Finding the nucleolus of any n-person cooperative game by a single linear program

In this paper we show a new method for calculating the nucleolus by solving a unique minimization linear program with Oð4n Þ constraints whose coefficients belong to f−1; 0; 1g. We discuss the need of having all these constraints and empirically prove that they can be reduced to Oðkmax2n Þ, where kmax is a positive integer comparable with the number of players. A computational experience shows the applicability of our method over (pseudo)random transferable utility cooperative games with up to 18 playersThe authors want to thank Javier Arin, Safae El Haj Ben Ali, Guillermo Owen and Johannes H. Reijnierse for their useful and valuable help. The research of the authors has been partially supported by the projects FQM-5849 (Junta de Andalucia \ FEDER), and by the project MTM2010-19576-C02-01 (MICINN, Spain). This paper was written while the second author was enjoying a grant for a short postdoctoral research visit at the Instituto Universitario de Investigacion de Matematicas de la Universidad de Sevilla (IMUS). Special thanks are due to one anonymous referee for his/her valuable comments.Puerto Albandoz, J.; Perea Rojas Marcos, F. (2013). Finding the nucleolus of any n-person cooperative game by a single linear program. Computers and Operations Research. 40(10):2308-2313. https://doi.org/10.1016/j.cor.2013.03.011S23082313401

Universal characterization sets for the nucleolus in balanced games

We provide a new mo dus op erandi for the computation of the nucleolus in co op- erative games with transferable utility. Using the concept of dual game we extend the theory of characterization sets. Dually essential and dually saturated coalitions determine b oth the core and the nucleolus in monotonic games whenever the core is non-empty. We show how these two sets are related with the existing charac- terization 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. We conclude with a sample computation of the nucleolus of bankruptcy games - the shortest of its kind

Finding and verifying the nucleolus of cooperative games

The nucleolus offers a desirable payoff-sharing solution in cooperative games, thanks to its attractive properties—it always exists and lies in the core (if the core is non-empty), and it is unique. The nucleolus is considered as the most ‘stable’ solution in the sense that it lexicographically minimizes the dissatisfactions among all coalitions. Although computing the nucleolus is very challenging, the Kohlberg criterion offers a powerful method for verifying whether a solution is the nucleolus in relatively small games (i.e. with the number of players n≤ 15). This approach, however, becomes more challenging for larger games because of the need to form and check a criterion involving possibly exponentially large collections of coalitions, with each collection potentially of an exponentially large size. The aim of this work is twofold. First, we develop an improved version of the Kohlberg criterion that involves checking the ‘balancedness’ of at most (n- 1) sets of coalitions. Second, we exploit these results and introduce a novel descent-based constructive algorithm to find the nucleolus efficiently. We demonstrate the performance of the new algorithms by comparing them with existing methods over different types of games. Our contribution also includes the first open-source code for computing the nucleolus for games of moderately large sizes. © 2020, The Author(s)

The complexity of the nucleolus in compact games

This is the author accepted manuscript. The final version is available from ACM via the DOI in this recordThe nucleolus is a well-known solution concept for coalitional games to fairly distribute the total available worth among the players. The nucleolus is known to be NP-hard to compute over compact coalitional games, that is, over games whose functions specifying the worth associated with each coalition are encoded in terms of polynomially computable functions over combinatorial structures. In particular, hardness results have been exhibited over minimum spanning tree games, threshold games, and flow games. However, due to its intricate definition involving reasoning over exponentially many coalitions, a nontrivial upper bound on its complexity was missing in the literature and looked for. This article faces this question and precisely characterizes the complexity of the nucleolus, by exhibiting an upper bound that holds on any class of compact games, and by showing that this bound is tight even on the (structurally simple) class of graph games. The upper bound is established by proposing a variant of the standard linear-programming based algorithm for nucleolus computation and by studying a framework for reasoning about succinctly specified linear programs, which are contributions of interest in their own. The hardness result is based on an elaborate combinatorial reduction, which is conceptually relevant for it provides a "measure" of the computational cost to be paid for guaranteeing voluntary participation to the distribution process. In fact, the pre-nucleolus is known to be efficiently computable over graph games, with this solution concept being defined as the nucleolus but without guaranteeing that each player is granted with it at least the worth she can get alone, that is, without collaborating with the other players. Finally, this article identifies relevant tractable classes of coalitional games, based on the notion of type of a player. Indeed, in most applications where many players are involved, it is often the case that such players do belong in fact to a limited number of classes, which is known in advance and may be exploited for computing the nucleolus in a fast way.Part of E. Malizia’s work was supported by the European Commission through the European Social Fund and by Calabria Regio

Interactive operational decision making:Purchasing situations & mutual liability problems

Three chapters of this dissertation deal with three different types of interactive purchasing situations, in which multiple buying organizations interact with similar (or possibly the same) suppliers for the procurement of the same commodity. Decisions to be made in interactive purchasing concern if and how to cooperate with other buying organizations. If so, one has to tackle the important question of how to allocate possible cost savings. And if not, how to interact with the supplier(s) on an individual strategic level, while taking into account the strategic behavior of the other purchasers

Investment Under Uncertainty, Market Evolution and Coalition Spillovers in a Game Theoretic Perspective.

The rationality assumption has been the center of neo-classical economics for more than half a century now. In recent years much research has focussed on models of bounded rationality. In this thesis it is argued that both full and bounded rationality can be used for different kind of problems. In the first part full rationality is assumed to analyse technology adoption by firms in a duopolistic and uncertain environment. In the second part, boundedly rational models are developed to study the evolution of market structure in oligopolistic markets as well as price formation on (possibly) incomplete financial markets. The third part of the thesis presents an alternative to the framework of Transferable Utility games in cooperative game theory. The model introduced here explicitly takes into account the outside options that players often have in real-life situations if they choose not to participate in a coalition.