The following two player game is an (inadequate) attempt to capture an idea I am very interested in exploring. If any of the learned folks here recognize the problem I'm playing with and can point me in the right direction for study, I would greatly appreciate it.
We start with a set. This can be anything.
Both players choose an element from the set without showing the other. After both have picked, they reveal their chosen elements to each other. If they are not the same element, they repeat the process. If they are the same element, they have won after $t$ turns. Players don't communicate beforehand. They are allowed random choices in their strategies.
Players would like to minimize $E[t]$.
The game is intended to be set up so that players end up following the same strategy if they don't make any arbitrary choices while selecting their strategy. E.g. if the elements of the set are just the points contained by a circle, they can't both arrive at "pick [some arbitrary point in the circle]" as their strategy, but they may arrive at "pick the center" as their strategy.
I think there is nothing specifically like this in the game theoretic literature. The reason is that the game you are describing has a trivial Nash equilibrium:
Whatever the set, both pick the same element immediately, thus minimizing $t$.*
What you describe is different from a coordination game where the players have preferences over the elements in the set, not over the time it takes to reach "agreement" as here. In your game, players don't care which element is chosen as long as there is agreement. In the typical coordination games there is usually a twist, say for set $(a,b)$, player 1 would prefer both to select $a$ while player 2 would prefer both to select $b$, inducing a small conflict of interest.
*How can they pick the same element from the set without prior communication you might ask? This is part of a Nash equilibrium, which requires all players to correctly guess the strategy of the opponent and best respond to it.
The most relevant topic for you, therefore, seems not to be what the equilibria are, but which of the many available ones is selected; i.e., the topic of equilibrium selection. Thomas Schelling did some work on that decades ago invoking the notion of "focal points". There is more recent work employing notions of risk dominance etc.