Shapley value

We use cooperative game theory to analyze the power structure in the pipeline network for Russian gas. If the assessment is narrowly focussed on the abilities to obstruct flows in the existing system, the main transit countries, Belarus and Ukraine, appear to be strong. Once investment options are accounted for, Russia achieves clear dominance. Competition between transit countries is of little strategic relevance compared to Russia's direct access to its customers. Comparing our theoretical results with empirical evidence, we find that the Shapley value explains the power of major transit countries better than the core and the nucleolus. JEL: L95, F50, D43

Transportation planning is an important part of the supply chain or wood flow chain in forestry. There are often several forest companies operating in the same region and collaboration between two or more companies is rare. However, there is an increasing interest in collaborative planning as the potential savings are large, often in the range 5-15%. There are several issues to agree on before such collaborative planning can be used in practice. A key question is how savings should be distributed among the participants. In this paper, we investigate a number of possibilities based on economic models including Shapley value, the nucleolus, separable and non-separable costs, shadow prices and volume weights. We also propose a new allocation method, with the aim that the participants relative profits are as equal as possible. A large case study comprising eight forest companies in Sweden, is described and analyzed.

In this paper we consider a cost allocation problem arising in a consortium for urban solid wastes collection and disposal. A classical allocation rule is the proportional division according to the volume of wastes collected; here we propose a different allocation method, rooted in game theory, based on a model developed by Fragnelli et al. [3], using the Shapley Value and the Owen Value as allocation rules. The model was applied to the data of the consortium Ovadese -Valle Scrivia, comparing our results and the actual ones.

Recently, a trend towards horizontal collaboration in the supply chain has been gaining ground. Collaborative supply chains promise to improve sustainability and reduce transportation costs at the same time. This paper examines the possibility of collaboration and its implementation through a description of the current situation and a case study of three large companies in Belgium who have expressed their willingness to collaborate. As collaboration assumes a long-term relationship that needs to be built, this paper investigates the advantages of moving beyond mere bundling of logistics streams and assumes that partners are willing to explore further optimization opportunities that arise from collaboration. Most notably, this paper evaluates the effects of increased flexible with respect to delivery dates, as well as allowing large orders to be split into several deliveries. The case study developed in this paper shows that a decrease in distribution costs of 25.83% can be achieved....

In this paper we consider standard fixed tree games, for which each vertex unequal to the root is inhabited by exactly one player. We present two weighted allocation rules, the weighted down-home allocation and the weighted neighbour-home allocation, both inspired by the painting story in Maschler et al. (1995) . We show, in a constructive way, that the core equals both the set of weighted down-home allocations and the set of weighted neighbour allocations. Since every weighted down-home allocation specifies a weighted Shapley value ) in a natural way, and vice versa, our results provide an alternative proof of the fact that the core of a standard fixed tree game equals the set of weighted Shapley values. The class of weighted neighbour allocations is a generalization of the nucleolus, in the sense that the latter is in this class as the special member where players have all equal weights.

This survey captures the main contributions in the area described by the title that were published up to 1997. (Unfortunately, it does not capture all of them.) The variations that are the subject of this chapter are those axiomatically characterized solutions which are obtained by ...

Let L be a D-lattice, i.e. a lattice ordered effect algebra, and let BV be the Banach space of all real-valued functions of bounded variation on L (vanishing at 0) endowed with the variation norm. We prove the existence of a continuous Aumann-Shapley value ϕ on bv NA, the subspace of BV spanned by all functions of the form f • µ, where µ : L → [0, 1] is a nonatomic σ-additive modular measure and f : [0, 1] → R is of bounded variation and continuous at 0 and at 1. Mathematics Subject Classification (2000). 28A12, 06C15, 91A12. A. Basile et al. Positivity (D1) a b is defined if and only if b ≤ a; (D2) If b = b c. Vol. 12 (2008) On the Aumann-Shapley value 615

This paper studies coalition formation, payoff division and expected payoffs in a "divide the dollar by majority rule" game with random proposers. A power index is called self-confirming if it can be obtained as an equilibrium of the game using the index itself as probability vector. Unlike the Shapley value and other commonly used power indices, the nucleolus has this property. The proof uses Kohlberg's (1971) characterization of the nucleolus, reinterpreting the balancing weights as probabilities in a mixed strategy equilibrium. As for arbitrary probability vectors, the nucleolus seems a more likely equilibrium than the Shapley value because it assigns competitive prices to the players.

In this paper we develop a polynomial method based on sampling theory that can be used to estimate the Shapley value (or any semivalue) for cooperative games. Besides analyzing the complexity problem, we examine some desirable statistical properties of the proposed approach and provide some computational results.

In this paper the cone of convex cooperative fuzzy games is studied. As in the classical case of convex crisp games, these games have a large core and the fuzzy Shapley value is the barycen- ter of the core. Surprisingly, the core and the Weber set coincide ∗ This paper was written while the authors were research fellows at the

We study the problem of allocation over time of total cost incurred by countries in a cooperative game of pollution reduction. We compute the characteristic function for all possible coalitions and apply the Shapley value to determine a fair distribution of the total cooperative cost among players. We design a mechanism for allocation over time of total individual cost so that the initial agreement remains valid for the whole duration of the game.

In this paper we view bargaining and cooperation as an interaction superimposed on a game in strategic form. A multistage bargaining procedure for N players, the "proposer commitment" procedure, is presented. It is inspired by Nash's two-player variable-threat model; a key feature is the commitment to "threats." We establish links to classical cooperative game theory solutions, such as the Shapley value in the transferable utility case. However, we show that even in standard pure exchange economies, the traditional coalitional function may not be adequate when utilities are not transferable. (JEL: C70, C71, C78, D70) 1. For a finite set Z, we denote by |Z| the number of elements of Z. 2. We write arg max for the set of maximizers.

A numerical method based on a relaxation algorithm and the Nikaido-Isoda function is presented for the calculation of Nash-Cournot equilibria in electricity markets. Nash equilibrium is attained through a relaxation procedure applied to an objective function, the Nikaido-Isoda function, which is derived from the existing profit maximization functions calculated by the generating companies. We also show how to use the relaxation algorithm to compute, and enforce, a coupled constraint equilibrium, which occurs if regulatory, generation, and distribution (and more) restrictions are placed on the companies and entire markets. Moreover, we use the relaxation algorithm to compute players' payoffs under several player configurations. This is needed for the solution of our game under cooperative game theory concepts, such as the bilateral Shapley value and the kernel. We show that the existence of both depends critically on demand price elasticity. The numerical method converges to a unique solution under rather specific but plausible concavity conditions. A case study from the IEEE 30-bus system, and a three-bus bilateral market example with a dc model of the transmission line constraints are presented and discussed.

Integrated Supply (I/S) has developed dramatically since its emergence in 1990s. But the buyers do not contact with the suppliers directly under general integrated supply model, which will lead to losing the direct control over them. I/S model by third-party logistics(TPL) may overcomes the disadvantage of traditional I/S model. A key problem of TPL integrated supply is to allocate the supply cost among the suppliers rationally. In this paper, Shapley value method is commended to solve the problem. To get over the two inherent limitations of Shapley value method, EOQ model of integrated supply is firstly adopted, which makes Shapley value method more practicable. Then, a weighting factor is used to adjust the Shapley value result which will lead to more reasonable result. Finally, an example proves that shapley value method is feasible and rational to allocate the I/S losgistics cost among the suppliers.

We investigate the impact and the possible consequences of the construction of the Nord Stream 2 pipeline. We model the European gas network as a cooperative game between regions as players over the pipeline network. Our model offers several novelties compared to earlier cooperative studies. Firstly, we focus on cost saving rather than on the profits of cooperation. Secondly, we introduce liquefied natural gas as a player. Thirdly, we apply an iterative linear program to account for the long term bilateral contracts that still drive the gas market. This modelling technique also allows us to identify individual gas flows. We focus on the change of influence of the players in three different scenarios. We investigate how the power of the agents shift when the Nord Stream pipeline is expanded, when the Ukrainian pipeline is shut down and finally when both of these happen. Our calculations show that when Nord Stream 2 is operational, Russia and Western Europe improve their position compared to the base scenario, while other suppliers, notably Norway, together with Central-Eastern-and Southern Europe suffer losses, especially when the Ukrainian route is dismissed. The results highlight that both the supporters and adversaries of Nord Stream 2 are governed by self-interest and solidarity and trust, the values proclaimed by the EU and the Energy Union, remain but a slogan.

Resumen-La teoría de juegos cooperativos se centra en el cálculo de beneficios y viabilidad de la cooperación entre diferentes agentes o actores. Estos agentes podrían ser personas que están negociando para un objetivo común, un grupo de empresas que buscan ampliar sus cuotas de mercado o un grupo de partes interesadas que comparten un mismo problema. Esto ha llevado a un mayor interés en el tema por los investigadores, los profesionales y los responsables políticos que buscan formas equitativas y eficientes de colaborar para una meta específica. El valor de Shapley proporciona una solución a un juego cooperativo de n jugadores que es justa y estable. Sin embargo, el cálculo de este valor es de una complejidad combinatoria, por lo que se han desarrollado numerosos algoritmos para obtener soluciones en un tiempo computacional razonable. En este trabajo se realiza una revisión de los algoritmos desarrollados en la literatura para calcular el valor de Shapley y se propone un algoritmo ...

The ability to explain in understandable terms, why a machine learning model makes a certain prediction is becoming immensely important, as it ensures trust and transparency in the decision process of the model. Complex models, such as ensemble or deep learning models, are hard to interpret. Various methods have been proposed that deal with this matter. Shapley values provide accurate explanations, as they assign each feature an importance value for a particular prediction. However, the exponential complexity of their calculation is dealt efficiently only in decision tree-based models. Another method is surrogate models, which emulate a black-box model's behavior and provide explanations effortlessly, since they are constructed to be interpretable. Surrogate models are model-agnostic, but they produce only approximate explanations, which cannot always be trusted. We propose a method that combines these two approaches, so that we can take advantage of the model-agnostic part of the surrogate models, as well as the explanatory power of the Shapley values. We introduce a new metric, Top j Similarity, that measures the similitude of two given explanations, produced by Shapley values, in order to evaluate our work. Finally, we recommend ways on how this method could be improved further.

This paper argues that the nucleolus can compete with the Shapley value as a measure of P-power. It currently has more solid noncooperative foundations for majority games. It also identifies a set of attractive coalitions that are expected to form (unlike the Shapley value, which is based on the values of all coalitions) and does better than the Shapley value at some postulates of voting power. On the negative side, it may give a payoff of zero to players that are not dummies, though this possibility is excluded for constant-sum weighted majority games. 2RD (United Kingdom); phone +44 115 9515468, fax +44 115 951 4159, e-mail [email protected]. This paper has benefitted from the comments of two referees.

We show how the quality of decisions based on the aggregated opinions of the crowd can be conveniently studied using a sample of individual responses to a standard IQ questionnaire. We aggregated the responses to the IQ questionnaire using simple majority voting and a machine learning approach based on a probabilistic graphical model. The score for the aggregated questionnaire, Crowd IQ, serves as a quality measure of decisions based on aggregating opinions, which also allows quantifying individual and crowd performance on the same scale.

This study attempts to carry out a comprehensive analysis of poverty, inequality and polarisation trends using Cameroon household surveys collected before and during the HIPC process. The theoretical decomposition frameworks propelling the study are motivated mainly by the Shapely Value. Empirical estimates are obtained from DAD 4.4 using both money-metric and child nutrition indicators, as well as poverty lines, with the monetary threshold derived non-parametrically. Effects within-zones account for much of monetary poverty changes than effects between-zones. The findings that inter-zone effects contribute in alleviating rural poverty, while aggravating urban poverty, suggests the potentials of rural-urban migration in alleviating rural poverty. Changes in money-metric poverty and health deprivation sharply contrast each other. While health poverty deteriorated, income poverty retreated. This is indication that economic growth may not necessarily engender significant reduction in a...

IVIE working papers offer in advance the results of economic research under way in order to encourage a discussion process before sending them to scientific journals for their final publication. * We want to thank an anonymous IVIE's referee for his/her comments. This research has been supported by the Spanish Ministerio de Ciencia y Tecnología under project BEC2000-0875, and by the Universidad del País Vasco under project UPV/EHU00031.321-HA-7918/2000. The first author also acknowledges financial support from the Spanish Ministerio de Ciencia y Tecnología under the Ramón y Cajal Program. Part of this paper was written while the second author was visiting the Departamento de Fundamentos del Análisis Económico in the Universidad de Alicante, whose hospitality is gratefully acknowledged.

This paper presents a linear programming model that allocates the waters of the Euphrates and Tigris rivers to irrigation, urban consumption, and on-stream hydroelectricity production in the three riparian countries (Turkey, Syria, and Iraq), by maximizing the aggregate net benefits from water uses while accounting for water-conveyance costs. The model represents, in network form, the system made of the two rivers and their various consumption, supply, and transshipment nodes, and accounts for evaporation and return flows. The constraints include water-conservation balances and maximum and minimum water consumption. The model is used to assess the economic consequences of various cooperation and noncooperation strategies that may be adopted by the riparian countries. Cooperative game-theory concepts (core and Shapley value) are used to identify stable water allocations, under which all three countries find it beneficial to cooperate. The results suggest that an allocation of the tot...

The equitable division of a joint cost (or a jointly produced output) among agents with different shares or types of output (or input) commodities, is a central theme of the theory of cooperative games with transferable utility. Ever since Shapley's seminal contribution in 1953, this question has generated some of the deepest axiomatic results of modern microeconomic theory.

A value for games with a coalition structure is introduced, where the rules guiding cooperation among the members of the same coalition are different from the interaction rules among coalitions. In particular, players inside a coalition exhibit a greater degree of solidarity than they are willing to use with players outside their coalition. The Shapley value is therefore used to compute the aggregate payoffs for the coalitions, and the solidarity value to obtain the payoffs for the players inside each coalition.

In this letter we present a simple and straightforward proof of a new formula for the Shapley value. Unlike the usual weighted-average-of-marginal-contributions expression, our formula emphasizes the relative worths of players within each coalition size.

Voting power in voting situations is measured by the probability of changing decisions by altering the cast `yes' or `no' votes. Recently this analysis has been extended by strategic abstention. Abstention, just as `yes' or `no' votes can change decisions. This theory is often applied to weighted voting situations, where voters can cast multiple votes. Measuring the power of a party in a national assembly seems to fit this model, but in fact its power comprises of votes of individual representatives each having a single vote. These representatives may vote yes or no, or may abstain, but in some cases they are not even there to vote. We look at absentees not due to a conscious decision, but due to illness, for instance. Formally voters will be absent, say, ill, with a certain probability and only present otherwise. As in general not all voters will be present, a thin majority may quickly melt away making a coalition that is winning in theory a losing one in practice. ...

We extend the Aumann-Shapley value to mixed action-set games, i.e., multilevel TU games where there are simultaneously two types of players: discrete players that possess a ®nite number of activity levels in which they can join a coalition, and continuous players that possess a continuum of levels.

We study a multicast game in ad-hoc wireless networks in which a source sends the same message or service to a set of receiving stations via multi-hop communications and the overall transmission cost is divided among the receivers according to given cost sharing methods. Receivers enjoy a benefit equal to the difference between the utility they get from the transmission and the shared cost they are asked to pay. Assuming a selfish and rational behavior, each user is willing to receive the transmission if and only if his shared cost does not exceed his utility. Moreover, given the strategies of the other users, he wants to select a strategy of minimum shared cost. A Nash equilibrium is a solution in which no user can increase his benefit by seceding in favor of a different strategy. We consider the following reasonable cost sharing methods: the overall transmission cost is equally shared among all the receivers (egalitarian), the cost of each intermediate station is divided among its downstream receivers equally (semi-egalitarian) or proportionally to the transmission powers they require to reach their next-hop stations (proportional), and finally each downstream user at an intermediate station equally shares only his required next-hop power among all the receivers asking at least such a level of power (Shapley value). We prove that, while the first three cost sharing methods in general do not admit a Nash equilibrium, the Shapley value yields games always converging to a Nash equilibrium. Moreover, we provide matching upper and lower bounds for the price of anarchy of the Shapley game with respect to two different global cost functions, that is the overall cost of the power assignment, that coincides with the sum of all the shared costs, and the maximum shared cost paid by the receivers. Finally, in both cases we show that finding the best Nash equilibrium is computationally intractable, that is NP-hard.

The paper studies cooperative sharing rules in fisheries coalition games and develops a new sharing rule which takes into account the stability of cooperation when externalities are present. The paper contributes to the literature by introducing a connection between cooperative games (sharing rules) and non-cooperative games (stability). For an illustrative example it sets up a discrete-time, deterministic, coalition game model among the major agents exploiting the cod stock in the Baltic Sea.

Deregulation and restructuring have become unavoidable trends to the power industry recently, in order to increase its efficiency, to reduce operation costs, or to provide customers a better service. The once centralized system planning and management must be remodeled to reflect the changes in the market environment. We propose and have developed a multi-agent based system to assist players, such as, owners of power generation stations, owners of transmission lines, and groups of consumers, to select partners to form coalitions. The system provides a cooperation plan and its associated cost allocation plan for the user to support its decision making process. Among several coalition formation and cost allocation criteria, we have selected the Bilateral Shapley Value (BSV) as the theoretical foundation to develop the system. We have tested the multi-agent system with a classical transmission expansion example.

Max Albert has argued that the theory of power indices “should not ... be considered as part of political science” and that “[v]iewed as a scientific theory, it is a branch of probability theory and can safely be ignored by political scientists”. Albert’s argument rests on a particular claim concerning the theoretical status of power indices, namely that the theory of power indices is not a positive theory, i.e., not one that has falsifiable implications. I re-examine the theoretical status of power indices and argue that it would be unwise for political scientists to ignore such indices. Although I agree with Albert that the theory of power indices is not a positive theory, I suggest that it is a theory of measurement that can usefully supplement other positive and normative social- scientific theories.
Preprint: http://personal.lse.ac.uk/LIST/PDF-files/List-EUPolitics.pdf

In this paper, we address the problem of botnet detection by correlating information from trusted hosts and network. Botnets are groups of compromised computers controlled by a botmaster through a command and control (C&C) channel. They are noted as one of the foremost security threat causing large scale attacks such as Distributed Denial of Service (DDoS), Spam, mass identity theft and click fraud. Various approaches are used to detect botnets and they range from network to host level detection. To enhance the detection rate, a correlation based model was proposed that combines both host and network level information. Such a model is valid in a network made of trusted hosts. The emergence of smartphones with the capability of mobility and being hosts in different networks, open the door of having untrusted hosts that can reveal fake information. As a solution, we propose a trust-based model that uses cooperative game theory to cluster trusted hosts. The trust is built using the reputation value and it is computed using the hosts' marginal contribution which is derived from Shapley value. Simulation results show that our model improves the detection score compared to the traditional correlation model. Where in one of the simulated scenarios we are able to detect a benign cluster of hosts faster than the traditional correlation model.

It is well known since Owen (1968) that the weights in the weighted Shapley value cannot be interpreted as a measure of power (i.e., of the ability to bargain) of the players. This paper proposes a new weight scheme for the Shapley value. Weights in this framework have to be interpreted as a measure of bargaining power. Two different axiomatic characterizations of this new value are proposed: one including the weights in the axioms and one without.

Single-valued solutions for the case of two-sided market games without product differentiation, also known as Bo¨hm-Bawerk horse market games, are analyzed. The nucleolus is proved to coincide with the s value, and is thus the midpoint of the core. The Shapley value is in the core only if the game is a square glove market, and in this case also coincides with the two aforementioned solutions.

This paper uses a contract theory model to argue that covenants ruling debt renegotiations are important to assure the sovereign willingness to pay. The model includes the following features: first, collective action clauses, exit consents, aggregation provisions and pari passu clauses play an important role in the post default "game" of negotiations and coalitions. These covenants are represented in reduced form by the endogenous probability of refinancing a defaulted sovereign debt. Second, the model has "endogenous bad luck" because the unfavorable state of nature where default occurs depends on the level of indebtedness, which is itself an endogenous variable. Third, "vultures", contrary to conventional wisdom, tend to improve the access of emerging economies to capital markets because they might help to rule out strategic defaults. And fourth, under special assumptions the model is able to analyze the possibility of post default discrimination between domestic and foreign bondholders.

We present a generalization of spatial power indexes able to overcome their main limitations, namely: i) the excessive concentration of power measures; ii) the too high sensitivity to players' location in the ideological space. Voters' propensity to support an issue is modeled via a random utility function with two additive terms: the deterministic term accounts for voters' preference-driven/predictable behavior; the random one is a catch-all term that accounts for all the idiosyncratic/unpredictable factors. The relative strength of the two terms gives rise to a continuum of cases ranging from the Shapley value, where all aggregation patterns are equally probable, to a standard spatial value, like the Owen-Shapley index, where instead the conditional order is fully deterministic. As an illustrative application, we analyze the distribution of power in the Council of Ministers under three different scenarios: i) EU15 Pre-Nice; ii) EU27 Nice Treaty; iii) EU27 Lisbon Treaty.

A new family of centrality measures, based on game theoretical concepts, is proposed for social networks. To reflect the interests that motivate the interactions among individuals in a network, a cooperative game in characteristic function form is considered. From the graph and the game, the graph-restricted game is obtained. Shapley value in a game is considered as actor's power. The difference between actor's power in the new game and his / her power in the original one is proposed as a centrality measure. Conditions are given to reach some desirable properties. Finally, a decomposition is proposed. 

This paper focuses on new characterizations of convex multi-choice games using the notions of exactness and superadditivity. Furthermore, level-increase monotonic allocation schemes (limas) on the class of convex multi-choice games are introduced and studied. It turns out that each element of the Weber set of such a game is extendable to a limas, and the (total) Shapley value for multi-choice games generates a limas for each convex multi-choice game.

We consider the problem of designing distribution rules to share 'welfare' (cost or revenue) among individually strategic agents. There are many known distribution rules that guarantee the existence of a (pure) Nash equilibrium in this setting, e.g., the Shapley value and its weighted variants; however, a characterization of the space of distribution rules that guarantee the existence of a Nash equilibrium is unknown. Our work provides an exact characterization of this space for a specific class of scalable and separable games, which includes a variety of applications such as facility location, routing, network formation, and coverage games. Given arbitrary local welfare functions W, we prove that a distribution rule guarantees equilibrium existence for all games (i.e., all possible sets of resources, agent action sets, etc.) if and only if it is equivalent to a generalized weighted Shapley value on some 'ground' welfare functions W', which can be distinct from W. However, if budget-balance is required in addition to the existence of a Nash equilibrium, then W' must be the same as W. We also provide an alternate characterization of this space in terms of 'generalized' marginal contributions, which is more appealing from the point of view of computational tractability. A possibly surprising consequence of our result is that, in order to guarantee equilibrium existence in all games with any fixed local welfare functions, it is necessary to work within the class of potential games.

We present a generalization of spatial power indexes able to overcome their main limitations, namely: i) the excessive concentration of power measures; ii) the too high sensitivity to players' location in the ideological space. Voters' propensity to support an issue is modeled via a random utility function with two additive terms: the deterministic term accounts for voters' preference-driven/predictable behavior; the random one is a catch-all term that accounts for all the idiosyncratic/unpredictable factors. The relative strength of the two terms gives rise to a continuum of cases ranging from the Shapley value, where all aggregation patterns are equally probable, to a standard spatial value, like the Owen-Shapley index, where instead the conditional order is fully deterministic. As an illustrative application, we analyze the distribution of power in the Council of Ministers under three different scenarios: i) EU15 Pre-Nice; ii) EU27 Nice Treaty; iii) EU27 Lisbon Treaty.