Perpetually Fair Assignments Via Balanced Sequences of Permutations

There is a set of n indivisible items (goods or chores), and a set of n players. Each day, a single item should be assigned to each player. Assignments based on latin squares guarantee fairness after every n days; our goal is to ensure fairness after every single day. We present two 'balance' conditions on latin squares. Informally, a latin square is balanced if its top rows and leftmost columns contain all n labels; this ensures that all n players receive one of the top items in one of the early days. One such condition can always be satisfied, but is arguably too weak; a second condition is strong, and can be satisfied for all n≤ 12, but cannot be satisfied for some larger values of n, including all n>108. We show that the second balance condition guarantees that the cumulative assignment is always proportional up to one item (PROP1), where proportionality holds in a strong ordinal sense -- for every valuations that are consistent with the item ranking. Finally, we present a weaker balance condition on a sequence, that guarantees ordinal proportionality up to two items (PROP2). Whether or not this condition can be satisfied for all n remains an open question.