(copied from math.SE)
BACKGROUND: A cake has to be divided among 3 people with possibly different tastes, such that each person receives a single connected piece, and no person prefers another person's piece. In other words, no participant should end up being jealous of any other participant. Symbolically, let $\ v_{jk}\ $ be the value (measure) of the $k$-th piece according to the measure of participant $\ j.\ $ Thus we would like:
$$\forall_{j=1\ 2\ 3}\ \ \ v_{jj}\ \ =\ \ \max(v(j\ 1)\,\ v(j\ 2)\,\ v(j\ 3))$$
This problem was unsolved for several tens of years, until Stromquist (1980) suggested the following division protocol:
A referee moves a sword from left to right over the cake, hypothetically dividing it into a small left piece and a large right piece. Each player holds a knife over what he considers to be the midpoint of the right piece. As the referee moves his sword, the players continually adjust their knives, always keeping them parallel to the sword. When any player shouts "cut", the cake is cut by the sword and by whichever of the players' knives happens to be the middle one of the three.
The player who shouted "cut" receives the left piece. He must be satisfied, because he knew what all three pieces would be when he said the word. Then the player whose knife ended nearest to the sword, if he didn't shout "cut", takes the centerpiece; and the player whose knife was farthest from the sword, if he didn't shout "cut", takes the right piece. The player whose knife was used to cut the cake, if he hasn't already taken the left piece, will be satisfied with whatever piece is left over. If ties must be broken - either because two or three players shout simultaneously or because two or three knives coincide - they may be broken arbitrarily.
Usually, protocols for cake-cutting among $\ n\ $ participants are considered fair if each player has a safe strategy, that is, each player can play in a way that guarantee that s/he receives at least $\ \frac1n\ $ value (measure) of the cake, regardless of what other players do. For example, consider the classic protocol for 2 players: "I cut, you choose". I (the cutter) have to cut the cake to two pieces that I consider to be of equal value, but, even if I cut the cake in a very strange manner to two very unequal pieces, you still have a safe strategy - you just pick the piece that you consider to be more valuable. So, the protocol is fair for you; furthermore, none of the two of them needs to be left jealous.
MY QUESTION: It is clear that Stromquist's three knives procedure is fair. But is Stromquist avoiding any jealousy on the part of any participant?
