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.
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