A Loss Calibarated Methods for Bipartite Rationing Problems

The standard problem of rationing a single overdemanded commodity (O’Neill [1982], Aumann and Maschler [1985], Young [1987]) has a natural bipartite extension with multiple types of a single commodity (e.g., jobs with different skill requirements), and each agent can only consume some types of the resource (e.g., workers have different skills). We define the new standard loss calibrated rationing methods, that equalize across agents the ratio of shares to (calibrated) losses (demand minus share). We extend them to bipartite methods that 1) are not affected by the elimination of an edge and the corresponding flow (Consistency), and 2) treat resource types with identical connectivity as a single type. They are essentially the only standard methods with a bipartite extension meeting the two properties above. Most of the parametric methods discussed in the literature (Young [1987], Young [1988]) do not admit such extension.