We had heard rumblings of multiplayer modes in BioShock Infinite, but very little detail about what they entailed. Now that Irrational Games as we knew it is no more, though, former employees are feeling more free to share details on what the modes were, and why they were ultimately axed.
According to employees speaking to Polygon, Irrational was working at length on two multiplayer modes while the single-player was still in its infancy. The first was Border Control, a tower-defense game with its own art style modeled after old political cartoons. It was to be set in Columbia, as part of the city's indoctrination for kids to learn racial bigotry. The other, Spec Ops, would have had up to four players cooperating against enemies, with loot and character progression mechanics. The more ambitious part of Spec Ops was going to be the randomized content, and the stages would progressively evolve throughout the months to tell a story about the war. Various other prototypes were in the works too.
Those modes were reportedly scrapped when Irrational head Ken Levine grew impatient with the slow pace of progress on the single-player campaign. At the time the studio was still iterating and experimenting with single-player.
This aligns with what we've heard before, including the broad strokes of the modes themselves. Shortly after they were eliminated, Levine confirmed the lack of multiplayer, which he later chalked up to lack of resources to finish them properly.