Waste Makes Haste

We consider the classic problem of envy-free division of a heterogeneous good (“cake”) among several agents. It is known that, when the allotted pieces must be connected, the problem cannot be solved by a finite algorithm for three or more agents. The impossibility result, however, assumes that the entire cake must be allocated. In this article, we replace the entire-allocation requirement with a weaker partial-proportionality requirement: the piece given to each agent must be worth for it at least a certain positive fraction of the entire cake value. We prove that this version of the problem is solvable in bounded time even when the pieces must be connected. We present simple, bounded-time envy-free cake-cutting algorithms for (1) giving each of n agents a connected piece with a positive value; (2) giving each of three agents a connected piece worth at least 1/3; (3) giving each of four agents a connected piece worth at least 1/7; (4) giving each of four agents a disconnected piece worth at least 1/4; and (5) giving each of n agents a disconnected piece worth at least (1 − ε)/n for any positive ε.

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