Theorem 3. **The online cut-and-choose 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: Consider the player cutting the cake. As they are risk averse, and **as there is**
a chance that they will have to take the slice of cake that they cut, **they will cut a slice**
that is at least 1k of the total remaining value where k **is the number of players still to be**
allocated cake. Similarly they will not cut a slice that is more than 1k of the total remaining value for fear that the next player to arrive will take it, leaving behind cake that if it is divided proportionally gives them a slice of small value. Hence, the procedure is forward proportional and weakly truthful. It is also immediately envy free since each slice that the cutting player sees being allocated has the same value. 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 i**th player (**i < n − **1) that assigns a value 0** to [0, ai**], a value 1 to [**ai, ai+1**], a value 0 to [**ai+1, ai+2], a value n − i to [ai+2, **1]. For the**
n − 1th player, we construct a valuation function that assigns a value 0 **to [0**, an−1**], and**
values of 1 to both [an−1, an] and [an, **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 or order monotonic consider 4 players and a cake in which the first player places a value of 3 units on [0, 1 4
], 1 unit on [ 14
,
3 4
] and 8 units on [ 34
, **1], the second**
player places a value of 0 units on [0, 1 4
], 4 units on [ 14
,
1 2
], 8 units on [ 12
,
5 8
], and 0 units on
[
5 8
, **1], the third player places a value of 6 units on [0**,
1 4
] 0 units on [ 14
,
1 2
], 1 unit on [ 12
,
5 8
],
2 units on [ 58
,
3 4
], and 3 units on [ 34
, **1], and the fourth player places a value of 0 units on**
[0, 1 4
], 9 units on [ 14
,
1 2
], 1 unit on [ 12
,
3 4
], and 2 units on [ 34
, 1].

If we apply the online cut-and-choose procedure, the first player **will cut off and keep**
the slice [0, 1 4
], the second player will cut off and keep [ 14
,
1 2
], The third player will now cut the cake into two pieces: [ 12
,
3 4
] and [ 34
, **1]. The fourth player will take the slice [** 34
, **1], leaving**
the third player with the slice [ 12
,
3 4
]
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 cutting the cake into the slice [ 14
,
5 8
]. Finally, the procedure is not order monotonic as the value of the cake allocated to the fourth player decreases from 2 units to 32 units when they arrive before the third player. ♥

## 7 Online Moving Knife

Another class of procedure for cutting cakes uses one or more moving knives. For example, in the Dubins-Spanier procedure for n **players [6], a knife is moved across the cake from left**
to right. When a player shouts "stop", the cake is cut and this player takes the piece to the left of the knife. The procedure then continues with the remaining n − **1 players until just**
one player is left (who takes whatever remains). This procedure is proportional but is not envy-free. However, only the first n−**2 players to be allocated slices of cake can be envious.**
We can use the Dubins-Spanier procedure as the basis of an online moving knife procedure. The first k players (k ≥ **2) to arrive perform one round of a moving knife procedure**