Expand the Shares Together: Envy-Free Mechanisms with a Small Number of Cuts

Abstract

We study the problem of fair division of a heterogeneous resource among strategic players. Given a divisible heterogeneous cake, we wish to divide the cake among n players to meet these conditions: (I) every player (weakly) prefers his allocated cake to any other player’s share (such notion is known as envy-freeness), (II) the allocation is dominant strategy-proof (truthful) (III) the number of cuts made on the cake is small. We provide methods for dividing the cake under different assumptions on the valuation functions of the players. First, we suppose that the valuation function of every player is a single interval with a special property, namely ordering property. For this case, we propose a process called expansion process and show that it results in an envy-free and truthful allocation that cuts the cake into exactly n pieces. Next, we remove the ordering restriction and show that for this case, an extended form of the expansion process, called expansion process with unlocking yields an envy-free allocation of the cake with at most $2(n-1)$ cuts. Furthermore, we show that in the expansion process with unlocking, the players may misrepresent their valuations to earn more share. In addition, we use a more complex form of the expansion process with unlocking to obtain an envy-free and truthful allocation that cuts the cake in at most $2(n-1)$ locations. We also, evaluate our expansion method in practice. In the empirical results, we compare the number of cuts made by our method to the number of cuts in the optimal solution ($n-1$). The experiments reveal that the number of cuts made by the expansion and unlocking process for envy-free division of the cake is very close to the optimal solution. Finally, we study piecewise constant and piecewise uniform valuation functions with m pieces and present the conditions, under which a generalized form of expansion process can allocate the cake via O(nm) cuts.