Consistent bilateral assignment

In the bilateral assignment problem, source a holds the amount ra of resource of type a, while sink i must receive the total amount xi of the various resources. We look for assignment rules meeting the powerful separability property known as Consistency: “every subassignment of a fair assignment is fair”. They are essentially those rules selecting the feasible flow minimizing the sum ∑i,aW(yia), where W is smooth and strictly convex

Scheduling with Opting Out: Improving upon Random Priority

In a scheduling problem where agents can opt out, we show that the familiar Random Priority (RP) a rule can be improved upon by another mechanism dubbed Probabilistic Serial (PS). Both mechanisms are nonmanipulable in a strong sense, but the latter is Pareto superior to the former and serves a larger (expected number of agents. The PS equilibrium outcome is easier to compute than the RP outcome; on the other hand RP is easier to implement than PS. We show that the improvement of PS over RP is significant but small: at most a couple of percentage points in the relative welfare gain and the relative difference in quantity served. We conjecture that the latter never exceeds 8.33 %. Both gains vanish when the number of agents is large.

Minimizing the Worst Slowdown: Off-Line and On-Line

Minimizing the slowdown (expected sojourn time divided by job size) is a key concern of fairness in scheduling and queuing problems where job sizes are very heterogeneous. We look for protocols (service disciplines) capping the worst slowdown (called here liability) a job may face no matter how large (or small) the other jobs are. In the scheduling problem (all jobs released at the same time), allowing the server to randomize the order of service cuts almost in half the liability profiles feasible under deterministic protocols. The same statement holds if cash transfers are feasible and users have linear waiting costs.

Normative Microeconomics and the Social Contract

More than thirty years ago, the advancing mathematical economics and the emerging game theory joined forces to attack an ambitious program of social engineering in the microeconomic scale often known "mechanism design", but more accurately described as "normative economics." It combines the tools of normative/axiomatic and strategic/equilibrium analysis to address, inter alia, the design of auctions and other trading mechanisms, the provision of public goods, the fair division of costs or manna--e.g., inheritance and bankruptcy settlements--, the rationing of overdemanded commodities and the scheduling of tasks. My goal in this lecture is to explore the methodological and ideological premises of normative microeconomics. I submit that this approach falls squarely in the three centuries old tradition in political philosophy known as the social contract doctrine, and provides powerful arguments against its intellectual nemesis, the minimal state doctrine. This controversial strand explains some of the resistance to the normative approach within the academic economic profession, and is likely to shape its development for the foreseeable future.

On Scheduling Fees to Prevent Merging, Splitting and Transferring of Jobs

A deterministic server is shared by users with identical linear waiting costs, requesting jobs of arbitrary lengths. Shortest jobs are served first for efficiency. The server can monitor the length of a job, but not the identity of its user, thus merging, splitting or partially transferring jobs offer cooperative strategic opportunities. Can we design cash transfers to neutralize such manipulations? We prove that merge-proofness and split-proofness are not compatible, and that it is similarly impossible to prevent all transfers of jobs involving three agents or more. On the other hand, robustness against pair-wise transfers is feasible, and essentially characterize a one-dimensional set of scheduling methods. This line is borne by two outstanding methods, the merge-proof S+ and the split-proof S?. Splitproofness, unlike Mergeproofness, is not compatible with several simple tests of equity. Thus the two properties are far from equally demanding.

Three Solutions to a Simple Commons Problem

We compare the equity and incentive properties of three efficient solutions to a simple problem of cooperative production with binary demands for a homogeneous service, when marginal cost is either monotonically increasing or monotonically decreasing. The solutions are the familiar competitive equilibrium with equal incomes. The Shapley value of the stand alone cooperative game, and the virtual price solution, applying the egalitarian equivalence idea to this particular model.

Split-Proof Probabilistic Scheduling

If shortest jobs are served first, splitting a long job into smaller jobs reported under different aliases can reduce the actual wait until completion. If longest jobs are served first, the dual maneuver of merging several jobs under a single reported identity is profitable. Both manipulations can be avoided if the scheduling order is random, and users care only about the expected wait until completion of their job. The Proportional rule stands out among rules immune to splitting and merging. It draws the job served last with probabilities proportional to size, then repeats among the remaining jobs. Among split-proof scheduling rules constructed in this recursive way, it is characterized by either one of the three following properties: an agent with a longer job incurs a longer delay; total expected delay is at most twice optimal delay; the worst expected delay of any single job is at most twice the smallest feasible worst delay. A similar result holds within the natural family of separable rules.

Size versus fairness in the assignment problem

When not all objects are acceptable to all agents, maximizing the number of objects actually assigned is an important design concern. We compute the guaranteed size ratio of the Probabilistic Serial mechanism, i.e., the worst ratio of the actual expected size to the maximal feasible size. It converges decreasingly to 1 − 1 e 63.2% as the maximal size increases. It is the best ratio of any Envy-Free assignment mechanism

On Demand Responsiveness in Additive Cost Sharing

We propose two new axioms of demand responsiveness for additive cost sharing with variable demands. Group Monotonicity requires that if a group of agents increase their demands, not all of them pay less. Solidarity says that if agent i demands more, j should not pay more if k pays less. Both axioms are compatible in the partial responsibility theory postulating Strong Ranking, i.e., the ranking of cost shares should never contradict that of demands. The combination of Strong Ranking , Solidarity and Monotonicity characterizes the quasi-proportional methods, under which cost shares are proportional to 'rescaled' demands. The alternative full responsibility theory is based on Separability, ruling out cross-subsidization when costs are additively separable. Neither the Aumann-Shapley nor the Shapley-Shubik method is group monotonic. On the otherhand, convex combinations of "nearby" fixed-path methods are group-monotonic: the subsidy-free serial method is the main example. No separable method meets Solidarity, yet restricting the axiom to submodular (or supermodular) cost functions leads to a characterization of the fixed-flow methods, containing the Shapley-Shubik and serial methods.

Bargaining among Groups: An Axiomatic Viewpoint

We introduce a model of bargaining among groups, and characterize a family of solutions using a Consistency axiom and a few other invariance and monotonicity properties. For each solution in the family, there exists some constant alpha >= 0 such that the "bargaining power" of a group is proportional to calpha where c is the cardinality of the group.