Some way into the story of Bioshock Infinite, you'll have to choose between two options for Elizabeth: whether to give her a pendant showing a bird or a cage. We'll explain the choice here, as well as what impact it has on the rest of the game.
Warning: the following contains spoilers for the plot of Bioshock Infinite.
Should I Choose the Bird or the Cage?
The honest truth is that it doesn't really matter, at least not in terms of gameplay, with more of a symbolic impact than anything mechanical. Choice, divergent paths, freedom and entrapment are running themes of Bioshock Infinite, and this is an extension of that idea, inviting you to make a decision that you don't get to see the results of. The necklace does play into the game later, but it doesn't actually change your overall experience or set you on a different path.
What Happens If You Choose the Bird or Cage?
Spoilers still, but this is less about you and more about the Lutece twins who offer you the choice, similar to the coin flip they gave you at the beginning. Being interdimensional explorers unlimited by time or space, the Luteces have likely seen this scenario play out hundreds of times, either in different timelines or by going back and looping the same events over and over. This is their way of keeping score of all the little differences and nuances. You could also make the point that it's a metanarrative comment about free will and player choice, but that's probably for a different article.
All that being said, there is a moment at the very final minutes of the game where Booker sees Elizabeth, only to suddenly exclaim that it isn't her. Looking closer reveals she's not wearing the pendant you picked out, presumably making her an Elizabeth from another universe.
It's a bit of a mind-bending answer, but should clear things up after a couple of days to process it. If you want something more straightforward in the meantime, why not check here to see how to crack all the Vox Populi ciphers and codes? Or if you want news on the upcoming Bioshock sequel, just follow this link here.