The Prophet and the Voronoi Diagram
For online decision-making problems with geometric payoffs, this provides the first constant-factor prophet inequality for Voronoi cell area, showing that simple strategies can compete with the optimal offline choice.
In a stream of n random points, a player must pick one point immediately upon arrival, with payoff being the area of its Voronoi cell in the final diagram. A simple strategy achieves, with high probability, a payoff within a constant factor of the optimal (prophet) choice, despite the optimal payoff being Θ(log n) times larger than the average.
Consider a stream of $n$ random points (say, from the unit square) arriving one by one, where a player has to make an irreversible immediate decision for each arriving point whether to pick it. The player has to pick a single point, and the payoff is the area of the cell of the picked point, in the final Voronoi diagram of \emph{all} the points. We show that there is a simple strategy so that with probability $\geq 1 - \tilde O(1/\sqrt{n})$, the player's payoff is only a constant factor smaller than the optimal choice (i.e., the one made by the prophet). This competitiveness is somewhat surprising, as this payoff is larger by a factor of $Θ( \log n)$ than the average payoff.