Fairness-Efficiency Tradeoffs in Dynamic Fair Division David Zeng, Alex Psomas • 𝑈 items arrive online, 𝑜 agents • Agent 𝑗 has value 𝑤 𝑗𝑢 ∈ [0,1] for item 𝑢 that we learn when the item arrives • Item must be allocated immediately and irrevocably • After 𝑈 rounds, we will have some allocation 𝐵 = (𝐵 1 , … , 𝐵 𝑜 ) • Additive valuations: 𝑤 𝑗 𝐵 𝑘 = σ 𝑢 ∈𝐵 𝑘 𝑤 𝑗𝑢 • Ideally, allocation is both fair and efficient
Adversary Model and Results 1 No 𝑜 + 𝜁 -Pareto efficient and sublinear envy allocation algorithm Non-adaptive Adaptive 𝑤 𝑢 ~𝐸 Ԧ 𝑤 𝑗𝑢 ~𝐸 𝑤 𝑗𝑢 ~𝐸 𝑗 adversary adversary Fairness and efficiency Fairness and efficiency compatible incompatible ex-post Pareto efficient and pairwise (EF1 or EF w.h.p.) allocation algorithm
Algorithm for Correlated Agents ( Ԧ 𝑤 𝑢 ~𝐸 ) • Reduce finding a • online ex-post Pareto efficient and (EF1 or EF w.h.p.) allocation algorithm to finding a • offline Pareto efficient and CISEF fractional allocation • Algorithm sketch Given online problem and distribution 𝐸 with support 𝛿 1 , … , 𝛿 𝑛 , use the • support of 𝐸 as the items for offline problem, scaling by the probabilities. Use the fractional allocation 𝑌 to guide our allocation in the online problem. • If 𝑌 𝑗𝑙 = 0.4 , if the item arriving at time 𝑢 has type 𝛿 𝑙 , allocate the item to agent 𝑗 with • probability 0.4 Treat cliques as one combined agent when doing randomized allocation • When item is allocated to the clique, give to unhappiest agent in clique •
CISEF Clique Identical Strongly Envy-Free • CISEF 1 2 Either agent 𝑗 strictly prefers her own • bundle to the bundle of agent 𝑘 Or 𝑗 and 𝑘 have identical allocations • and the same value (up to a scaling factor) for all the items that are allocated to either of them • How to find CISEF and Pareto Identical allocations efficient allocation? and valuations 3 Start with solution to Eisenberg-Gale • convex program Strongly envy-free If agent 𝑗 is indifferent to agent 𝑘 , • (carefully) move items from 𝑘 to 𝑗 to create strong envy-free edges
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