Distributed envy minimization for resource allocation

Minimizing envy in distributed discrete resource or task allocation, is an unusual distributed optimization challenge, since the quality of the allocation for each of the agents is dependent, not only on its own allocation, but on the allocation of others as well. Thus, in order to perform distributed search for allocations with minimal envy there is a need to design innovative algorithms that can cope with the challenging constraint structure of an envy minimization problem. Distributed methods for minimizing envy among agents in indivisible resource allocation problems are presented. First, Distributed Envy Minimization Problems (DEMP) are formulated as Distributed Constraint Reasoning problems. When the DEMPs are large, and cannot be solved by a complete search an incomplete local search algorithm is presented. Each transfer of a good from one agent to another involves the change of state of more than one agent. Thus, a minimizing envy local search algorithm must build upon actions (transfers) that include multiple agents. Since DEMPs are particularly susceptible to local minima during local search, the paper proposes an algorithm that alternates between two different hill climbing search phases. The first phase uses one-transfer steps while the other exploits envy cycle elimination steps. An algorithm that minimizes envy while preserving efficiency, is proposed. The proposed algorithm finds a Pareto optimal allocation with low envy. In the context of resource allocation problems, a Pareto optimal solution is particularly desirable since it presents a stable solution. The proposed algorithm first finds a divisible Pareto optimal envy-free allocation using a Fisher market equilibrium. This allocation is transferred into an indivisible allocation of goods while maintaining the Pareto optimal characteristic of the allocation and a low envy level among agents.