Compromise values in cooperative game theory

Bargaining;game theory

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

Fair risk allocation in illiquid markets

Let us consider a financially constrained leveraged financial firm having some divisions which have invested into some risky assets. Using c oherent measures of risk the sum of the capital requirements of the divisions is larger tha n the capital requirement of the firm itself, there is some diversification benefit that should b e allocated somehow for proper performance evaluation of the divisions. In this pa per we use cooperative game theory and simulation to assess the possibility to jointly sat isfy three natural fairness requirements for allocating risk capital in illiquid markets: Core C ompatibility, Equal Treatment Property and Strong Monotonicity. Core Compatibility can be viewed as the allocated r isk to each coalition (subset) of divisions should be at least as much as the risk increment th e coalition causes by joining the rest of the divisions. Equal Treatment Property guarantees that if two divisions have the same stand� alone risk and also they contribute the same risk t o all the subsets of divisions not containing them, then the same risk capital should be allocate d to them. Strong Monotonicity requires that if a division weakly reduces its stand�alone r isk and also its risk contribution to all the subsets of the other divisions, then as an incentiv e its allocated risk capital should not increase. Analyzing the simulation results we concl ude that in most of the cases it is not possible to allocate risk in illiquid markets satis fying the three fairness notions at the same time, one has to give up at least one of them

Path monotonicity, consistency and axiomatizations of some weighted solutions

On the domain of cooperative games with transferable utility, we introduce path monotonicity, a property closely related to fairness (van den Brink, in Int J Game Theory 30:309-319, 2001). The principle of fairness states that if a game changes by adding another game in which two players are symmetric, then their payoffs change by the same amount. Under efficiency, path monotonicity is a relaxation of fairness that guarantees that when the worth of the grand coalition varies, the players' payoffs change according to some monotone path. In this paper, together with the standard properties of projection consistency (Funaki, in Dual axiomatizations of solutions of cooperative games. Mimeo, New York, 1998) and covariance, we show that path monotonicity characterizes the weighted surplus division solutions. Interestingly, replacing projection consistency by either self consistency (Hart and Mas-Colell, in Econometrica 57:589-614, 1989) or max consistency (Davis and Maschler, in Nav Res Logist Q 12:223-259, 1965) we obtain new axiomatic characterizations of the weighted Shapley values and the prenucleolus, respectively. Finally, by the duality approach we provide a new axiomatization of the weighted egalitarian non-separable contribution solutions using complement consistency (Moulin, in J Econ Theory 36:120-148, 1985

Externalities and the nucleolus

In most economic applications, externalities prevail: the worth of a coalition depends on how the other players are organized. We show that there is a unique natural way of extending the nucleolus from (coalitional) games without externalities to games with externalities. This is in contrast to the Shapley value and the core for which many different extensions have been proposed

Externalities and the nucleolus

In most economic applications, externalities prevail: the worth of a coalition depends on how the other players are organized. We show that there is a unique natural way of extending the nucleolus from (coalitional) games without externalities to games with externalities. This is in contrast to the Shapley value and the core for which many different extensions have been proposed

The semireactive bargaining set of a cooperative game

bargaining;cooperative games