Dividing the Indivisible: Procedures for Allocating Cabinet Ministries to Political Parties in a Parliamentary System

Political parties in Northern Ireland recently used a divisor method of apportionment to choose, in sequence, ten cabinet ministries. If the parties have complete information about each others' preferences, we show that it may not be rational for them to act sincerely by choosing their most-preferred ministry that is available. One consequence of acting sophisticatedly is that the resulting allocation may not be Pareto-optimal, making all the parties worse off. Another is nonmonotonicty-choosing earlier may hurt rather than help a party. We introduce a mechanism that combines sequential choices with a structured form of trading that results in sincere choices for two parties. Although there are difficulties in extending this mechanism to more than two parties, other approaches are explored, such as permitting parties to making consecutive choices not prescribed by an apportionment method. But certain problems, such as eliminating envy, remain.APPORTIONMENT METHODS; CABINETS; SEQUENTIAL ALLOCATION; MECHANISM DESIGN; FAIRNESS

Distributed Fair Allocation of Indivisible Goods

International audienceDistributed mechanisms for allocating indivisible goods are mechanisms lacking central control, in which agents can locally agree on deals to exchange some of the goods in their possession. We study convergence properties for such distributed mechanisms when used as fair division procedures. Specifically, we identify sets of assumptions under which any sequence of deals meeting certain conditions will converge to a proportionally fair allocation and to an envy-free allocation, respectively. We also introduce an extension of the basic framework where agents are vertices of a graph representing a social network that constrains which agents can interact with which other agents, and we prove a similar convergence result for envy-freeness in this context. Finally, when not all assumptions guaranteeing envy-freeness are satisfied, we may want to minimise the degree of envy exhibited by an outcome. To this end, we introduce a generic framework for measuring the degree of envy in a society and establish the computational complexity of checking whether a given scenario allows for a deal that is beneficial to every agent involved and that will reduce overall envy

Minimizing and balancing envy among agents using Ordered Weighted Average

International audienceIn the problem of fair resource allocation, envy freeness is one of the most interesting fairness criterion as it ensures that no agent prefers the bundle of another agent. However, when considering indivisible goods, an envy-free allocation may not exist. In this paper, we investigate a new relaxation of envy freeness consisting in minimizing the Ordered Weighted Average (OWA) of the envy vector. The idea is to choose the allocation that is fair in the sense of the distribution of the envy among agents. The OWA aggregator is a well-known tool to express fairness in multiagent optimization. In this paper, we focus on fair OWA operators where the weights of the OWA are decreasing. When an envy-free allocation exists, minimizing OWA will return this allocation. However, when no envy-free allocation exists, one may wonder how fair min OWA allocations are. After some definitions and description of the model, we show how to formulate the computation of such a min OWA allocation as a Mixed Integer Program. Then, we investigate the link between the min OWA allocation and other well-known fairness measures such as max min share and envy freeness up to one good or to any good

Computing Pareto-Optimal and Almost Envy-Free Allocations of Indivisible Goods

We study the problem of fair and efficient allocation of a set of indivisible goods to agents with additive valuations using the popular fairness notions of envy-freeness up to one good (EF1) and equitability up to one good (EQ1) in conjunction with Pareto-optimality (PO). There exists a pseudo-polynomial time algorithm to compute an EF1+PO allocation and a non-constructive proof of the existence of allocations that are both EF1 and fractionally Pareto-optimal (fPO), which is a stronger notion than PO. We present a pseudo-polynomial time algorithm to compute an EF1+fPO allocation, thereby improving the earlier results. Our techniques also enable us to show that an EQ1+fPO allocation always exists when the values are positive and that it can be computed in pseudo-polynomial time. We also consider the class of $k$-ary instances where $k$ is a constant, i.e., each agent has at most $k$ different values for the goods. For such instances, we show that an EF1+fPO allocation can be computed in strongly polynomial time. When all values are positive, we show that an EQ1+fPO allocation for such instances can be computed in strongly polynomial time. Next, we consider instances where the number of agents is constant and show that an EF1+PO (likewise, an EQ1+PO) allocation can be computed in polynomial time. These results significantly extend the polynomial-time computability beyond the known cases of binary or identical valuations. We also design a polynomial-time algorithm that computes a Nash welfare maximizing allocation when there are constantly many agents with constant many different values for the goods. Finally, on the complexity side, we show that the problem of computing an EF1+fPO allocation lies in the complexity class PLS.Comment: 23 pages. A preliminary version appeared at AAAI 202