The second line

What this game-theory puzzle tests

This is a medium-difficulty puzzle that combines strategic reasoning, expectation minimization, and common-knowledge equilibrium. It asks you to find the stable outcome when all 100 people make a simultaneous, rational decision about whether to switch queues, knowing that everyone else is doing the same.

The crux is that a person's incentive to switch depends on how many others switch—which in turn depends on their expectations about others' behaviour. You must reason backwards from the structure of the problem: given that switchers land uniformly at random among the n positions they occupy, when is switching indifferent in expectation? When is it strictly better or worse to deviate? And what configuration is stable under common knowledge of rationality?

• Symmetric mixed-strategy equilibrium
• Expectation and variance trade-offs
• Self-referential reasoning (what others expect others to do)
• The role of indifference in pinning equilibrium size