A Pumpkin Dilemma

This is what I am aiming for this year! A beautiful big, round, red pumpkin. This is the variety I am growing this year Rouge Vif D'Etamps - seed kindly donated by Petunia (the shoes on the left! And so far it looks like all I am getting is this!

Here's the dilemma - I realise that my wonderful trailing pumpkin leaves cannot support an infinite number of mature pumpkins, but most of my little yellow pumpkins are just withering and rotting. A few weeks ago I counted a total of about 10 female flowers on one vine - I know they will not grow into 10 giant pumpkins, but who decides?
I've been watering and feeding the plants and the leaves and stalks look fantastic, but the little pumpkins nearest the root are withering and dying, the ones at the end of the growing tip seem to be OK. Are there rules to this phenomenon? Should I thin them out?