Is there research to prove that humans achieve more when challenging ourselves or when we challenge others? Given that there is a single player game, let's say, Tetris. I play the game for 5 days and my friend is also playing the game for 5 days. At the end of the 5 days we would like to see who has the highest score and that person wins some prize.

Is it better that we see the scores of our opponent and try to better it, or is better that I do not see my friend's score and focus on just constantly beating my own score?

Which of the 2 scenarios is likely to yield the higher score at the end of the 5 days? And why?

I'm doing some research on gamification and how competition can help us progress a skill.
Generally, being observed raises performance on simple tasks and can hinder performance on complex tasks: https://en.wikipedia.org/wiki/Social_facilitation

For trivia, notice that the experiment widely regarded to be the first in social psychology was of observed vs. unobserved cycling. Being watched made people cycling faster: https://www.simplypsychology.org/Social-Facilitation.html excerpt:


  "According to Cottrell (1968), it’s not the presence of other people that is important for social facilitation to occur but the apprehension about being evaluated by them. We know that approval and disapproval are often dependent on others’ evaluations and so the presence of others triggers an acquired arousal drive based on evaluation anxiety."