Values on regular games under Kirchhoff’s laws

In cooperative game theory, the Shapley value is a central notion defining a rational way to share the total worth of a game among players. In this paper, we address a general framework leading to applications to games with communication graphs, where the set of feasible coalitions forms a poset where all maximal chains have the same length. We first show that previous definitions and axiomatizations of the Shapley value proprosed by Faigle and Kern, and Bilbao and Edelman still work. Our main contribution is then to propose a new axiomatization avoiding the hierarchical strength axiom of Faigle and Kern, and considering a new way to define the symmetry among players. Borrowing ideas from electric networks theory, we show that our symmetry axiom and the classical efficiency axiom correspond actually to the two Kirchhoff’s laws in the resistor circuit associated to the Hasse diagram of feasible coalitions. We finally work out a weak form of the monotonicity axiom which is satisfied by the proposed value.Regular set systems; regular games; Shapley value; probabilistic efficient values; regular values; Kirchhoff’s laws.

Interaction transform for bi-set functions over a finite set

Set functions appear as a useful tool in many areas of decision making and operations research, and several linear invertible transformations have been introduced for set functions, such as the Möbius transform and the interaction transform. The present paper establish similar transforms and their relationships for bi-set functions, i.e. functions of two disjoint subsets. Bi-set functions have been recently introduced in decision making (bi-capacities) and game theory (bi-cooperative games), and appear to open new areas in these fields.Set function; Bi-set function; Möbius transform; Interaction transform

Games on lattices, multichoice games and the Shapley value: a new approach

Multichoice games have been introduced by Hsiao and Raghavan as a generalization of classical cooperative games. An important notion in cooperative game theory is the core of the game, as it contains the rational imputations for players. We propose two definitions for the core of a multichoice game, the first one is called the precore and is a direct generalization of the classical definition. We show that the precore coincides with the definition proposed by Faigle, and that it contains unbounded imputations, which makes its application questionable. A second definition is proposed, imposing normalization at each level, causing the core to be a convex closed set. We study its properties, introducing balancedness and marginal worth vectors, and defining the Weber set and the pre-Weber set. We show that the classical properties of inclusion of the (pre)core into the (pre)-Weber set as well as their equality remain valid. A last section makes a comparison with the core defined by van den Nouweland et al.multichoice game ; lattice ; core

Values on regular games under Kirchhoff's laws

In cooperative game theory, the Shapley value is a central notion defining a rational way to share the total worth of a game among players. In this paper, we address a general framework, namely regular set systems, where the set of feasible coalitions forms a poset where all maximal chains have the same length. We first show that previous definitions and axiomatizations of the Shaphey value proposed by Faigle and Kern and Bilbao and Edelman still work. our main contribution is then to propose a new axiomatization avoiding the hierarchical strength axiom of Faigle and Kern, and considering a new way to define the symmetry among players. Borrowing ideas from electric networks theory, we show that our symmetry axiom and the classical efficiency axiom correspond actually to the two Kirchhoff's laws in the resistor circuit associated to the Hasse diagram of feasible coalitions. We finally work out a weak form of the monotonicity axiom which is satisfied by the proposed value.Regular set systems, regular games, Shapley value, probabilistic efficient values, regular values, Kirchhoff's laws.

Power indices expressed in terms of minimal winning coalitions

A voting situation is given by a set of voters and the rules of legislation that determine minimal requirements for a group of voters to pass a motion. A priori measures of voting power, such as the Shapley-Shubik index and the Banzhaf value, show the influence of the individual players. We used to calculate them by looking at marginal contributions in a simple game consisting of winning and losing coalitions derived from the rules of the legislation. We introduce a new way to calculate these measures directly from the set of minimal winning coalitions. This new approach logically appealing as it writes measures as functions of the rules of the legislation. For certain classes of games that arise naturally in applications the logical shortcut drastically simplifies calculations. The technique generalises directly to all semivalues. Keywords. Shapley-Shubik index, Banzhaf index, semivalue, minimal winning coalition, Möbius transform.Shapley-Shubik index, Banzhaf index, semivalue, minimal winning coalition, Möbius transform.

Values on regular games under Kirchhoff's laws

URL des Cahiers : https://halshs.archives-ouvertes.fr/CAHIERS-MSE Voir aussi l'article basé sur ce document de travail paru dans "Mathematical Social Sciences", Elsevier, 2009, 58, (3), pp. 322-340Cahiers de la MSE n° 2006.87 - Série bleue (CERMSEM) - ISSN 1624-0340In cooperative game theory, the Shapley value is a central notion defining a rational way to share the total worth of a game among players. In this paper, we address a general framework, namely regular set systems, where the set of feasible coalitions forms a poset where all maximal chains have the same length. We first show that previous definitions and axiomatizations of the Shaphey value proposed by Faigle and Kern and Bilbao and Edelman still work. our main contribution is then to propose a new axiomatization avoiding the hierarchical strength axiom of Faigle and Kern, and considering a new way to define the symmetry among players. Borrowing ideas from electric networks theory, we show that our symmetry axiom and the classical efficiency axiom correspond actually to the two Kirchhoff's laws in the resistor circuit associated to the Hasse diagram of feasible coalitions. We finally work out a weak form of the monotonicity axiom which is satisfied by the proposed value.En théorie des jeux coopératifs, la valeur de Shapley est une notion centrale permettant de définir d'une manière rationnelle le moyen de partager la valeur de la grande coalition entre tous les joueurs. Dans le cadre général de ce papier, l'ensemble des coalitions faisables où est défini le jeu forme un ensemble ordonné (par l'inclusion) dont toutes les chaînes maximales ont la même longueur. Nous montrons d'abord que certaines définitions et axiomatisations précédemment étudiées par Faigle et Kern de la valeur de Shapley restent valables. Notre principale contribution est de proposer une nouvelle axiomatisation qui évite l'axiome de force hiérarchique de Faigle et Kern (difficilement interprétable), considérant un nouveau moyen de généraliser l'axiome d'anonymat entre joueurs. Des idées de la théorie des réseaux électriques sont ensuite empruntées, où nous montrons que notre axiome d'anonymat (regularity axiom) ainsi que l'axiome bien connu d'efficacité (efficiency axiom) correspondent en fait aux deux lois de Kirchhoff d'un circuit électrique résistif (les noeuds étant données par les coalitions faisables et les branches par les couples de coalitions se précédant). Plus précisément, des analogies sont données entre l'axiome d'efficacité et la loi des nœuds entre l'axiome d'anonymat et la loi des mailles. Nous établissons enfin une forme plus faible de l'axiome de monotonie qui est satisfait par la valeur proposée

New axiomatizations of the Shapley interaction index for bi-capacities

International audienceBi-capacities are a natural generalization of capacities (or fuzzy measures) in a context of decision making where underlying scales are bipolar. They are able to capture a wide variety of decision behaviours. After a short presentation of the basis structure, we introduce the Shapley value and the interaction index for capacities. Afterwards, the case of bi-capacities is studied with new axiomatizations of the interaction index

Power indices expressed in terms of minimal winning coalitions

A voting situation is given by a set of voters and the rules of legislation that determine minimal requirements for a group of voters to pass a motion. A priori measures of voting power, such as the Shapley-Shubik index and the Banzhaf value, show the influence of the individual players in a voting situation and are calculated by looking at marginal contributions in a simple game consisting of winning and losing coalitions derived from the legislative rules. We introduce a new way to calculate these measures directly from the set of minimal winning coalitions and derive explicit formulae for the Shapley-Shubik and Banzhaf values. This new approach logically appealing as it writes measures as functions of the rules of the legislation. For certain classes of games that arise naturally in applications the logical shortcut drastically simplifies the numerical calculations to obtain the indices. The technique generalises directly to all semivalues

Values on regular games under Kirchhoff’s laws

Abstract. In cooperative game theory, the Shapley value is a central notion defining a rational way to share the total worth of a game among players. In this paper, we address a general framework leading to applications to games with communication graphs, where the set of feasible coalitions forms a poset where all maximal chains have the same length. We first show that previous definitions and axiomatizations of the Shapley value proprosed by Faigle and Kern, and Bilbao and Edelman still work. Our main contribution is then to propose a new axiomatization avoiding the hierarchical strength axiom of Faigle and Kern, and considering a new way to define the symmetry among players. Borrowing ideas from electric networks theory, we show that our symmetry axiom and the classical efficiency axiom correspond actually to the two Kirchhoff's laws in the resistor circuit associated to the Hasse diagram of feasible coalitions. We finally work out a weak form of the monotonicity axiom which is satisfied by the proposed value

Az erős,a gyenge, meg a ravasz: Hatalom és stratégiai viselkedés szavazási játékokban = The Strong, the Weak and the Cunning: Power and Strategy in Voting Games

Kutatási eredményeink három téma köré csoportosíthatók. Ezek közül az első a stratégiai megfontolások vizsgálata. Megmutatjuk, hogy a szavazók növelhetik befolyásukat, ha veszekednek más szavazókkal és a stratégiai hatalmi indexek jól definiáltak a játékok bizonyos osztályaira. Egy másik vonal a kooperatív játékok olyan kulcsfontosságú tulajdonságait vizsgálja, mint a konvexitás, vagy az egzaktság. Bizonyos esetekben a nyerő koalíciók halmaza külső okok miatt korlátozott: erre a leggyakoribb példa, mikor egy hálózaton elhelyezkedő csúcsok helyzeti befolyását vizsgáljuk. A csúcsok csak az őket összekötő élek mentén kommunikálhatnak és csak szomszédaikkal. Több érték és index is kiterjesztésre, illetve bevezetésre kerül ilyen hálózati játékokra, illetve az értékekhez axiomatikus karakterizációt adunk. Végül a hatalmi indexeket olyan játékokra is kiterjesztjük, ahol egyes szavazók hiányozhatnak. A nem stratégiai hiányzást vizsgáljuk és a Shapley értéket teljesen karakterizáljuk az általánosított súlyozott szavazási játékok osztályán. Modellünket különböző parlamentekre alkalmazzuk, illetve az elméleti módszerek több egyéb alkalmazását is vizsgáltuk, úgymint a Lisszaboni Szerződés hatását a Miniszterek Tanácsában folyó súlyozott szavazás hatalmi viszonyaira. | The results of the project centre around three themes. The first is strategic considerations. We have shown that voters are able to increase their power by strategic quarrelling and the strategic power indices are well defined for certain classes of games. Additional papers provide tests on key properties, such as convexity and exactness of cooperative games. In some situations the set of feasible (winning) coalitions is restricted exogenously. The most common example is to study positional power over a network where the voters are located at the nodes and can only communicate with their neighbours. Several values and indices are introduced and characterised for games over networks. At last we generalised power indices to weighted voting games where representatives may be absent. We study non-strategic absenteeism and characterise the Shapley value for the class of generalised weighted voting games. We have also studied applications studying the effect of absent voters in various national parliaments or the effect of the Lisbon Treaty of the European Union to the power balance in the Council of Ministers