Stromquist Moving-knives Procedure

Definition

Stromquist moving-knives procedure

Let the cake be the interval $C=[0,1]$, and let three agents have nonatomic, additive value measures $V_1,V_2,V_3$. The Stromquist moving-knives procedure continuously finds an envy-free division of $C$ into three connected pieces.

A referee moves a sword from left to right, with position $s$. Each agent $i$ continuously keeps a knife at $k_i(s)\ge s$ that bisects the cake remaining to the right of the sword:

$$V_i([s,k_i(s)])=V_i([k_i(s),1])=\frac{1}{2}{V}_i([s,1]).$$

Write $k_{(1)}(s)\le k_{(2)}(s)\le k_{(3)}(s)$ for the ordered knife positions. The sword and the median knife define

$$L=[0,s],M=[s,k_{(2)}(s)],R=[k_{(2)}(s),1].$$

Each agent calls cut when $L$ becomes as valuable as the piece they would receive by remaining silent: $M$ for the holder of $k_{(1)}$, $R$ for the holder of $k_{(3)}$, and either piece for the holder of $k_{(2)}$, who values $M$ and $R$ equally. At the first call, the caller receives $L$; among the two remaining agents, the one whose knife is nearer the sword receives $M$, and the other receives $R$. Ties are broken arbitrarily. Under these continuous stopping strategies, the resulting allocation $(X_1,X_2,X_3)$ is connected and envy-free:

$$V_i(X_i)\ge V_i(X_j)\text{for every }i,j\in \{1,2,3\}.$$