The Shapley value for directed graph games

The Shapley value for directed graph (digraph) games, TU games with limited cooperation introduced by an arbitrary digraph prescribing the dominance relation among the players, is introduced. It is dened as the average of marginal contribution vectors corresponding to all permutations that do not violate the subordination of players. We assume that in order to cooperate players may join only coalitions containing no players dominating them. Properties of this solution are studied and a convexity type condition is provided that guarantees its stability with respect to an appropriately dened core concept. An axiomatization for cycle digraph games for which the digraphs are directed cycles is obtained

The average covering tree value for directed graph games

We introduce a single-valued solution concept, the so-called average covering tree value, for the class of transferable utility games with limited communication structure represented by a directed graph. The solution is the average of the marginal contribution vectors corresponding to all covering trees of the directed graph. The covering trees of a directed graph are those (rooted) trees on the set of players that preserve the dominance relations between the players prescribed by the directed graph. The average covering tree value is component efficient, and under a particular convexity-type condition it is stable. For transferable utility games with complete communication structure the average covering tree value equals to the Shapley value of the game. If the graph is the directed analog of an undirected graph the average covering tree value coincides with the gravity center solution

Pemodelan Keputusan Stok Dan Impor Bawang Merah

Untuk mengoptimalkan fungsi rantai pasok dan menghadapi tantangan berupa kendala-kendala di dalam proses operasionalnya, setiap pelaku rantai pasok perlu untuk bekerjasama mengintegrasikan masing-masing fungsinya. Salah satu permasalahan dalam rantai pasok adalah ketidakseimbangan kuantitas dari penjualan dengan kuantitas pasokan. Penelitian ini bertujuan untuk mengusulkan model sederhana untuk menghadapi tantangan adanya ketidakseimbangan kuantitas penjualan dan kuantitas pasokan. Ketidakseimbangan tersebut menyebabkan overstock atau loss. Permasalahan tersebut diusulkan untuk diselesaikan dengan menggunakan volume buffer. Penelitian ini memodelkan model keputusan jumlah pasokan dan mengangkat studi kasus impor bawang merah di Indonesia, yang merupakan salah satu komoditas yang berpengaruh pada inflasi. Meskipun kebutuhan konsumsi cukup stabil dari sepanjang tahun, namun jumlah pasokan sangat dipengaruhi faktor ketidakpastian yang terjadi selama proses tanam hingga panen. Ketika pasokan diprediksi tidak mencukupi kebutuhan pasar, maka pemerintah akan mengambil keputusan untuk mengimpor komoditas tersebut. Oleh karena itu, pemerintah seharusnya membutuhkan mekanisme volume buffer untuk memastikan kebutuhan konsumsi terpenuhi sekaligus harga pasar terkendali. Permulaan dari model ini mempertimbangkan nilai-nilai stokastik dan model lanjutan dibuat untuk memperoleh harga pasar. Pricing model dalam penelitian ini mengacu pada prinsip metode Shapley value yang dimodifikasi. Variabel-variabel utama dalam penelitian ini adalah prediksi jumlah pasokan, prediksi jumlah kebutuhan, buffer dan kebutuhan pasokan yang diperoleh dari petani (stock needed), kebutuhan aktual, dan harga untuk setiap pelaku rantai pasok. Model dalam penelitian ini dirancang secara sekuensial. Hasil dari validasi penelitian menunjukkan harga pada model berbeda signifikan dengan data sekunder yang dibandingkan. Oleh karena itu, model dapat digunakan untuk menentukan jumlah pasokan yang dibutuhkan dan dapat digunakan untuk referensi harga pasar. ================================================================= To optimize supply chain role, the pl ayers of supply chain need to integrate its function. One of the general problems in supply chain was the unbalanced quantity of sales and quantity of supply. This paper focused on modeling a simple method to manage the gap between the demand and the supply. The gap may causing an overstock or a loss. This paper propose a buffer quantity in order to handle the gap by using import decision. The case study was about shallot supply - demand in Indonesia. In this study we model the supply decisions of shallot i n Indonesia. While the demand is quite stable over time, the supply is very much affected by the yield from the farms. The shortage can result in the government importing shallot from other countries. Hence, the government also needs to have a proper buffe ring mechanism in order to ensure the supply is sufficient and the price is quite stable. The initial model of this research was built by stochastic parameters and the extended model to gain pricing mechanism was built by Shapley value principal with modif ication. The primary variables were supply quantity prediction, demand quantity prediction, buffer and purchased quantity (stock needed), actual consumption, and price for three players. The validation proved that the result of price at each player presented a significant difference. Therefore, the model could be applied to decide the stock quantity needed and to keep the price stable at each player especially at the end player which was influencing the market price

Structural restrictions in cooperation

Cooperative games with transferable utilities, or simply TU-games, refer to the situations where the revenues created by a coalition of players through cooperation can be freely distributed to the members of the coalition. The fundamental question in cooperative game theory deals with the problem of how much payoff every player should receive. The classical assumption for TU-games states that every coalition is able to form and earn the worth created by cooperation. In the literature, there are several different modifications of TU-games in order to cover the cases where cooperation among the players is restricted. The second chapter of this monograph provides a characterization of the average tree solution for TU-games where the restricted cooperation is represented by a connected cycle-free graph on the set of players. The third chapter considers TU-games for which the restricted cooperation is represented by a directed graph on the set of players and introduces the average covering tree solution and the dominance value for this class of games. Chapter four considers TU-games with restricted cooperation which is represented by a set system on the set of players and introduces the average coalitional tree solution for such structures. The last two chapters of this monograph belong to the social choice theory literature. Given a set of candidates and a set of an odd number of individuals with preferences on these candidates, pairwise majority comparison of the candidates yields a tournament on the set of candidates. Tournaments are special types of directed graphs which contain an arc between any pair of nodes. The Copeland solution of a tournament is the set of candidates that beat the maximum number of candidates. In chapter five, a new characterization of the Copeland solution is provided that is based on the number of steps in which candidates beat each other. Chapter six of this monograph is on preference aggregation which deals with collective decision making to obtain a social preference. A sophisticated social welfare function is defined as a mapping from profiles of individual preferences into a sophisticated social preference which is a pairwise weighted comparison of alternatives. This chapter provides a characterization of Pareto optimal and pairwise independent sophisticated social welfare functions

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. 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The Shapley value for directed graph games

The Shapley value for directed graph (digraph) TU games with limited cooperation induced by a digraph prescribing the dominance relation among the players is introduced. It is defined as the average of the marginal contribution vectors corresponding to all permutations which do not violate the induced subordination of players. We study properties of this solution and its core stability. For digraph games with the digraphs being directed cycles an axiomatization of the solution is obtained