Fairness-Efficiency Tradeoffs in Dynamic Fair Division David Zeng, - - PowerPoint PPT Presentation

Fairness-Efficiency Tradeoffs in Dynamic Fair Division

• 𝑈 items arrive online, 𝑜 agents
• Agent 𝑗 has value 𝑤𝑗𝑢 ∈ [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 = (𝐵1, … , 𝐵𝑜)
• Additive valuations: 𝑤𝑗 𝐵𝑘 = σ𝑕𝑢∈𝐵𝑘 𝑤𝑗𝑢
• Ideally, allocation is both fair and efficient

David Zeng, Alex Psomas

Adversary Model and Results

𝑤𝑗𝑢~𝐸 𝑤𝑗𝑢~𝐸𝑗 Ԧ 𝑤𝑢~𝐸 Non-adaptive adversary Adaptive adversary Fairness and efficiency compatible Fairness and efficiency incompatible ex-post Pareto efficient and pairwise (EF1 or EF w.h.p.) allocation algorithm No

1 𝑜 + 𝜁 -Pareto efficient and sublinear

envy allocation algorithm

Algorithm for Correlated Agents ( Ԧ 𝑤𝑢~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 𝛿1, … , 𝛿𝑛 , 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 𝑌𝑗𝑙 = 0.4, if the item arriving at time 𝑢 has type 𝛿𝑙, 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

Strongly envy-free Identical allocations and valuations

1 2 3

Clique Identical Strongly Envy-Free

• CISEF
• 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

efficient allocation?

• Start with solution to Eisenberg-Gale

convex program

• If agent 𝑗 is indifferent to agent 𝑘,

(carefully) move items from 𝑘 to 𝑗 to create strong envy-free edges