Gift Problem

In the Philippines it's traditional and a responsibility for godparents to give gifts to their godchildren at Christmas. My increasing brood of godchildren is putting the problem of gift giving at the forefront. Everybody loves the uncle or the aunt that seems to know what you want for Christmas. But more often than not, these guys have low information costs (i.e. most probably they see their godchildren often). What's an absentee gift giver to do?

Assume there are only 2 toys (X, Y) in the world, assume further they are priced the same (a price of $1). Child Z values one toy more than the other (say by double, $2). He has no clue what he's going to get from absentee godparent. But child Z's level of happiness is expressed by building expectation of the form 1/2*X + 1/2*Y he's assuming absentee gift giver flips a fair coin before buying a toy. Child Z's expected happiness then is 1 and 1/2. So if an absentee gift giver sends money and let the child Z buy the toy, happiness of the child is 2 that's more than the expected happiness. If the cost for the child of having to buy his toy (i.e. walking to the store) is less than 1/2, then child Z is better off getting $1 cash. Hey, we can't all be Santa :)