Robert Lutece

Robert Lutece is a time-travelling physicist and a supporting character in BioShock Infinite. Along with his female counterpart, Rosalind, they escort Booker DeWitt to and across the city of Columbia offering him aid. Additionally, they organize several experiments along the way, presenting him with choices and attempting to predict the outcomes. Robert frequently bets against his "sister", and consequently loses most of the time. Though the Luteces actions mainly help Booker, they also aid Zachary Hale Comstock, and in fact supplied him with all the technology necessary for the creation of Columbia. Unlike his sister, Robert is not a fatalist. Rosalind describes this by saying that where Robert sees a blank page, she sees King Lear.

History
Robert Lutece is a pioneer in the field of quantum physics living in the same timeline with Booker DeWitt. He is contacted by Rosalind Lutece, his female counterpart from an alternate reality and "sister", after she discovers a way to mechanically engineer Tears in her world and across multiple timelines, correspondingly, and open a tear for each other to meet face to face. Soon, the twins acted as agents for Zachary Hale Comstock, in which Robert is tasked to absolve Booker DeWitt's debts in exchange for his infant daughter, Anna.

From then on, the twins work together for four years and help to develop technology that aids in Comstock's creation and development of Columbia. However, as they experiment further with tears, they foresee Comstock's future in his efforts to guide Anna, now renamed as Elizabeth, into leading Columbia to the destruction of New York City several decades after. Realizing the mistakes they committed in creating Elizabeth's fate, Robert manages to convince Rosalind into returning Elizabeth back to her original timeline after providing an ultimatum that he will part with his sister if she is not willing to undo what they have done, despite the latter's laments of their course of action and immensity of the task.

In  efforts to cover up Elizabeth's origins and secure the image of the "Prophet's Seed". Comstock assigned Jeremiah Fink to sabotage their tear-manipulation machine. As the result, they were killed but as a side effect now exist across all of space time. This allows them to appear wherever, and whenever they want, in which Rosalind is content with as she can stay together with her "brother", free of restraint from life, death and all divided barriers of possibility. However, Robert himself is still not satisfied with the matter regarding Elizabeth's fate still unresolved and that their circumstances made their task even more difficult as they could risk disrupting realities even further if they directly interfere now that they are "dead". Robert then proposes that for their unfinished business to be resolved, they need bring Booker DeWitt to Comstock's universe to retrieve his daughter.

Bioshock Infinite
So, the twins meet Booker again at a point 20 years after the loss of his daughter and lead him into Comstock's timeline in order to save Elizabeth, though the latter begins to recreate memories from his old ones as a result of his past guilt over his daughter and the side effect to his crossing over (In which Robert remarks "I should know. I lived it"). Afterwards, the twins take Booker, now with no memories of Anna except the name itself, across the waters to the lighthouse and provides him a casket regarding his "job" to clear his "debts", and thus lead upon to the events of Booker's encounters in Columbia.

The results of their actions did not go unnoticed though as Comstock becomes aware of Booker's arrival and continue to fend him off with every resources and manpower possible. And so the twins continue to assist and guide Booker, using their abilities to appear to every Booker in different timelines that attempted to or succeeded in freeing Elizabeth and observe their journey to find the truth and erase the timelines that carry Comstock's plans for Elizabeth. Their fate after Booker sacrifices himself to ensure that Comstock cannot exist is unknown.

Trivia

 * The Lutece siblings share similarities with Tweedle Dee and Tweedle Dum, characters from Alice in Wonderland.
 * Robert Lutece is voiced by Oliver Vaquer.
 * After receiving the Telegram from Rosalind Lucete, walk towards the telescope to the left. After viewing it, look down towards the end of the street near the trashcan and you can see Robert Lutece juggling. After you exit the telescope, the Lutece twins disappear.