  In the assignment problem, the goal is to assign indivisible items to agents
who have ordinal preferences, efficiently and fairly, in a strategyproof
manner. In practice, first-choice maximality, i.e., assigning a maximal number
of agents their top items, is often identified as an important efficiency
criterion and measure of agents' satisfaction. In this paper, we propose a
natural and intuitive efficiency property,
favoring-eagerness-for-remaining-items (FERI), which requires that each item is
allocated to an agent who ranks it highest among remaining items, thereby
implying first-choice maximality. Using FERI as a heuristic, we design
mechanisms that satisfy ex-post or ex-ante variants of FERI together with
combinations of other desirable properties of efficiency (Pareto-efficiency),
fairness (strong equal treatment of equals and sd-weak-envy-freeness), and
strategyproofness (sd-weak-strategyproofness). We also explore the limits of
FERI mechanisms in providing stronger efficiency, fairness, or
strategyproofness guarantees through impossibility results.
