Pyramidal values

We propose and analyze a new type of values for cooperative TU-games, which we call pyramidal values. Assuming that the grand coalition is sequentially formed, and all orderings are equally likely, we define a pyramidal value to be any expected payoff in which the entrant player receives a salary, and the rest of his marginal contribution to the just formed coalition is distributed among the incumbent players. We relate the pyramidal-type sharing scheme we propose with other sharing schemes, and we also obtain some known values by means of this kind of pyramidal procedures. In particular, we show that the Shapley value can be obtained by means of an interesting pyramidal procedure that distributes nonzero dividends among the incumbents. As a result, we obtain an alternative formulation of the Shapley value based on a measure of complementarity between two players. Finally, we introduce the family of proportional pyramidal values, in which an incumbent receives a dividend in proportion to his initial investment, measured by means of his marginal contribution

Pyramidal values

We propose a new type of values for cooperative TU-games, which we call pyramidal values. Assuming that the grand coalition is sequentially formed, and all orderings are equally likely, we define a pyramidal value to be any expected payoff in which the entrant player receives a salary and the right to get part of the benefits derived from subsequent incorporations to the just formed coalition, whereas the remaining benefit is distributed among the incumbent players. To be specific, we consider some parametric families of pyramidal values: the egalitarian pyramidal family, which coincides with the a-consensus value family introduced by Ju et al. in (2007), the proportional pyramidal family, and the weighted pyramidal family, which in turn includes the other two families as special cases. We also analyze the properties of these families, as well as their relationships with other previously defined values.This research has been supported by I+D+i research project MTM2011-27892 from the Government of Spain

The Shapley group value

Following the original interpretation of the Shapley value (Shapley, 1953a) as a priori evaluation of the prospects of a player in a multi-person iteraction situation, we propose a group value, which we call the Shapley group value, as a priori evaluation of the prospects of a group of players in a coalitional game when acting as a unit. We study its properties and we give an axiomatic characterization. We motivate our proposal by means of some relevant applications of the Shapley group value, when it is used as an objective function by a decision maker who is trying to identify an optimal group of agents in a framework in which agents interact and the attained benefit can be modeled by means of a transferable utility game. As an illustrative example we analyze the problem of identifying the set of key agents in a terrorist network.This research has been supported by I+D+i research project MTM2011-27892 from the Government of Spai

A game theoretic approach to group centrality

This paper is centered in the valuation of the centrality of groups following aproblem-specific approach (Friedkin, 1991). Assuming a TU-game that reflects theinterests which motivate the interactions among individuals in a network, we extend thegame theoretic centrality measure of Gomez et al. (2003) to the case of groups, anddefine the game theoretic group centrality of a group as the variation of its value orpower due to their social relations. We rely on the Shapley group value (Flores et al.,2014) for measuring the value of a group in a game without any restriction, and weintroduce the Myerson group value in order to measure the value when the socialstructure is considered.This research has been supported by I+D+i research project MTM2011-27892 from the Government of Spai

Networks and collective action

Given a social network, we are interested in the problem of measuring the influence of a group of agents to lead the society to adopt their behavior. Motivated by the description of terrorist movements, we provide a markovian dynamical model for non-symmetric societies, which takes into account two special features: the hard core terrorist group cannot be influenced, and the remaining agents may change from active to non-active and vice versa during the process. In this setting, we interpret the absorption time of the model, which measures how quickly the terrorist movement achieve the support of all society, as a group measure of power. In some sense, our model generalizes the classical approach of DeGroot to consensus formatio

The position value as a centrality measure in social networks

The position value, introduced by Meessen (1988), is a solution concept for cooperative games in which the value assigned to a player depends on the value of the connections or links he has with other players. This concept has been studied by Borm et al. (1992) and characterised by Slikker (2005). In this paper, we analyse the position value from the point of view of the typical properties of a measure of centrality in a social network. We extend the analysis already developed in Gomez et al. (2003) for the Myerson centrality measure, where the symmetric effect on the centralities of the end nodes of an added or removed edge is a fundamental part of its characterisation. However, the Position centrality measure, unlike the Myerson centrality measure, responds in a more versatile way to such addition or elimination. After studying the aforementioned properties, we will focus on the analysis and characterisation of the Position attachment centrality given by the position value when the underlying game is the attachment game. Some comparisons are made with the attachment centrality introduced by Skibski et al. (2019).Depto. de Estadística e Investigación OperativaFac. de Ciencias MatemáticasInstituto de Matemática Interdisciplinar (IMI)FALSEMinisterio de Ciencia e Innovaciónunpu

Spatio-temporal variability of stable isotopes (18 O and 2H) in soil and xylem waters under Mediterranean conditions.

Soil profiles and trees twigs were sampled in the Can Vila Mediterranean catchment (0.56 km2; Vallcebre Research catchments, NE Spain) to evaluate the spatial variability of the isotopic signature (18O and 2H) of xylem and bulk soil waters at the plot scale and between different locations within the catchment. During two one day sampling campaigns with different antecedent soil moisture conditions, soil samples (0-10, 10-20, 20-30, 40-50 and 90-100 cm) and xylem samples (3 trees per plot) were collected in six Scots pine stands distributed throughout the catchment. Moreover, the water stable isotopes analysed were collected in rainfall, groundwater and streamwater at the catchment outlet during and between the sampling campaigns. Water from soil and xylem samples was extracted by cryogenic vacuum distillation and isotope analyses were obtained by infrared spectroscopy. Stable isotopes ratios of bulk soil water and xylem water fell below the local meteoric water line (LMWL) in both sampling campaigns. In contrast, groundwater ratios fell along the LMWL, being well mixed with stream water. A marked vertical variation in soil water isotopes was observed for the dry campaign in all profiles, with enriched shallow horizons indicating evaporation. This variation was not observed for the wet campaign. Moreover, the spatial variation across the catchment was much greater for the dry campaign compared to the wet campaign. A marked variability in the xylem isotopic signature among trees of the same plot was observed for both sampling campaigns. Finally, in some plots and for both campaigns, the isotopic signature of xylem water was more evaporated than that of bulk soil water. There was no clear pattern relating the topographic index, as an indicator of saturation conditions of the sampling location within the catchment, with soil water isotopic signature. Nor was there a clear relationship found between the isotopic signature of pines¿ xylem and tree characteristics, such as DBH, height, or tree competition index

Some game theoretic marketing attribution models

In this paper, we propose and analyse two game theoretic approaches to design attribution mechanisms for multi-channel marketing campaigns. Both approaches are based on a key performance index function that provides the benefit obtained in each of the observed paths to conversion. The first approach considers the problem as a cooperative transferable utility game, and the proposed attribution mechanisms are based on the Shapley value. The second approach models the problem as a bankruptcy problem and the proposed attribution mechanism is based on the constrained equal-losses rule. We also extend the above approaches to deal with the cases in which the position or the repetition of the channels on the paths to conversion are taken into account