Three players place points on a gold bar of length 1, one after another. Each gets the portion closest to their point (a 1D Voronoi split). Everyone plays optimally. What can Player 1 guarantee? Start from the end. Player 3 (last mover) will always place themselves to steal the largest available gap. They react, not commit. Player 2 knows this. So instead of chasing the biggest immediate slice, they position to limit how much Player 3 can steal. Now Player 1 has to think two steps ahead: “Where do I place so that even after both optimal responses, I still get the most?” Key insight: If Player 1 does NOT start in the middle, they lose control. * Suppose Player 1 shifts left. * Player 2 moves toward the larger right side. * Player 3 then jumps to the far right endpoint. * Result: Player 1 gets squeezed into a smaller region. This always punishes imbalance. But if Player 1 starts exactly at 1/2: * Player 2’s best move is symmetric (slightly left or right of center) * Player 3 ta...