At present the final conclusion is still being worked out.

What my original post contributed is translating The Button into the formal language of game theory... defining the space of possible actions and formalizing utility functions for two different simple player types.

With that accomplished, mathematical theorems from game theory can be used to prove that there exists something called a 'Mixed Strategy Nash Equilibrium' for this simple version of the game.

What that means is that there exists a stable set of strategies. 'Stable' in the sense that if you gave players additional information about other players strategies, nobody would change their own strategy. In this case, a strategy might involve every player choosing to act in a somewhat random manner.

One interesting question is whether real human behavior in this game is somehow approaching or approximating the behavior predicted by Nash Equilibrium theory. It's not quite clear what that would demonstrate, other than showing how interesting group dynamics can emerge from the interaction of different individuals. It would also be interesting from the perspective of a mathematical theory being able to describe or predict human behavior in a somewhat complicated situation.

/u/puzzlednerd has offered one possible solution to this problem statement in terms of a possible Nash equilibrium strategy. With this achieved, it's then possible to compare the predictions of this analysis to actual behavior. But there are a lot of other issues, since the way this version of the problem is defined is a lot simpler than the range of different strategies and goals in the real version of The Button.