Nodes of directed graphs ranked by solutions defined on cooperative games

Hierarchical structures, transportation systems, communication networks and even sports competitions can be modeled by means of directed graphs. Since digraphs without a predefined game are considered, the main part of the work is devoted to establish conditions on cooperative games so that they can be used to measure accessibility to the nodes. Games that satisfy desirable properties are called test games. Each ranking on the nodes is then obtained according to a pair formed by a test game and a solution defined on cooperative games whose utilities are given for every ordered coalition. Solutions here proposed are extensions of the wide family of semivalues to games in generalized characteristic function form.Postprint (published version

Data Banzhaf: A Robust Data Valuation Framework for Machine Learning

Data valuation has wide use cases in machine learning, including improving data quality and creating economic incentives for data sharing. This paper studies the robustness of data valuation to noisy model performance scores. Particularly, we find that the inherent randomness of the widely used stochastic gradient descent can cause existing data value notions (e.g., the Shapley value and the Leave-one-out error) to produce inconsistent data value rankings across different runs. To address this challenge, we introduce the concept of safety margin, which measures the robustness of a data value notion. We show that the Banzhaf value, a famous value notion that originated from cooperative game theory literature, achieves the largest safety margin among all semivalues (a class of value notions that satisfy crucial properties entailed by ML applications and include the famous Shapley value and Leave-one-out error). We propose an algorithm to efficiently estimate the Banzhaf value based on the Maximum Sample Reuse (MSR) principle. Our evaluation demonstrates that the Banzhaf value outperforms the existing semivalue-based data value notions on several ML tasks such as learning with weighted samples and noisy label detection. Overall, our study suggests that when the underlying ML algorithm is stochastic, the Banzhaf value is a promising alternative to the other semivalue-based data value schemes given its computational advantage and ability to robustly differentiate data quality.Comment: AISTATS 2023 Ora

Coalitional control in the framework of cooperative game theory

[EN] Coalitional control is a fairly new branch of distributed control where the agents merge dynamically into coalitions according to the enabled/disabled communication links at each time instant. Therefore, with these schemes there is a reduction of the communication burden without compromising the system performance. In this tutorial, the main features of these schemes will be introduced in the framework of cooperative game theory, being the game related to the cost function that is optimized by the control approach, and with the players corresponding to either the communication links or the agents involved. In this context, several cooperative game theory tools will be considered in order to: rank the players, impose constraints on them, provide more effcient ways of calculation, perform system partitioning, etc., hence analyzing the main features related to each tool.[ES] El control coalicional es una rama incipiente del control distribuido donde los distintos agentes se agrupan de forma dinámica en coaliciones en función de los enlaces de comunicación activos/inactivos en cada instante de tiempo. Gracias a ello, se reduce la carga de comunicación sin comprometer las prestaciones del sistema. En este tutorial, se analizan las principales características de estos esquemas dentro del marco de la teoría de juegos cooperativos, estando el juego definido por la función de coste a optimizar en el esquema de control, y correspondiendo los jugadores bien a los enlaces de comunicación o bien a los propios agentes. En este contexto, se estudiarán diversas herramientas de teoría de juegos cooperativos, con objeto de clasificar jugadores, imponer restricciones en los mismos, proponer vías de cálculo más eficientes, realizar particionado de sistemas, etc., examinando las características más relevantes presentadas por cada herramienta.Este estudio ha sido parcialmente financiado por los proyectos de investigación OCONTSOLAR, (H2020 ADG-ERC, ID 789051), C3PO (MINECO, DPI2017-86918-R), y GESVIP (Junta de Andalucía, US-1265917). Asimismo, se agradece a Jose María Maestre, Encarnación Algaba y Eduardo F. Camacho las innumerables discusiones mantenidas a lo largo de los anos de doctorado que me ayudaron a dominar los conceptos presentados en este tutorial. Es también de destacar los comentarios del Editor y los revisores anónimos que han contribuido a la mejora sustancial del manuscrito. Finalmente, se dedica este artículo a Lloyd S. Shapley (1923-2016), ya que su concepto de solución (Shapley, 1953b) ha inspirado todo mi trabajo.Muros, FJ. (2021). El control coalicional en el marco de la teoría de juegos cooperativos. Revista Iberoamericana de Automática e Informática industrial. 18(2):97-112. https://doi.org/10.4995/riai.2020.13456OJS97112182Alamo, T., Normey-Rico, J. E., Arahal, M. R., Limon, D., Camacho, E. F., June 2006. Introducing linear matrix inequalities in a control course. In: Proceedings of the 7th IFAC Symposium on Advances in Control Education (ACE 2006). Madrid, Spain, pp. 205-210. https://doi.org/10.3182/20060621-3-ES-2905.00037Algaba, E., Fragnelli, V., Sánchez-Soriano, J. (Eds.), December 2019. The Handbook of the Shapley Value. CRC Press Series in Operations Research. Chapman & Hall/CRC, Boca Ratón, Florida, USA. https://doi.org/10.1201/9781351241410Aranda-Escolástico, E., Guinaldo, M., Heradio, R., Chacon, J., Vargas, H., Sánchez, J., Dormido, S., March 2020. 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An axiomatization of the Banzhaf value. International Journal of Game Theory 17 (2), 89-99. https://doi.org/10.1007/BF01254541Loehman, E. T., Whinston, A. B., 1976. A generalized cost allocation scheme. In: Stevens, A., Lin, Y. (Eds.), Theory and Measurement of Economic Externalities. Academic Press, New York, USA, pp. 87-101. https://doi.org/10.1016/B978-0-12-450450-9.50013-0Lopez-Rodriguez, F., Maestre, J. M., Muros, F. J., Camacho, E. F., July 2020. A modular feedback approach for distributed control. In: Proceedings of the 21st IFAC World Congress (IFAC 2020). Berlin, Germany, pp. 4086-4091.Lucchetti, R., Moretti, S., Patrone, F., Radrizzani, P., August 2010. The Shapley and Banzhaf values in microarray games. Computers & Operations Research 37 (8), 1406-1412. https://doi.org/10.1016/j.cor.2009.02.020Maestre, J. M., November 2010. Distributed model predictive control based on game theory. Ph.D. thesis, Department of Systems and Automation Engineering, University of Seville, Seville, Spain.Maestre, J. M., Ishii, H., October 2017. A PageRank based coalitional control scheme. International Journal of Control, Automation and Systems 15 (5), 1983-1990. https://doi.org/10.1007/s12555-016-0336-8Maestre, J. M., Lopez-Rodriguez, F., Muros, F. J., Ocampo-Martinez, C., February 2021. Modular feedback control of networked systems by clustering: A drinking water network case study. Processes 9 (2), 389. https://doi.org/10.3390/pr9020389Maestre, J. M., Muñoz de la Peña, D., Camacho, E. F., Alamo, T., 2011. Distributed model predictive control based on agent negotiation. Journal of Process Control 21 (5), 685-697. https://doi.org/10.1016/j.jprocont.2010.12.006Maestre, J. M., Muñoz de la Peña, D., Jiménez Losada, A., Algaba, E., Camacho, E. F., September/October 2014. A coalitional control scheme with applications to cooperative game theory. 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A POWER INDEX BASED FRAMEWORKFOR FEATURE SELECTION PROBLEMS

One of the most challenging tasks in the Machine Learning context is the feature selection. It consists in selecting the best set of features to use in the training and prediction processes. There are several benefits from pruning the set of actually operational features: the consequent reduction of the computation time, often a better quality of the prediction, the possibility to use less data to create a good predictor. In its most common form, the problem is called single-view feature selection problem, to distinguish it from the feature selection task in Multi-view learning. In the latter, each view corresponds to a set of features and one would like to enact feature selection on each view, subject to some global constraints. A related problem in the context of Multi-View Learning, is Feature Partitioning: it consists in splitting the set of features of a single large view into two or more views so that it becomes possible to create a good predictor based on each view. In this case, the best features must be distributed between the views, each view should contain synergistic features, while features that interfere disruptively must be placed in different views. In the semi-supervised multi-view task known as Co-training, one requires also that each predictor trained on an individual view is able to teach something to the other views: in classification tasks for instance, one view should learn to classify unlabelled examples based on the guess provided by the other views. There are several ways to address these problems. A set of techniques is inspired by Coalitional Game Theory. Such theory defines several useful concepts, among which two are of high practical importance: the concept of power index and the concept of interaction index. When used in the context of feature selection, they take the following meaning: the power index is a (context-dependent) synthesis measure of the prediction\u2019s capability of a feature, the interaction index is a (context-dependent) synthesis measure of the interaction (constructive/disruptive interference) between two features: it can be used to quantify how the collaboration between two features enhances their prediction capabilities. An important point is that the powerindex of a feature is different from the predicting power of the feature in isolation: it takes into account, by a suitable averaging, the context, i.e. the fact that the feature is acting, together with other features, to train a model. Similarly, the interaction index between two features takes into account the context, by suitably averaging the interaction with all the other features. In this work we address both the single-view and the multi-view problems as follows. The single-view feature selection problem, is formalized as the problem of maximization of a pseudo-boolean function, i.e. a real valued set function (that maps sets of features into a performance metric). Since one has to enact a search over (a considerable portion of) the Boolean lattice (without any special guarantees, except, perhaps, positivity) the problem is in general NP-hard. We address the problem producing candidate maximum coalitions through the selection of the subset of features characterized by the highest power indices and using the coalition to approximate the actual maximum. Although the exact computation of the power indices is an exponential task, the estimates of the power indices for the purposes of the present problem can be achieved in polynomial time. The multi-view feature selection problem is formalized as the generalization of the above set-up to the case of multi-variable pseudo-boolean functions. The multi-view splitting problem is formalized instead as the problem of maximization of a real function defined over the partition lattice. Also this problem is typically NP-hard. However, candidate solutions can be found by suitably partitioning the top power-index features and keeping in different views the pairs of features that are less interactive or negatively interactive. The sum of the power indices of the participating features can be used to approximate the prediction capability of the view (i.e. they can be used as a proxy for the predicting power). The sum of the feature pair interactivity across views can be used as proxy for the orthogonality of the views. Also the capability of a view to pass information (to teach) to other views, within a co-training procedure can benefit from the use of power indices based on a suitable definition of information transfer (a set of features { a coalition { classifies examples that are subsequently used in the training of a second set of features). As to the feature selection task, not only we demonstrate the use of state of the art power index concepts (e.g. Shapley Value and Banzhaf along the 2lines described above Value), but we define new power indices, within the more general class of probabilistic power indices, that contains the Shapley and the Banzhaf Values as special cases. Since the number of features to select is often a predefined parameter of the problem, we also introduce some novel power indices, namely k-Power Index (and its specializations k-Shapley Value, k-Banzhaf Value): they help selecting the features in a more efficient way. For the feature partitioning, we use the more general class of probabilistic interaction indices that contains the Shapley and Banzhaf Interaction Indices as members. We also address the problem of evaluating the teaching ability of a view, introducing a suitable teaching capability index. The last contribution of the present work consists in comparing the Game Theory approach to the classical Greedy Forward Selection approach for feature selection. In the latter the candidate is obtained by aggregating one feature at time to the current maximal coalition, by choosing always the feature with the maximal marginal contribution. In this case we show that in typical cases the two methods are complementary, and that when used in conjunction they reduce one another error in the estimate of the maximum value. Moreover, the approach based on game theory has two advantages: it samples the space of all possible features\u2019 subsets, while the greedy algorithm scans a selected subspace excluding totally the rest of it, and it is able, for each feature, to assign a score that describes a context-aware measure of importance in the prediction process

Algorithmic and complexity aspects of simple coalitional games

Simple coalitional games are a fundamental class of cooperative games and voting games which are used to model coalition formation, resource allocation and decision making in computer science, artificial intelligence and multiagent systems. Although simple coalitional games are well studied in the domain of game theory and social choice, their algorithmic and computational complexity aspects have received less attention till recently. The computational aspects of simple coalitional games are of increased importance as these games are used by computer scientists to model distributed settings. This thesis fits in the wider setting of the interplay between economics and computer science which has led to the development of algorithmic game theory and computational social choice. A unified view of the computational aspects of simple coalitional games is presented here for the first time. Certain complexity results also apply to other coalitional games such as skill games and matching games. The following issues are given special consideration: influence of players, limit and complexity of manipulations in the coalitional games and complexity of resource allocation on networks. The complexity of comparison of influence between players in simple games is characterized. The simple games considered are represented by winning coalitions, minimal winning coalitions, weighted voting games or multiple weighted voting games. A comprehensive classification of weighted voting games which can be solved in polynomial time is presented. An efficient algorithm which uses generating functions and interpolation to compute an integer weight vector for target power indices is proposed. Voting theory, especially the Penrose Square Root Law, is used to investigate the fairness of a real life voting model. Computational complexity of manipulation in social choice protocols can determine whether manipulation is computationally feasible or not. The computational complexity and bounds of manipulation are considered from various angles including control, false-name manipulation and bribery. Moreover, the computational complexity of computing various cooperative game solutions of simple games in dierent representations is studied. Certain structural results regarding least core payos extend to the general monotone cooperative game. The thesis also studies a coalitional game called the spanning connectivity game. It is proved that whereas computing the Banzhaf values and Shapley-Shubik indices of such games is #P-complete, there is a polynomial time combinatorial algorithm to compute the nucleolus. The results have interesting significance for optimal strategies for the wiretapping game which is a noncooperative game defined on a network