After the first round, I wouldn't think there's much reason to guess higher than the previous highest number (or lower than the previous third number), which suggests convergence. If everyone updates based just on what other people did last time, then won't they cycle progressively closer around the initial second number? (Actually, I'm pretty sure the same holds even if people anticipate other's responses, provided they all reason forward the same number of steps.)

ETA: What happens If there is a tie for the highest number? Does the third highest guess win, or the two highest together? What if everyone guesses the same thing?