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.     

Networks and outside options

We introduce and characterize a component efficient value for TU games with a cooperation structure which in contrast to the Myerson (Math Oper Res 2:225–229, 1977) value accounts for outside options. It is based on the idea that the distribution of the worth within a component should be consistent with some connected graph which reflects the outside options of the component’s players.     

Famine,Food Insecurity,Poverty and Public Action

Book reviewed in this article: Hunger and Public Action. By Jean Drèze and Amartya Sen The Political Economy of Hunger. Edited by Jean Drèze and Amartya Sen.     

The consensus value: a new solution concept for cooperative games

To generalize the standard solution for 2-person TU games into n-person cases, this paper introduces a recursive two-sided negotiation process to establish cooperation between all players. This leads to a new solution concept for cooperative games: the consensus value. An explicit comparison with the Shapley value is provided, also at the axiomatic level. Moreover, a class of possible generalizations of the consensus value is introduced and axiomatized with the Shapley value at one end and the equal surplus solution at the other. Finally, we discuss a non-cooperative mechanism which implements the consensus value.     

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     

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.     

Measuring influence in command games

In the paper, we study a relation between command games proposed by Hu and Shapley and an influence model. We show that our framework of influence is more general than the framework of the command games. We define several influence functions which capture the command structure. These functions are compatible with the command games, in the sense that each commandable player for a coalition in the command game is a follower of the coalition under the command influence function. Some of the presented influence functions are equivalent to the command games, that is, they are compatible with the command games, and additionally each follower of a coalition under the command influence function is also a commandable player for that coalition in the command games. For some influence functions we define the equivalent command games. We show that not for all influence functions the compatible command games exist. Moreover, we propose a more general definition of the influence index and show that under some assumptions, some power indices, which can be used in the command games, coincide with some expressions of the weighted influence indices. Both the Shapley–Shubik index and the Banzhaf index are equal to a difference between the weighted influence indices under some influence functions, and the only difference between these two power indices lies in the weights for the influence indices. An example of the Confucian model of society is broadly examined. The authors wish to gratefully thank two anonymous referees for useful suggestions concerning this paper.     

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     

An extension of Bowen’s dynamic voting rule to many dimensions

 We ask in this paper about the effect on social decisions of limiting the size of changes that voters may propose each time in an otherwise standard dynamic social choice model. The voting rule we study can be seen as an extension of Bowen’s dynamic “majority voting” rule, and is closely related to the dynamic procedures for public good allocation in the literature (Drèze and de la Vallée Poussin 1971; Malinvaud 1971; Laffont and Maskin 1983; Chander 1993). Under general assumptions we prove existence and Pareto efficiency of equilibrium, and show that our rule motivates voters not to misrepresent preferences (more precisely, the rule is Strongly Locally Individually Incentive Compatible). Under Euclidean preferences we find that electoral cycles do not arise (i.e., the rule is convergent), that there is a unique equilibrium, and that the equilibrium coincides with the solution to an old problem of geometry, first addressed by Fermat, Torricelli, and Cavallieri. Received: 20 September 1994/Accepted: 6 August 1996     

Generic efficiency and collusion-proofness in exchange economies

The purpose of this paper is to study the kind of efficient allocations that can be achieved in exchange economies with asymmetric information, by means of a decentralized mechanism robust to coalitional, strategic deviations. To this end, we define a new strategic equilibrium concept – called strong collusion-proof contract – designed to characterize stable communication agreements in games with differential information against non-binding, self-enforcing and incentive compatible deviations by coalitions. We then construct a strategic market mechanism which, for quasi-linear economies, is such that its strong collusion-proof contracts generically induce the incentive compatible interim efficient allocations. Moreover, we show that these allocations can be achieved by strong collusion-proof contracts. We show that the internally consistent extension of the strong collusion-proof contracts generically yields the same set of efficient allocations. Received: 22 January 2001/Accepted: 15 April 2002 RID="*" ID="*"  This author was working at CORE when this paper was written. We wish to thank Claude d'Aspremont, David Martimort, Jean-Fran?ois Mertens and Heracles Polemarchakis for helpful comments on an earlier version. The usual disclaimer applies.     

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.     

Monotone-path Dutta-Ray solutions on convex games

It is well known that on the domain of convex games, the Dutta-Ray solution satisfies many desirable properties such as population-monotonicity, max consistency, converse max consistency, and conditional self-consistency. In this paper, we define a family of possibly non-symmetric and non-homogeneous generalizations of the Dutta-Ray solution, which we refer to as “monotone-path Dutta-Ray solutions.” We show that above four properties are preserved by our generalizations. Received: 25 August 2000/Accepted: 11 September 2001     

Expected fair allocation in farsighted network formation

I consider situations in which a group of players extracts a value if they organise themselves in different network structures, and I define a solution concept to describe the decentralised decision that determines the network formation process and the allocation of the value. I demonstrate that there is a solution concept satisfying discounted expected versions of pairwise stability (Jackson and Wolinsky J Econ Theory 71:44–74, 1996) and fairness (Myerson Math Oper Res 2:225–229, 1977a) jointly with the requirement that the allocation rule be component efficient if the players’ discount factor is sufficiently low.     

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.     

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.     

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     

Social and Psychological Challenges of Poker

Poker is a competitive, social game of skill and luck, which presents players with numerous challenging strategic and interpersonal decisions. The adaptation of poker into a game played over the internet provides the unprecedented opportunity to quantitatively analyze extremely large numbers of hands and players. This paper analyzes roughly twenty-seven million hands played online in small-stakes, medium-stakes and high-stakes games. Using PokerTracker software, statistics are generated to (a) gauge the types of strategies utilized by players (i.e. the ‘strategic demography’) at each level and (b) examine the various payoffs associated with different strategies at varying levels of play. The results show that competitive edges attenuate as one moves up levels, and tight-aggressive strategies––which tend to be the most remunerative––become more prevalent. Further, payoffs for different combinations of cards, varies between levels, showing how strategic payoffs are derived from competitive interactions. Smaller-stakes players also have more difficulty appropriately weighting incentive structures with frequent small gains and occasional large losses. Consequently, the relationship between winning a large proportion of hands and profitability is negative, and is strongest in small-stakes games. These variations reveal a meta-game of rationality and psychology which underlies the card game. Adopting risk-neutrality to maximize expected value, aggression and appropriate mental accounting, are cognitive burdens on players, and underpin the rationality work––reconfiguring of personal preferences and goals––players engage into be competitive, and maximize their winning and profit chances.     

Sequential entry in many-to-one matching markets

We study sequential bargaining in many-to-one matching markets. We show that there is an advantage to entering late in the market, and that the last agent to enter the market will receive his or her best partner in a stable matching, extending the results of Blum and Rothblum (J Econ Theory 103(2):429–443, 2002) and Cechlárová (Randomized matching mechanism revisited. Mimeo, Safarik University, 2002) for the marriage model. We also discuss the relation between sequential bargaining and a possible alternative formulation based on the NTU Shapley value. We thank Peter Biró and Utku ünver for their advice on the related literature. We also thank an anonymous referee for comments.     

Random conjugates of bankruptcy rules

This article introduces and analyzes random conjugates of bankruptcy rules. A random conjugate is a rule which is derived from the definition of the underlying rule for two-claimant problems. For example, the random conjugate of the Aumann–Maschler rule yields an extension of concede-and-divide: the basic solution for bankruptcy problems with two claimants. Using the concept of random conjugates an alternative characterization of the proportional rule is provided. It turns out that the procedural definition of a random conjugate extends several of the properties of the underlying rule for two-claimant problems to the general domain of problems with an arbitrary number of claimants.     

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.