How to Make Envy Vanish Over Time

We study the dynamic fair division of indivisible goods. Suppose T items arrive online and must be allocated upon arrival to one of n agents, each of whom has a value in [0,1] for the current item. Our goal is to design allocation algorithms that minimize the maximum envy at time T , ENVYT, defined as the maximum difference between any agent's overall value for items allocated to another agent and to herself. We say that an algorithm has vanishing envy if the ratio of envy over time, ENVYT/T, goes to zero as T goes to infinity. We design a polynomial-time, deterministic algorithm that achieves ENVYT ın ~O ( √T/n ), and show that this guarantee is asymptotically optimal. We also derive tight (in T ) bounds for a more general setting where items arrive in batches.

[1]  Allan Borodin,et al.  On the power of randomization in on-line algorithms , 2005, Algorithmica.

[2]  Elchanan Mossel,et al.  On approximately fair allocations of indivisible goods , 2004, EC '04.

[3]  Steven J. Brams,et al.  Fair division - from cake-cutting to dispute resolution , 1998 .

[4]  Ariel D. Procaccia,et al.  The Unreasonable Fairness of Maximum Nash Welfare , 2016, EC.

[5]  Toby Walsh,et al.  Online Cake Cutting , 2010, ADT.

[6]  W. Stromquist.  How to Cut a Cake Fairly , 1980 .

[7]  F. Su.  Rental Harmony: Sperner's Lemma in Fair Division , 1999 .

[8]  Ariel D. Procaccia,et al.  Beyond Dominant Resource Fairness , 2015, ACM Trans. Economics and Comput..

[9]  A E Bostwick,et al.  THE THEORY OF PROBABILITIES. , 1896, Science.

[10]  Ariel D. Procaccia,et al.  No agent left behind: dynamic fair division of multiple resources , 2013, AAMAS.

[11]  Sébastien Bubeck,et al.  Regret Analysis of Stochastic and Nonstochastic Multi-armed Bandit Problems , 2012, Found. Trends Mach. Learn..

[12]  Ariel D. Procaccia.  Cake Cutting Algorithms , 2016, Handbook of Computational Social Choice.

[13]  Xiaotie Deng,et al.  Algorithmic Solutions for Envy-Free Cake Cutting , 2012, Oper. Res..

[14]  Peter Sanders.  On the Competitive Analysis of Randomized Static Load Balancing , 2017 .

[15]  JOEL SPENCER,et al.  Balancing games , 1977, J. Comb. Theory B.

[16]  Toby Walsh,et al.  Online Fair Division: Analysing a Food Bank Problem , 2015, IJCAI.

[17]  Paul R. Milgrom,et al.  Designing Random Allocation Mechanisms: Theory and Applications , 2013 .

[18]  Eric J. Friedman,et al.  Controlled Dynamic Fair Division , 2017, EC.

[19]  Toby Walsh,et al.  Pure Nash Equilibria in Online Fair Division , 2017, IJCAI.