however, value the cake allocated to players who have already departed more than our own cake. This models situations where, for instance, we do not envy cake we don't see being allocated, or players eat their cake before departing and we do not **envy cake that has already** been eaten. Note that forward envy freeness implies forward proportionality but not vice versa. Similarly, envy freeness implies forward envy freeness but not vice versa. An even weaker form of envy freeness is when a player does not envy cake that is allocated to other players whilst they are in the room. A cake cutting procedure is immediately envy free iff no player values the pieces of cake allocated to another player after their arrival and before their departure more than their own. Note that forward envy freeness implies immediate envy freeness but not vice versa.

## 3.3 Equitability

Another fairness property is equitability. A cake cutting procedure is equitable **iff all players** assign the same value to the pieces of cake to which they are allocated (and so no player envies anothers valuation). For 3 or more player, equitability and envy freeness can be incompatible [2]. Equitability is a difficult property to achieve, even more **so in our online** setting. Unlike proportionality or envy freeness, there seems little **merit in considering** weaker forms of equitability. Either all players assign the same value **to their allocated cake** or they do not. There is no advantage to ignoring the value of the cake allocated to players who have already departed.

## 3.4 Efficiency

Another important notion is efficiency. Efficiency is also called Pareto optimality. A cake cutting procedure is Pareto optimal **iff there is no other allocation to the one returned** that is more valuable for one player and at least as valuable for the others. Note that Pareto optimality does not in itself ensure fairness since allocating all **the cake to one player** is Pareto optimal. A cake cutting procedure is weakly Pareto optimal **iff there is no other** allocation to the one returned that is more valuable for all players. A **cake cutting procedure** that is Pareto optimal is weakly Pareto optimal but not vice versa.

## 3.5 Strategy Proofness

Another consideration is whether players have an incentive to act truthfully. A cake cutting procedure is weakly truthful **iff a player will do at least as well by telling the truth whatever**
valuations are held by the other players [1]. A stronger notion (often called strategy proofness in social choice) is that players must not be able to profit even when they know how others value the cake. As in [3], we say that a cake cutting procedure is truthful **iff there** are no valuations where a player will do better by lying.

## 4 Other Properties

We consider some other properties of (online) cake cutting procedures.

## 4.1 Surjectivity

This property has been studied in social choice but appears to have **received less attention** in fair division. It indicates whether the cake can be divided in every possible way. A cake cutting procedure is surjective **iff there are valuation functions for the players such that**
every possible partition of the cake into n **pieces is possible. Note that this definition only**