The analysis reveals $( ext{ln}(n)+2)$-approximate Pareto efficiency in allocation tasks, indicating efficiency bounds under cardinal preferences.
Key Points
The Probabilistic Serial mechanism shows $( ext{ln}(n)+2)$-approximate Pareto efficiency under cardinal preferences, addressing a significant performance gap.
A polynomial-time algorithm computes a fair allocation that is envy-free and approximately Pareto-efficient in $O( ext{sqrt}(n))$ welfare degradation.
This research establishes that the PS mechanism can also handle chore allocations, ensuring $n$-approximately Pareto-efficient outcomes.
The results offer the first bounds for fair, efficient allocation in cardinal preference scenarios, contributing to the assignment problem framework.