scholarly journals Dividing bads under additive utilities

2018 ◽  
Vol 52 (3) ◽  
pp. 395-417 ◽  
Author(s):  
Anna Bogomolnaia ◽  
Hervé Moulin ◽  
Fedor Sandomirskiy ◽  
Elena Yanovskaia
Keyword(s):  
Additive Utilities

scholarly journals When Do Envy-Free Allocations Exist?

2019 ◽  
Vol 33 ◽  
pp. 2109-2116 ◽  
Author(s):  
Pasin Manurangsi ◽  
Warut Suksompong
Keyword(s):  
Phase Transition ◽  
High Probability ◽  
Fair Division ◽  
The Other ◽  
Other Hand ◽  
Additive Utilities ◽  
The One

We consider a fair division setting in which m indivisible items are to be allocated among n agents, where the agents have additive utilities and the agents’ utilities for individual items are independently sampled from a distribution. Previous work has shown that an envy-free allocation is likely to exist when m = Ω (n log n) but not when m = n + o (n), and left open the question of determining where the phase transition from non-existence to existence occurs. We show that, surprisingly, there is in fact no universal point of transition— instead, the transition is governed by the divisibility relation between m and n. On the one hand, if m is divisible by n, an envy-free allocation exists with high probability as long as m ≥ 2n. On the other hand, if m is not “almost” divisible by , an envy-free allocation is unlikely to exist even when m = Θ(n log n)/log log n).

Additive Utilities on Densely Ordered Sets

2002 ◽  
Vol 46 (5) ◽  
pp. 515-530 ◽  
Author(s):  
Yutaka Nakamura
Keyword(s):  
Ordered Sets ◽  
Additive Utilities

scholarly journals Computing an Approximately Optimal Agreeable Set of Items

2017 ◽  
Author(s):  
Pasin Manurangsi ◽  
Warut Suksompong
Keyword(s):  
Polynomial Time ◽  
Approximation Ratio ◽  
Small Subset ◽  
Worst Case ◽  
Ordinal Preferences ◽  
Additive Utilities

We study the problem of finding a small subset of items that is agreeable to all agents, meaning that all agents value the subset at least as much as its complement. Previous work has shown worst-case bounds, over all instances with a given number of agents and items, on the number of items that may need to be included in such a subset. Our goal in this paper is to efficiently compute an agreeable subset whose size approximates the size of the smallest agreeable subset for a given instance. We consider three well-known models for representing the preferences of the agents: ordinal preferences on single items, the value oracle model, and additive utilities. In each of these models, we establish virtually tight bounds on the approximation ratio that can be obtained by algorithms running in polynomial time.

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