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