Definition
Stromquist moving-knives procedure
Let the cake be the interval
, and let three agents have nonatomic, additive value measures . The Stromquist moving-knives procedure continuously finds an envy-free division of into three connected pieces. A referee moves a sword from left to right, with position
. Each agent continuously keeps a knife at that bisects the cake remaining to the right of the sword: Write
for the ordered knife positions. The sword and the median knife define Each agent calls cut when
becomes as valuable as the piece they would receive by remaining silent: for the holder of , for the holder of , and either piece for the holder of , who values and equally. At the first call, the caller receives ; among the two remaining agents, the one whose knife is nearer the sword receives , and the other receives . Ties are broken arbitrarily. Under these continuous stopping strategies, the resulting allocation is connected and envy-free: