The value of a player in n-person games

. The article decomposes the Shapley value into a value matrix which gives the value of every player to every other player in n-person games. Element Φ_ij(v) in the value matrix is positive, zero, or negative, dependent on whether row player i is beneficial, has no impact, or is not beneficial for column player j. The elements in each row and in each column of the value matrix sum up to the Shapley value of the respective player. The value matrix is illustrated by the voting procedure in the European Council of Ministers 1981–1995. Received: 9 September 1998/Accepted: 11 February 2000     

Axiomatizations of the normalized Banzhaf value and the Shapley value

A cooperative game with transferable utilities– or simply a TU-game – describes a situation in which players can obtain certain payoffs by cooperation. A solution concept for these games is a function which assigns to every such a game a distribution of payoffs over the players in the game. Famous solution concepts for TU-games are the Shapley value and the Banzhaf value. Both solution concepts have been axiomatized in various ways. An important difference between these two solution concepts is the fact that the Shapley value always distributes the payoff that can be obtained by the `grand coalition' consisting of all players cooperating together while the Banzhaf value does not satisfy this property, i.e., the Banzhaf value is not efficient. In this paper we consider the normalized Banzhaf value which distributes the payoff that can be obtained by the `grand coalition' proportional to the Banzhaf values of the players. This value does not satisfy certain axioms underlying the Banzhaf value. In this paper we introduce some new axioms that characterize the normalized Banzhaf value. We also provide an axiomatization of the Shapley value using similar axioms. Received: 10 April 1996 / Accepted: 2 June 1997     

Reconciling marginalism with egalitarianism: consistency, monotonicity, and implementation of egalitarian Shapley values

One of the main issues in economic allocation problems is the trade-off between marginalism and egalitarianism. In the context of cooperative games this trade-off can be framed as one of choosing to allocate according to the Shapley value or the equal division solution. In this paper we provide three different characterizations of egalitarian Shapley values being convex combinations of the Shapley value and the equal division solution. First, from the perspective of a variable player set, we show that all these solutions satisfy the same reduced game consistency. Second, on a fixed player set, we characterize this class of solutions using monotonicity properties. Finally, towards a strategic foundation, we provide a non-cooperative implementation for these solutions which only differ in the probability of breakdown at a certain stage of the game. These characterizations discover fundamental differences as well as intriguing connections between marginalism and egalitarianism.     

Income inequality games

The paper explores different applications of the Shapley value for either inequality or poverty measures. We first investigate the problem of source decomposition of inequality measures, the so-called additive income sources inequality games, based on the Shapley value, introduced by Chantreuil and Trannoy (1999) and Shorrocks (1999). We show that multiplicative inequality games provide dual results compared with Chantreuil and Trannoy’s ones. We also investigate the case of multiplicative poverty games for which indices are non additively decomposable in order to capture contributions of sub-indices, which are multiplicatively connected with, as in the Sen-Shorrocks-Thon poverty index. We finally show, in the case of additive poverty indices, that the Shapley value may be equivalent to traditional methods of decomposition such as subgroup consistency and additive decomposition.     

The apex power measure for directed networks

In this paper we provide an axiomatization of the Shapley value restricted to the class of apex games using an equal loss property which states that the payoff of an apex player and a non-apex player decrease by the same amount if we make this particular non-apex player a null player. We also generalize this axiomatization to the class of games that can be obtained as sums of apex games. After discussing these axiomatizations we apply apex games and their Shapley values in measuring relational power in directed networks. We conclude by mentioning how these results can be adapted to give axiomatizations of the Banzhaf value. Received: 17 February 1999/Accepted: 2 October 2002     

Graph monotonic values

A disturbing feature of most of the solution concepts for TU games with incomplete communication is that payments of players may decrease when they activate a new link. That can be considered as a drawback which does not occur for the Myerson value (Math Oper Res 2:225–229, 1977) of superadditive games. The present article offers a new axiomatic characterization of the Myerson value. We show that the Myerson value is the unique solution for games with communication structures verifying a set of properties including monotonicity with respect to the graph and coinciding with the Shapley value when the communication is complete.     

A Matrix Approach to TU Games with Coalition and Communication Structures

The aim of this article is to present a technique to construct extensions of the Shapley value. Only basic matrix algebra is used. We concentrate on TU games with coalition structures and with communication structures. We define an efficient Aumann–Drèze value and an efficient Myerson value. We also define two families of values for TU games, the first being a convex combination of the efficient Aumann–Drèze value and of the Shapley value and the second a convex combination of the efficient Myerson value and of the Shapley value. We show that the Myerson value, the Aumann–Drèze value, the Shapley value and the four new solutions above are linked by a relationship of “similarity”.     

Mixed refinements of Shapley's saddles and weak tournaments

We investigate refinements of two solutions, the saddle and the weak saddle, defined by Shapley (1964) for two-player zero-sum games. Applied to weak tournaments, the first refinement, the mixed saddle, is unique and gives us a new solution, generally lying between the GETCHA and GOTCHA sets of Schwartz (1972, 1986). In the absence of ties, all three solutions reduce to the usual top cycle set. The second refinement, the weak mixed saddle, is not generally unique, but, in the absence of ties, it is unique and coincides with the minimal covering set. Received: 14 August 1998/Accepted: 12 November 1999     

THE CHALLENGE OF REVENUE SHARING WITH BUNDLED PRICING: AN APPLICATION TO MUSIC

Although bundling can substantially increase profits relative to standalone pricing, particularly for zero‐marginal‐cost information products, it has one major problem: bundling produces revenue that is not readily attributable to particular pieces of intellectual property, creating a revenue division problem. We evaluate several possible solutions using unique song valuation survey data. We find the Shapley value, a well‐motivated theoretical solution, is universally incentive compatible (all bundle elements fare better inside the bundle than under standalone pricing), but revenue‐sharing schemes feasible with readily available consumption data are not. Among feasible schemes, Ginsburgh and Zang's modified Shapley value performs best. (JEL C71, D79, L14)     

Core concepts for share vectors

A value mapping for cooperative games with transferable utilities is a mapping that assigns to every game a set of vectors each representing a distribution of the payoffs. A value mapping is efficient if to every game it assigns a set of vectors which components all sum up to the worth that can be obtained by all players cooperating together.? An approach to efficiently allocate the worth of the ‘grand coalition’ is using share mappings which assign to every game a set of share vectors being vectors which components sum up to one. Every component of a share vector is the corresponding players' share in the total payoff that is to be distributed among the players. In this paper we discuss a class of share mappings containing the (Shapley) share-core, the Banzhaf share-core and the Large Banzhaf share-core, and provide characterizations of this class of share mappings. Received: 9 August 1999/Accepted: 25 April 2000     

Satisficing in strategic environments: A theoretical approach and experimental evidence

The satisficing approach is generalized and applied to finite n-person games. We formally define the concept of satisficing and propose a theory that allows satisficing players to make “optimal” decisions without being equipped with any prior. We also review some experiments on strategic games illustrating and partly supporting our theoretical approach.     

Strategy-proofness versus efficiency on restricted domains of exchange economies

 Strategy-proofness has been shown to be a strong property, particularly on large domains of preferences. We therefore examine the existence of strategy-proof and efficient solutions on restricted, 2-person domains of exchange economies. On the class of 2-person exchange economies in which agents have homothetic, strictly convex preferences we show, as Zhou (1991) did for a larger domain, that such a solution is necessarily dictatorial. As this proof requires preferences exhibiting high degrees of complementarity, our search continues to a class of linear preferences. Even on this “small” domain, the same negative result holds. These two results are extended to many superdomains, including Zhou’s. Received: 9 June 1995/Accepted: 8 January 1996     

Game Theoretical Formulations of The Olson Problem*

Abstract This paper takes up one of the basic themes of Mancur Olson's Logic of Collective Action (Harvard University Press 1965). that is a group size as a cause of suboptimal provision of collective or public goods. A general framework is developed for classifying collective action situations involving public goods provisions. This framework focuses on the two characteristics: relations between contribution and provision, and rivalness or jointness in consumption of the collective goods. This framework distinguishes six types of collective actions, for each of which a game theoretical formulation is developed to obtain. models concerning social movements against (or for) new legislations, a petition for the recall of an official, a strike, lobbying, building a lighthouse, creation of a database, etc. These models, formulated either as an N-person chicken game or as an N-person prisoner's dilemma game. are examined with respect to how a group size affects non-cooperative equilibria and their Paretooptimality. There is no group size effect in the collective action situations formulated as an N-person chicken game, while large groups may suffer from suboptimal provision of the public goods in the collective action situations formulated as an N-person prisoner's dilemma game. Two types of the group size effect in N-person prisoner's dilemmas must be distinguished. In some cases. “no contribution” is the equilibrium regardless of the group size. but increase in the group size makes the equilibrium Pareto-deficient. In other cases, increase in the group size changes the equilibrium from the Pareto-efficient one with N contributors to the deficient one with no contributors.     

Lorenz undominated allocations for TU-games: The weighted Coalitional Lorenz Solutions

In this paper we introduce and study the w-Coalitional Lorenz Solutions to identify the similarities and differences between the prenucleolus and the Shapley value. The similarity is that they both use egalitarian criteria over coalitions. The two main differences are: the prenucleolus and the Shapley value use different egalitarian criteria, and they weight the coalitions differently when applying the criteria. Received: 27 October 2000/Accepted: 2 October 2001     

The minimal overlap rule revisited

This paper provides an analysis of the Minimal Overlap Rule, a solution for bankruptcy problems introduced by O’Neill (1982). We point out that this rule can be understood as a composition of Ibn Ezra’s proposal and the recommendation given by the Constrained Equal Loss Rule. Following an interpretation of bankruptcy problems in terms of TU games, we show that the Minimal Overlap Value is the unique solution for bankruptcy games which satisfies Anonymity and Core Transition Responsiveness.     

Inequality decomposition values: the trade-off between marginality and efficiency

This paper presents a general procedure for decomposing income inequality measures by income sources. The methods of decomposition proposed are based on the Shapley value and extensions of the Shapley value of transferable utility cooperative games. In particular, we find that Owen’s value can find an interesting application in this context.We show that the axiomatization by the potential of Hart and Mas-Colell remains valid in the presence of the domain restriction of inequality indices. We also examine the properties of these decomposition rules and perform a comparison with Shorrocks’ decomposition rule properties.     

A value for games with coalition structures

This paper presents an axiomatization of a value for games with coalition structures which is an alternative to the Owen Value. The motor of this new axiomatization is a consistency axiom based on an associated game, which is not a reduced game. The new value of an n-player unanimity game is the compound average of the new values of all the (n-1)-player unanimity games. The new value of a unanimity game allocates to bigger coalitions a larger share of the total wealth. Note that the Owen value allocates to all the coalitions the same share independently of their size.     

Egalitarian property for power indices

In this study, we introduce and examine the Egalitarian property for some power indices on the class of simple games. This property means that after intersecting a game with a symmetric or anonymous game the difference between the values of two comparable players does not increase. We prove that the Shapley–Shubik index, the absolute Banzhaf index, and the Johnston score satisfy this property. We also give counterexamples for Holler, Deegan–Packel, normalized Banzhaf and Johnston indices. We prove that the Egalitarian property is a stronger condition for efficient power indices than the Lorentz domination.     

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.