Temporal Fair Division of Indivisible Items

We study a fair division model where indivisible items arrive sequentially, and must be allocated immediately and irrevocably. Previous work on online fair division has shown impossibility results in achieving approximate envy-freeness under these constraints. In contrast, we consider an informed setting where the algorithm has complete knowledge of future items, and aim to ensure that the cumulative allocation at each round satisfies approximate envy-freeness -- which we define as temporal envy-freeness up to one item (TEF1). We focus on settings where items can be exclusively goods or exclusively chores. For goods, while TEF1 allocations may not always exist, we identify several special cases where they do -- two agents, two item types, generalized binary valuations, unimodal preferences -- and provide polynomial-time algorithms for these cases. We also prove that determining the existence of a TEF1 allocation is NP-hard. For chores, we establish analogous results for the special cases, but present a slightly weaker intractability result. We also establish the incompatibility between TEF1 and Pareto-optimality, with the implication that it is intractable to find a TEF1 allocation that maximizes any $p$-mean welfare, even for two agents.

Temporal Elections: Welfare, Strategyproofness, and Proportionality

We investigate a model of sequential decision-making where a single alternative is chosen at each round. We focus on two objectives-utilitarian welfare (Util) and egalitarian welfare (Egal)-and consider the computational complexity of the associated maximization problems, as well as their compatibility with strategyproofness and proportionality. We observe that maximizing Util is easy, but the corresponding decision problem for Egal is NP-complete even in restricted cases. We complement this hardness result for Egal with parameterized complexity analysis and an approximation algorithm. Additionally, we show that, while a mechanism that outputs a Util outcome is strategyproof, all deterministic mechanisms for computing Egal outcomes fail a very weak variant of strategyproofness, called non-obvious manipulability (NOM). However, we show that when agents have non-empty approval sets at each timestep, choosing an Egal-maximizing outcome while breaking ties lexicographically satisfies NOM. Regarding proportionality, we prove that a proportional (PROP) outcome can be computed efficiently, but finding an outcome that maximizes Util while guaranteeing PROP is NP-hard. We also derive upper and lower bounds on the price of proportionality with respect to Util and Egal.