There has not been one reply that has even been close to being on point. The question is: When ticket prices are as high as in New York, why are Londoners, unlike New Yorkers, willing to pay up to an additional 4 quid for a slight brochure full of adverts?
Because there is no other option.
If I had the choice between a tatty free Playbill or paying £4 for a programme (or even, £10 for a 'souvenir brochure') then I'd go with the Playbill. For me, I get programmes because I find it useful to be able to look back and see who I've seen in what (my programmes all have scribbles on to show where I've seen understudies / alternates).
So, as no-one is giving me the option of a free Playbill here, I choose to spend £4 on a programme, because I'd rather that than nothing.