones and portable electronic devices, which can help police trace their goods if stolen." Anyone with information is asked to contact police on 0131 311 3131.