Optimizing Social Welfare in Social Networks

We study the computational complexity of envy minimization and maximizing the social welfare of graph-envy-free allocations in social networks. Besides the already known \(\mathrm {NP}\)-completeness of finding allocations with maximal utilitarian social welfare we prove that \(\mathrm {NP}\)-completeness is in general also given for the egalitarian social welfare and the Nash product. Moreover, we focus on an extended model, based on directed social relationship graphs and undirected social trading graphs, and analyze the computational complexity of reaching a graph-envy-free allocation by trades with so-called don’t care agents and without money.

[1]  Atila Abdulkadiroglu,et al.  RANDOM SERIAL DICTATORSHIP AND THE CORE FROM RANDOM ENDOWMENTS IN HOUSE ALLOCATION PROBLEMS , 1998 .

[2]  Gerhard J. Woeginger,et al.  Good Things Come to Those Who Swap Objects on Paths , 2019, ArXiv.

[3]  Rolf Niedermeier,et al.  Envy-Free Allocations Respecting Social Networks , 2018, AAMAS.

[4]  Edith Elkind,et al.  Fair Division of a Graph , 2017, IJCAI.

[5]  Ioannis Caragiannis,et al.  Knowledge, Fairness, and Social Constraints , 2018, AAAI.

[6]  Yann Chevaleyre,et al.  Allocating Goods on a Graph to Eliminate Envy , 2007, AAAI.

[7]  Abram Burk.  A Reformulation of Certain Aspects of Welfare Economics , 1938 .

[8]  Richard M. Karp,et al.  Reducibility Among Combinatorial Problems , 1972, 50 Years of Integer Programming.

[9]  M. O. Lorenz,et al.  Methods of Measuring the Concentration of Wealth , 1905, Publications of the American Statistical Association.

[10]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[11]  Jörg Rothe,et al.  Complexity Theory and Cryptology. An Introduction to Cryptocomplexity , 2005, Texts in Theoretical Computer Science. An EATCS Series.

[12]  Laurent Gourvès,et al.  Object Allocation via Swaps along a Social Network , 2017, IJCAI.

[13]  Jörg Rothe,et al.  Fairness and Rank-Weighted Utilitarianism in Resource Allocation , 2015, ADT.

[14]  Jörg Rothe,et al.  Fair Division of Indivisible Goods , 2016, Economics and Computation.

[15]  Yann Chevaleyre,et al.  Issues in Multiagent Resource Allocation , 2006, Informatica.

[16]  Jörg Rothe,et al.  Approximation and complexity of the optimization and existence problems for maximin share, proportional share, and minimax share allocation of indivisible goods , 2018, Autonomous Agents and Multi-Agent Systems.