Finding the nucleoli of large cooperative games

Abstract

Highlights• We propose a new nested-LPs-based approach for finding the nucleolus• We resolve subtle issues in handling multiple optimal solutions in each LP• The approach involves fewer and smaller LPs compared to existing methods• The approach proves to work well in several large combinatorial games AbstractThe nucleolus is one of the most important solution concepts in cooperative game theory as a result of its attractive properties -it always exists (if the imputation is non-empty), is unique, and is always in the core (if the core is non-empty). However, computing the nucleolus is very challenging because it involves the lexicographical minimization of an exponentially large number of excess values. We present a method for computing the nucleoli of large games, including some structured games with more than 50 players, using nested linear programs (LP). Although different variations of the nested LP formulation have been documented in the literature, they have not been used for large games because of the large size and number of LPs involved. In addition, subtle issues such as how to deal with multiple optimal solutions and with tight constraint sets need to be resolved in each LP in order to formulate and solve the subsequent ones. Unfortunately, this technical issue has been largely overlooked in the literature. We treat these issues rigorously and provide a new nested LP formulation that is smaller in terms of the number of large LPs and their sizes. We provide numerical tests for several games, including the general flow games, the coalitional skill games and the weighted voting games, with up to 100 players.

Finding and verifying the nucleolus of cooperative games

“…There are structured games such as weighted voting games, network flow games and coalitional skill games in which Step 2 can be executed efficiently. We refer the readers to [20] for details.…”

“…Instead, since these games are of a very peculiar nature from an algorithmic perspective, they carry a theoretical interest from a computational point of view. For games with structures like this, we expect further improvement by Table 5 Computing the nucleolus of type I games exploiting the structure in a similar way to [20]. However, within the scope of this paper, we want to provide a like-for-like comparison and hence leave further improvements for future research.…”

“…The nucleolus can also be found by solving a sequence of LPs. However, either the number of LPs involved is exponentially large [15,16] or the sizes of the LPs are exponential [17][18][19][20]. Our second aim is to directly solve the lexicographical minimization problem via introducing a new descent-based approach.…”

Weighted nucleoli and dually essential coalitions

“…In general, a linearly weighted nucleolus can be determined by the very same methods as the (standard) nucleolus, only straightforward adjustments are needed that only negligibly effect the performance. This is particularly true for the most frequently applied sequential linear programming approach pioneered by Kopelowitz (1967) (for a recent implementation finely tuned even for large games, see (Nguyen and Thomas 2016)). On the other hand, and in contrast to the rich literature on the computation of the (standard) nucleolus in specific classes of games, we can only mention the algorithm by Huijink et al (2015) that computes the per-capita nucleolus in bankruptcy games.…”

Finding the Nucleoli of Large Cooperative Games: A Disproof with Counter-Example

“…1 INTRODUCTION Nguyen and Thomas (2016) claimed that they have invented a method for the extremely challenging task of computing the nucleoli of games with more than 50 players using nested linear programs (LP). Unfortunately, their claim is false.…”