You have the essence of the strategy: Unless there is a limit on the number of times a gift can be stolen, always steal a gift valued above the average of all

You have the essence of the strategy: Unless there is a limit on the number of times a gift can be stolen, always steal a gift valued above the average of all currently opened gifts. Limits on the number of times a gift can be stolen are popular, making the strategy more useful. In that case, you should open a gift if none of the available gifts are above average.