Sequential Fair Allocation: Achieving the Optimal Envy-Efficiency Tradeoff Curve
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by
Sean R. Sinclair, Siddhartha Banerjee, Christina Lee Yu
2021
Abstract
We consider the problem of dividing limited resources to individuals arriving
over T rounds. Each round has a random number of individuals arrive, and
individuals can be characterized by their type (i.e. preferences over the
different resources). A standard notion of `fairness' in this setting is that
an allocation simultaneously satisfy envy-freeness and efficiency. The former
is an individual guarantee, requiring that each agent prefers her own
allocation over the allocation of any other; in contrast, efficiency is a
global property, requiring that the allocations clear the available resources.
For divisible resources, when the number of individuals of each type are known
upfront, the above desiderata are simultaneously achievable for a large class
of utility functions. However, in an online setting when the number of
individuals of each type are only revealed round by round, no policy can
guarantee these desiderata simultaneously, and hence the best one can do is to
try and allocate so as to approximately satisfy the two properties.
We show that in the online setting, the two desired properties (envy-freeness
and efficiency) are in direct contention, in that any algorithm achieving
additive envy-freeness up to a factor of L_T necessarily suffers an
efficiency loss of at least 1 / L_T. We complement this uncertainty principle
with a simple algorithm, HopeGuardrail, which allocates resources based on an
adaptive threshold policy. We show that our algorithm is able to achieve any
fairness-efficiency point on this frontier, and moreover, in simulation
results, provides allocations close to the optimal fair solution in hindsight.
This motivates its use in practical applications, as the algorithm is able to
adapt to any desired fairness efficiency trade-off.
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