Design a society — set what each class earns — but you will be dropped into it at random, and you do not get to pick your seat. Behind that veil, would you still build a tower with a gutter at its foot? Drag the class bars, then drop yourself in, and see where a fair chooser learns to care about the worst-off.
You set each class's payoff and its share of the population; then you are dropped in at random with probability equal to each class's size, so your expected outcome behind the veil is Σ payoff·share (computed live) while your realised life is the luck of the draw. A chooser who cannot pick their seat and is loss-averse weights the worst-off position heavily — Rawls's maximin — and the readout tracks the floor and flags when it is a gutter. The expectation and the random assignment are exact.
Rawls's veil is a thought experiment about justice, not a measured fact; whether a rational chooser behind it maximises the average (utilitarian) or the minimum (maximin) is itself contested. The instrument makes the structure honest — you design the society, then live a random life in it — but 'the fair choice' is a figure, argued not proven.