to select a slice of the cake. Whoever chooses this slice, departs. At this point, if all players have arrived, we continue the moving knife procedure with k − **1 players. Alternatively the**
next player arrives and we start again a moving knife procedure with k **players. As before,**
we assume that all players know how many players will arrive.

Running Example:. Consider again the example in which there are three players, **the first**
player values only [
1 2
, 1]**, the second player values only** [
1 3
, 1]**, and the third player values only**
[0, 3 4
]**. If we operate the online version of the moving knife procedure, the first two players**
will arrive and perform one round of the moving knife procedure. The second player will be the first to call "cut" and will depart with the slice [0, 5 9
] **(as this has** 13 of the total value of the cake for them). The third player will now arrive and perform a round of the moving knife procedure with the first player using the remaining cake, [
5 9
, 1]**. The third player will**
be the first to call "cut" and will depart with the slice [
5 9
,
47 72 ] **(as this has** 12 the total value of the remaining value for them). The first player will then depart with what remains, the slice [
47 72 ]**. It can be claimed that everyone is happy as the second and third players received**
a "fair"' proportion of the cake that was left when they first arrived, whilst the first player received an even greater proportional value.

This online version of the moving knife procedure has the same desirable properties as the online version of the cut-and-choose procedure.

Theorem 4. The online moving knife procedure is forward proportional, **immediately envy**
free, weakly truthful, surjective, scale invariant and sequential. However, it is not proportional, (forward) envy free, equitable, (weakly) Pareto optimal, truthful or order monotonic.

Proof: Suppose j **players (**j > **1) have still to be allocated cake. Consider any player who**
has arrived. They will call "cut" as soon as the knife reaches 1 j of the value of the cake left for fear that they will will receive cake of less value at a later stage. **Hence, the procedure**
is weakly truthful and forward proportional. The procedure is also **immediately envy free** as they will assign less value to any slice that is allocated after their arrival and before their departure. To demonstrate surjectivity, consider the partition that allocates the i**th player**
with the slice [ai, ai+1] where a1 = 0 and an+1 **= 1. We construct a valuation for the** ith player (i < n**) that assigns a value 0 to [0**, ai**], a value 1 to [**ai, ai+1], a value n−i to [ai+1, 1].

Finally, we construct a valuation function for the n**th player that assigns a value 0 to [0**, an], and a value of 1 to [an, 1]. With these valuation functions, the i**th player gets the slice**
[ai, ai+1**]. Finally, it is easy to see that the procedure is scale invariant and sequential.**
To show that this procedure is not proportional, (forward) envy free, equitable, (weakly)
Pareto optimal truthful consider again the example with 4 players used in the last proof.

We suppose that k **= 2 (i.e. at any one time, two players are watching the knife). The** first player calls "cut" and departs with the slice [0, 1 4
]. The second player calls "cut" and departs with the slice [ 14
,
1 2
]. Finally, the third player calls "cut" and departs with the slice
[
1 2
,
2 4
], leaving the fourth player with the slice [ 34
, 1].

The procedure is not proportional as the fourth player only receives 16 of the total value of the cake, not (forward) envy free as the first player envies the fourth player, and not equitable as players receive cake of different value. The procedure **is not (weakly) Pareto**
optimal as allocating the first player with [ 34
, **1], the second player with [** 12
,
3 4
], the third player with [0, 1 4
] and the fourth player with [ 14
,
1 2
] gives all players a slice of greater value.

The proceure is not truthful as the second player can get a larger **and more valuable**
slice by misrepresenting their preferences and not calling "cut" until the knife is about to reach 58 th of the way along the cake.

Finally, to show that the procedure is not order monotonic consider **3 players and a cake**
in which the first player places a value of 2 units on each of [0, 1 3
], [ 13
,
2 3
], and [ 23
, **1], the**
second player places a value of 0 units on [0, 1 3
], 3 units on each of [ 13
,
2 3
] and [ 23
, **1], and the**
7