The Nucleolus, the Kernel, and the Bargaining Set: An Update

One of David Schmeidler’s many important contributions in his distinguished career was the introduction of the nucleolus, one of the central single-valued solution concepts in cooperative game theory. This paper is an updated survey on the nucleolus and its two related supersolutions, i.e., the kernel and the bargaining set. As a first approach to these concepts, we refer the reader to the great survey by Maschler (1992); see also the relevant chapters in Peleg and Sudholter (2003). Building on the notes of four lectures on the nucleolus and the kernel delivered by one of the authors at the Hebrew University of Jerusalem in 1999, we have updated Maschler’s survey by adding more recent contributions to the literature. Following a similar structure, we have also added a new section that covers the bargaining set. The nucleolus has a number of desirable properties, including nonemptiness, uniqueness, core selection, and consistency. The first way to understand it is based on an egalitarian principle among coalitions. However, by going over the axioms that characterize it, what comes across as important is its connection with coalitional stability, as formalized in the notion of the core. Indeed, if one likes a single-valued version of core stability that always yields a prediction, one should consider the nucleolus as a recommendation. The kernel, which contains the nucleolus, is based on the idea of “bilateral equilibrium” for every pair of players. And the bargaining set, which contains the kernel, checks for the credibility of objections coming from coalitions. In this paper, section 2 presents preliminaries, section 3 is devoted to the nucleolus, section 4 to the kernel, and section 5 to the bargaining set.Iñarra acknowledges research support from the Spanish Government grant ECO2015-67519-P, and Shimomura from Grant-in-Aid for Scientific Research (A)18H03641 and (C)19K01558

Modelling human fairness in cooperative games : a goal programming approach

The issues of rationality in human behavior and fairness in cooperation have gained interest in various economic studies. In many prescriptive models of games, rationality of human decision makers implicitly assumes exchange-ability. This means that real people are assumed to adopt the beliefs of a player as expressed in the game when placed in the shoes of that particular player. However, it is a well debated topic in the literature that this modeling assumption is not in accordance to what behavioral economists have observed in some games played with real human subjects. Even when assuming the role of the same player in the game, different people think differently about the fairness of a particular outcome. People also view fairness as an essential ingredient of their decision making processes in games on cooperation. The aim of this research is to develop a new modeling approach to decision making in games on cooperation in which fairness is an important consideration. The satisficing and egilitarian philosophies on which weighted and Chebyshev Goal Programming (GP) rely, seem to offer an adequate and natural way for modeling human decision processes in at least the single-shot games of coordination that are investigated in this work. The solutions returned by the proposed GP approach aim to strike the right balance on several dimensions of con icting goals that are set by players themselves and that arise in the mental models these players have of other relevant players.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Game theoretic optimisation in process and energy systems engineering: A review

Game theory is a framework that has been used by various research fields in order to represent dynamic correlation among stakeholders. Traditionally, research within the process and energy systems engineering community has focused on the development of centralised decision making schemes. In the recent years, decentralised decision-making schemes have attracted increasing attention due to their ability to capture multi-stakeholder dynamics in a more accurate manner. In this article, we survey how centralised and decentralised decision making has been facilitated by game theoretic approaches. We focus on the deployment of such methods in process systems engineering problems and review applications related to supply chain optimisation problems, design and operations, and energy systems optimisation. Finally, we analyse different game structures based on the degree of cooperation and how fairness criteria can be employed to find fair payoff allocations

On the set of imputations induced by the k-additive core

An extension to the classical notion of core is the notion of $k$-additive core, that is, the set of $k$-additive games which dominate a given game, where a $k$-additive game has its M\"obius transform (or Harsanyi dividends) vanishing for subsets of more than $k$ elements. Therefore, the 1-additive core coincides with the classical core. The advantages of the $k$-additive core is that it is never empty once $k\geq 2$, and that it preserves the idea of coalitional rationality. However, it produces $k$-imputations, that is, imputations on individuals and coalitions of at most $k$ inidividuals, instead of a classical imputation. Therefore one needs to derive a classical imputation from a $k$-order imputation by a so-called sharing rule. The paper investigates what set of imputations the $k$-additive core can produce from a given sharing rule