Hi Amy, thanks for the post, awesome collection of stats in there.
It can be pretty tough getting brands' marketing managers to understand that Facebook isn't merely a channel they can push their branded promotions out on, but that instead they need to be focusing more on brand-experience content - stuff that ties in loosely with their brand's offering and image but doesn't directly scream "I'M A PROMOTION!!"
And you're right, Amy, giveaways definitely do encourage engagement and draw more Likes to a page - I guess this is a good strategy as a page is growing its audience, but there's probably also a point that brands need to get more sophisticated with its content strategy (so implementing more brand-experience focused content).
BUT - that's definitely easier said than done - the brand has to know what it wants it's identity/personality to be before us channel managers and strategists can reach this degree of sophistication!
Take a look at YO! Sushi and Nando's on Facebook:
... They're doing a great job of promoting the brand experience without banging on about their promotions - and they tend not to rely on giveaways, competitions or discounts.