Some do. Mainly, the theaters where the bar is not in the theater itself (the Golden, for example). You just have to ask.
If you reuse your Jujamcyn cup at any Jujamcyn theater, you'll save $5 on your drink. I'm pretty sure the bartenders are told to explain that, so they can't refuse to refill it when you bring it back.
Also, DON'T put those cups or their lids in the dishwasher. They will warp.
It seems like a cheap money grab (and the cheap plastic cups probably are overpriced), but they are meant to reduce messy spills in the theater, and it does help a lot.
Stand-by Joined: 10/16/14
I love collecting the souvenir cups from the bars. My collection is only at 10 (only started seeing shows a couple years ago). I thought this would be a good thread to give people the heads up on which shows carry a souvenir cup at the bars if you were wondering ahead of time.
My collection: Lion King, Newsies, Aladdin, Kinky Boots, Wicked, Big Fish, Spiderman, Cabaret (large and small), If/Then, Gigi
Your best bet is either eBay (where I've located a couple over the years for a decent price) and the Broadway Cares Flea Market.
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