On stratified sampling for estimating coalitional values

This paper addresses two sampling methodologies to respectively estimate the Owen value and the Banzhaf–Owen value for TU-games with a priori unions. Both proposals are based on stratified sampling on the set of those coalitions that are compatible with the system of unions according to their cardinalities. These sampling methodologies are analysed in terms of the theoretical properties and of the establishment of bounds for the absolute error from a statistical point of view. Finally, we evaluate the performance of these tools on several real well-known examples in the literatureThe author acknowledges the financial support of Ministerio de Economía y Competitividad of the Spanish government under grants MTM2017-87197-C3-2-P and PID2021-124030NB-C32, and of Xunta de Galicia through the ERDF (Grupos de Referencia Competitiva) ED431C 2021/24. Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.S

On the centrality analysis of covert networks using games with externalities

The identification of the most potentially hazardous agents in a terrorist organisation helps to prevent further attacks by effectively allocating surveillance resources and destabilising the covert network to which they belong. In this paper, several mechanisms for the overall ranking of covert networks members in a general framework are addressed based on their contribution to the overall relative effectiveness in the event of a merger. In addition, the possible organisation of agents outside of each possible merger naturally influences their relative effectiveness score, which motivates the innovative use of games in partition function form and specific ranking indices for individuals. Finally, we apply these methods to analyse the effectiveness of the hijackers of the covert network supporting the 9/11 attacksThis work is part of the R+D+I project grants MTM2017-87197-C3-3-P and PID2021-124030NB-C32, funded byMCIN/AEI/10.13039/501100011033/ and by “ERDF A way of making Europe”/EU. This research was also funded by Grupos de Referencia Competitiva ED431C-2021/24 from the Consellería de Cultura, Educación e Universidades, Xunta de Galicia.S

Analysis of the impact of DMUs on the overall efficiency in the event of a merger

This paper addresses several mechanisms for overall ranking Decision Making Units (DMUs) according to the contribution of DMUs to the relative efficiency score of a merger considering aggregate units The possible organization of agents outside each possible merger naturally influences the relative efficiency score, which motivates the use of games in partition function form and specific ranking indices for DMUs based on the Shapley value. Several computational problems arise in their exact computation when the number of DMUs increases. We describe two sampling alternatives to reduce these drawbacks. Finally, we apply these methods to analyse the efficiency of the hotel industry in SpainThis work has been supported by FEDER/Ministerio de Ciencia, Innovación y Universidades - Agencia Estatal de Investigación, Spain under grants MTM2017-87197-C3-2-P and MTM2017-87197-C3-3-P, and by the Xunta de Galicia through the European Regional Development Fund (Grupos de Referencia Competitiva ED431C-2017/38) and by the Consellería de Cultura, Educación e Universidades, Xunta de Galicia, Spain (Grupos de Referencia Competitiva ED431C-2020/03).S

Risk analysis sampling methods in terrorist networks based on the Banzhaf value

This article introduces the Banzhaf and the Banzhaf–Owen values as novel measures of risk analysis of a terrorist attack, determining the most dangerous terrorists in a network. This new approach counts with the advantage of integrating at the same time the complete topology (i.e., nodes and edges) of the network and a coalitional structure on the nodes of the network. More precisely, the characteristics of the nodes (e.g., terrorists) of the network and their possible relationships (e.g., types of communication links), as well as coalitional information (e.g., level of hierarchies) independent of the network. First, for these two new measures of risk analysis, we provide and implement approximation algorithms. Second, as illustration, we rank the members of the Zerkani network, responsible for the attacks in Paris (2015) and Brussels (2016). Finally, we give a comparison between the rankings established by the Banzhaf and the Banzhaf–Owen values as measures of risk analysisMinisterio de Ciencia e Innovación, Grant/Award Numbers: PGC2018-097965-B-I00, PID2021-124030NB-C32; Xunta de Galicia, Grant/Award Number: ED431C 2021/24; Ministerio de Ciencia, Innovación y Universidades, Grant/Award Number: MTM2017-87197-C3-3-PS

Sequencing Situations and Games with Non-Linear Cost Functions

This paper studies sequencing situations with non-linear cost functions. We show that the neighbor switching gains are now time-dependent, in contrast to the standard sequencing situations with linear cost functions, which complicate finding an optimal order and stable allocations. We derive conditions on the time-dependent neighbor switching gains in a (general) sequencing situation to guarantee convexity of the associated sequencing game. Moreover, we provide two procedures that uniquely specify a path from the initial order to an optimal order and we define two corresponding allocation rules that divide the neighbor switching gains equally in every step of the path. We show that the same conditions on the gains also guarantee stability for the allocations prescribed by these rules

Contrubicións en teoría de xogos cooperativos e aplicacións

The analysis of situations of multi-agent cooperation has received much attention in recent years. The study of these is based on the determination of the optimal policy for the involved agents in each scenario, which, from an economic point of view, has to reduce the joint costs. Game theory is the discipline distributing the associated benefits/costs among the cooperating agents. According to this, there are multiple situations to be analyzed, among others, centralized inventory problems or sequencing problems. First, new multi-agent inventory situations in which costs for product storage are not considered are studied. Nagarajan and Sosic (2008), Dror and Hartman (2011) or Fiestras-Janeiro et al. (2012) are some references in inventory problems. Authors study situations where a group of agents, who face individual inventory problems, cooperate to joint new orders of product. The determination of their sizes and their frequency bases the study. Once the optimal policy is obtained, the associated cost games are analyzed. From a theoretical point of view, the compliance of certain properties of interest will be studied, as well as the definition of distribution rules for the costs generated. The study of cooperative multi-agent models is completed with the analysis of a new sequencing situation in the processing of jobs by a machine. In a generic way, the existence of several independent jobs to be performed sequentially in a single machine is assumed. The main goal is to determine the optimal order of the processing in order to minimize the overall costs. Curiel et al. (1989) and Borm et al. (2002), among many others, analyze this class of problems. In particular, this work analyzes situations such as those described in which the cost of processing a job is an exponential or logarithmic function of its completion time. Fixed an initial order for the jobs, the resulting profit from the reordering of the job in their processing are analyzed. So, a savings game arises that will be studied from a game theoretical approach. Cooperative Game Theory is based on the formation of coalitions of agents which cooperate. The objective consists in the definition of values for sharing the costs / benefits by cooperation of the agents involved. Among others, the value of Shapley or the value of Banzhaf are two of the well-known mechanisms in the literature. They were extended to situations with a priori union systems for the agents and so, it appears the Owen value and the Banzhaf-Owen value, respectively. Finally, statistical procedures are proposed for the estimation of values. In those cases in which the number of agents involved is large, since that the direct calculation is computationally difficult, several authors propose methods for estimating them by sampling. Maleki (2015) and Castro et al. (2009) define mechanisms based on simple random sampling in order to obtain an estimation of the Shapley value of a TU game. Bachrach et al. (2010) proposes analogous procedures for the estimation of Banzhaf value. They extend their proposals to the games with a priori unions, focusing on the estimation of the Owen value and the Banzhaf-Owen value. In these cases, aspects such as the estimation error is studied and its application is illustrated on different known examples.El análisis de situaciones de cooperación multi-agente ha experimentado un gran incremento en los últimos años. El estudio de las mismas se basa en la determinación de la política óptima a seguir por los agentes involucrados en cada escenario lo que, desde un punto de vista económico, debe reducir los costes conjuntos. La teoría de juegos es la disciplina que permite distribuir los beneficios/costes asociados entre los agentes que cooperan. De acuerdo con esto, son múltiples las situaciones que pueden ser analizadas destacando, entre otros, los problemas de inventario centralizados o los problemas de secuenciación. En primer lugar, se analizan nuevas situaciones de inventario multi-agente para las que no se consideran costes por almacenamiento de producto. Nagarajan y Sosic (2008), Dror y Hartman (2011) o Fiestras-Janeiro et al. (2012) son algunas referencias en problemas de inventario. Estos autores estudian situaciones en las que un conjunto de agentes, que se enfrentan a problemas de inventario individuales, cooperan para realizar pedidos de producto de manera conjunta. La determinación de sus tamaños y la frecuencia de los mismos fundamentan el estudio. Determinada la política óptima a seguir, los juegos de costes asociados a los problemas planteados son analizados. Desde un punto de vista teórico, se estudiará el cumplimiento de ciertas propiedades de interés, así como la definición de reglas de reparto para los costes generados. El estudio de modelos cooperativos multi-agente se completa con el análisis de una nueva situación de secuenciación en el procesado de tareas por una máquina. De forma genérica, se asume la existencia de varias tareas independientes a realizar de manera secuencial en una única máquina. El objetivo fundamental pasa por determinar el orden óptimo de su procesado con el fin de minimizar los costes globales. Curiel et al. (1989) y Borm et al. (2002), entre muchos otros, analizan problemas de esta clase. En particular, en este trabajo se analizan situaciones como las descritas en las que el coste de procesado de una tarea es una función exponencial o logarítmica de su tiempo de finalización. Fijado un orden inicial para las tareas, se analizan los beneficios resultantes de la reordenación de las tareas para su procesado. Con esto, surge un juego de ahorro asociado que será estudiado desde el punto de vista de la teoría de juegos. La teoría de los juegos cooperativos se basa en la formación de coaliciones de agentes que actúan de manera coordinada. Uno de sus objetivos pasa por la definición de valores para el reparto de los costes/beneficios resultantes de dicha cooperación entre los agentes involucrados. Entre ellos, el valor de Shapley o el valor de Banzhaf figuran como dos de los mecanismos más conocidos en la literatura para este fin. Su extensión a situaciones con sistemas de uniones a priori para los agentes ha permitido la definición del valor de Owen y del valor de Banzhaf-Owen, respectivamente. Por último, se proponen procedimientos estadísticos para la estimación de valores. En aquellos casos en los que el número de agentes involucrados es elevado, dado que su cálculo directo resulta computacionalmente más difícil, varios autores proponen métodos para su estimación por muestreo. Maleki (2015) y Castro et al. (2009) definen mecanismos basados en el muestreo aleatorio simple para obtener una estimación del valor de Shapley de un juego TU. Bachrach et al. (2010) propone procedimientos análogos para la estimación del valor de Banzhaf. Se extienden sus propuestas al caso de valores con uniones a priori, centrándonos en el valor de Owen y el valor de Banzhaf-Owen. En estos casos, se estudian cuestiones tales como el error cometido en la estimación, ilustrando su aplicación sobre diferentes ejemplos conocidos.A análise de situacións de cooperación multi-axente experimentou un gran incremento nos últimos anos. O estudo das mesmas baséase na determinación da política óptima a seguir polos axentes involucrados en cada escenario o que, dende un punto de vista económico, debe reducir os costes conxuntos. A teoría de xogos é a disciplina que permite distribuir los beneficios/costes asociados entre os axentes que cooperan. Dacordo con esto, son múltiples as situacións que poden ser analizadas destacando, entre outros, os problemas de inventario centralizados ou os problemas de secuenciación. En primeiro lugar, analízanse novas situacións de inventario multi-axente para as que non se consideran costes por almacenamento de produto. Nagarajan e Sosic (2008), Dror e Hartman (2011) ou Fiestras-Janeiro et al. (2012) son algunhas referencias en problemas de inventario. Estos autores estudan situacións nas que un conxunto de axentes, que se enfrentan a problemas de inventario individuais, cooperan para realizar pedidos de produto de maneira conxunta. A determinación dos seus tamaños e a frecuencia dos mesmos fundamentan o estudo. Determinada a política óptima a seguir, os xogos de costes asociados ós problemas plantexados son analizados. Dende un punto de vista teórico, estudiarase o cumplimento de certas propiedades de interese, así como a definición de regras de reparto para os costes xerados. O estudo de modelos cooperativos multi-axente complétase coa análise dunha nova situación de secuenciación no procesado de tarefas por unha máquina. De forma xenérica, asúmese a existencia de varias tarefas independentes a realizar de maneira secuencial nunha única máquina. O obxectivo fundamental pasa por determinar a orde óptima do seu procesado co fin de minimizar os costes globales. Curiel et al. (1989) e Borm et al. (2002), entre moitos outros, analizan problemas desta clase. En particular, neste traballo analízanse situacións como as descritas nas que o coste de procesado dunha tarefa é unha función exponencial ou logarítmica do seu tempo de finalización. Fixado unha orde inicial para as tarefas, analízanse os beneficios resultantes da reordenación das tarefas para o seu procesado. Con esto, surxe un xogo de aforro asociado que será estudado dende o punto de vista da teoría de xogos. A teoría dos xogos cooperativos baséase na formación de coalicións de axentes que actúan de maneira coordinada. Un dos obxectivos pasa pola definición de valores para o reparto dos costes/beneficios resultantes de dita cooperación entre os axentes involucrados. Entre eles, o valor de Shapley ou o valor de Banzhaf figuran como dous dos mecanismos máis coñecidos na literatura para este fin. A súa extensión a situacións con sistemas de unións a priori para os axentes permitiu a definición do valor de Owen e do valor de Banzhaf-Owen, respectivamente. Por último, propóñense procedementos estatísticos para a estimación de valores. Naqueles casos nos que o número de axentes involucrados é elevado, dado que o seu cálculo directo resulta computacionalmente máis difícil, varios autores propoñen métodos para a súa estimación por mostraxe. Maleki (2015) e Castro et al. (2009) definen mecanismos baseados na mostraxe aleatoria simple para obter unha estimación do valor de Shapley dun xogo TU. Bachrach et al. (2010) propoñen procedementos análogos para a estimación do valor de Banzhaf. Exténdense as súas propostas ó caso de valores con unións a priori, centrándonos no valor de Owen e o valor de Banzhaf-Owen. Nestos casos, estúdanse cuestións tales como o erro cometido na estimación, ilustrando a súa aplicación sobre diferentes exemplos coñecidos.Ministerio de Economía y Competitividad de España | Ref. MTM2014-53395-C3-3-PMinisterio de Economía y Competitividad de España | Ref. MTM2017-87197-C3-2-PXunta de Galicia | Ref. ED431C 2016-04

On interactive sequencing situations with exponential cost functions

This paper addresses interactive one-machine sequencing situations in which the costs of processing a job are given by an exponential function of its completion time. The main difference with the standard linear case is that the gain of switching two neighbors in a queue is time-dependent and depends on their exact position. We illustrate that finding an optimal order is complicated in general and we identify specific subclasses, which are tractable from an optimization perspective. More specifically, we show that in these subclasses, all neighbor switches in any path from the initial order to an optimal order lead to a non-negative gain. Moreover, we derive conditions on the time-dependent neighbor switching gains in a general interactive sequencing situation to guarantee convexity of the corresponding cooperative game. These conditions are satisfied within our specific subclasses of exponential interactive sequencing situations

On Interactive Sequencing Situations with Exponential Cost Functions

This paper addresses interactive one-machine sequencing situations in which the costs of processing a job are given by an exponential function of its completion time. The main difference with the standard linear case is that the gain of switching two neighbors in a queue is time-dependent and depends on their exact position. We illustrate that finding an optimal order is complicated in general and we identify specific subclasses, which are tractable from an optimization perspective. More specifically, we show that in these subclasses, all neighbor switches in any path from the initial order to an optimal order lead to a non-negative gain. Moreover, we derive conditions on the time-dependent neighbor switching gains in a general interactive sequencing situation to guarantee convexity of the corresponding cooperative game. These conditions are satisfied within our specific subclasses of exponential interactive sequencing situations

On interactive sequencing situations with exponential cost functions

This paper addresses interactive one-machine sequencing situations in which the costs of processing a job are given by an exponential function of its completion time. The main difference with the standard linear case is that the gain of switching two neighbors in a queue is time-dependent and depends on their exact position. We illustrate that finding an optimal order is complicated in general and we identify specific subclasses, which are tractable from an optimization perspective. More specifically, we show that in these subclasses, all neighbor switches in any path from the initial order to an optimal order lead to a non-negative gain. Moreover, we derive conditions on the time-dependent neighbor switching gains in a general interactive sequencing situation to guarantee convexity of the corresponding cooperative game. These conditions are satisfied within our specific subclasses of exponential interactive sequencing situations