Characterization sets for the nucleolus in balanced games

Abstract

We provide a new modus operandi for the computation of the nucleolus in cooperative games with transferable utility. Using the concept of dual game we extend the theory of characterization sets. Dually essential and if the game is monotonic dually saturated coalitions determine both the core and the nucleolus whenever the core is non-empty. We show how these two sets are related to the existing characterization sets. In particular we prove that if the grand coalition is vital then the intersection of essential and dually essential coalitions forms a characterization set itself.

Sharing Loading Costs for Multi Compartment Vehicles

“…Another interesting area for research is to utilize recent work on characterization sets for the nucleolus (for instance [24,25]) to restrict further the number of constraints that need to be tested at each step. This would improve the speed of the approach as well as possibly building the solution calculation into the algorithm.…”

“…The nucleolus solution can be chosen to allocate the cost to each shipper. Computation of the nucleolus in general is NP-hard; it can be made somewhat more tractable by techniques introduced fairly recently [19,20,[24][25][26]. In any event, its computation could be done after the algorithm described above is completed and all subset cost estimates are available.…”

Weighted nucleoli and dually essential coalitions

“…We say that the core of balanced game (N , v) is fulldimensional if dim Co(v) = |N | − 1, or equivalently, if N is the only tight coalition over the entire core. Solymosi and Sziklai (2016) proved (in terminology of this paper) that if the core of balanced game (N , v) is full-dimensional then all inessential coalitions and the complements of all dually inessential coalitions can be ignored when we compute the standard (pre)nucleolus and the standard least core from the values of v. For a better understanding of the key issues, we supplement that paper by the following example. We demonstrate that without full-dimensionality of the core we might need to keep some of the aforementioned (individually redundant) coalitions.…”

“…Let us illustrate why this happens and why this can happen only when the core is not full-dimensional, see (Solymosi and Sziklai 2016) for a general proof. We dropped, for example, coalition 124 because it is weakly majorized by its partition v(12)…”

“…We omit the details, but underline that the key for being able to combine the proofs of Theorem 1 and 2 (rewritten in terms of v) is also what Solymosi and Sziklai (2016) proved: if the core of the balanced game v is full-dimensional then for every inessential coalition there is a weakly majorizing partition in E (v) ∩ F (v) and for the complement of every dually inessential coalition there is a weakly majorizing antipartition in…”