The axiomatic approach to three values in games with coalition structure (original) (raw)

2009, European Journal of Operational Research

https://doi.org/10.1016/J.EJOR.2010.05.014

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Abstract

We study three values for transferable utility games with coalition structure, including the Owen coalitional value and two weighted versions with weights given by the size of the coalitions. We provide three axiomatic characterizations using the properties of Efficiency, Linearity, Independence of Null Coalitions, and Coordination, with two versions of Balanced Contributions inside a Coalition and Weighted Sharing in Unanimity Games, respectively.

Non-Transferable Utility Coalitional Games via Mixed-Integer Linear Constraints

Journal of Artificial Intelligence Research, 2010

Coalitional games serve the purpose of modeling payoff distribution problems in scenar- ios where agents can collaborate by forming coalitions in order to obtain higher worths than by acting in isolation. In the classical Transferable Utility (TU) setting, coalition worths can be freely distributed amongst agents. However, in several application scenarios, this is not the case and the Non-Transferable Utility

On the logic of coalitional games

Proceedings of the fifth international joint conference on Autonomous agents and multiagent systems - AAMAS '06, 2006

We develop a logic for representing and reasoning about coalitional games without transferable payoffs. Although a number of logics of cooperation have been proposed over the past decade (notably Coalition Logic [14] and Alternating-time Temporal Logic [1]), these logics focused primarily on the issue of strategic cooperative ability -what states a coalition can effectively enforce -and have tended to ignore the essential issue of the preferences that agents have over such states; in addition, the connection between such logics and coalitional games, in the sense of cooperative game theory, is left implicit. The Coalitional Game Logic (CGL) that we develop in this paper differs from such previous logics in two important respects. First, CGL includes operators that make it directly possible to represent an agent's preferences over outcomes. Second, we interpret formulae of CGL directly with respect to coalitional games without transferable payoff, thereby establishing an explicit link between formulae of the logic and properties of coalitional games. We show that these coalitional games cannot be seen directly as models for Coalition Logic. We give a complete axiomatization of CGL, prove that it is expressively complete with respect to coalitional games without transferable payoff, show that the satisfiability problem for the logic is NP-complete, and to illustrate its use, we show how the logic can be used to characterise axiomatically a number of well-known solution concepts for coalitional games, including for example non-emptiness of the core.

Preserving coalitional rationality for non-balanced games

International Journal of Game Theory, 2014

In cooperative games, the core is one of the most popular solution concept since it ensures coalitional rationality. For non-balanced games however, the core is empty, and other solution concepts have to be found. We propose the use of general solutions, that is, to distribute the total worth of the game among groups rather than among individuals. In particular, the k-additive core proposed by Grabisch and Miranda is a general solution preserving coalitional rationality which distributes among coalitions of size at most k, and is never empty for k ≥ 2. The extended core of Bejan and Gomez can also be viewed as a general solution, since it implies to give an amount to the grand coalition. The k-additive core being an unbounded set and therefore difficult to use in practice, we propose a subset of it called the minimal bargaining set. The idea is to select elements of the k-additive core mimimizing the total amount given to coalitions of size greater than 1. Thus the minimum bargaining set naturally reduces to the core for balanced games. We study this set, giving properties and axiomatizations, as well as its relation to the extended core of Bejan and Gomez. We introduce also the notion of unstable coalition, and show how to find them using the minimum bargaining set. Lastly, we give a method of computing the minimum bargaining set.

The coalitional value in …nite-type continuum games

The coalitional value [Owen, Values of games with a priori unions. In: Hein R, Moeschlin O (Eds), Essays in Mathematical Economics and Game Theory. Springer Verlag, 1977] is de…ned for the class of continuos games with a …nite type of players. A formula for its computation is provided jointly with an axiomatic characterization of it. The properties used are a natural extension in this setting of the properties used in the characterization of the Owen's coalitional value for games with a …nite set of players.

A solution for bargaining problems with coalition structure

Mathematical Social Sciences, 2007

In this paper we study the restriction, to the class of bargaining problems with coalition structure, of several values which have been proposed on the class of non-transferable utility games with coalition structure. We prove that all of them coincide with the solution independently studied in Chae and Heidhues [Chae, S., Heidhues, P., 2004. A group bargaining solution. Mathematical Social Sciences 48, 37–53] and Vidal-Puga [Vidal-Puga, J.J., 2005a. A bargaining approach to the Owen value and the Nash solution with coalition structure. Economic Theory 25, 679–701]. Several axiomatic characterizations and two non-cooperative mechanisms are proposed.

Axiomatization of values of cooperative games using a fairness property

Applicationes Mathematicae, 2005

We propose new systems of axioms which characterize four types of values of cooperative games: the Banzhaf value, the Deegan-Packel value, the least square prenucleolus and the least square nucleolus. The common element used in these axiomatizations is a fairness property. It requires that if to a cooperative game we add another game in which two given players are symmetric, then their payoffs change by the same amount. In our analysis we will use an idea applied by R. van den Brink (2001) to obtain an axiomatic characterization of the Shapley value.

A value for cooperative games with a coalition structure!

Discussion Papers in Economic Behaviour, 2011

A value for games with a coalition structure is introduced, where the rules guiding the cooperation among the members of the same coalition are di¤erent 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 [Shapley, 1953] is therefore used to compute the aggregate payo¤s of the coalitions, and the Solidarity value [Nowak and Radzik, 1994] to obtain the payo¤s of the players inside each coalition.

A bargaining approach to the Owen value and the Nash solution with coalition structure

Economic Theory, 2005

The mechanism by Hart and Mas-Colell (1996) for non-transferable utility (NTU) games is generalized so that a coalition structure among players is taken into account. The new mechanism yields the Owen value for transferable utility (TU) games with coalition structure as well as the consistent value (Maschler and Owen 1989, 1992) for NTU games with trivial coalition structure. Furthermore, we obtain a solution for pure bargaining problems with coalition structure which generalizes the Nash (1950) bargaining solution.

Solutions for Games with General Coalitional Structure and Choice Sets

SSRN Electronic Journal, 2013

In this paper we introduce the concept of quasi-building set that may underlie the coalitional structure of a cooperative game with restricted communication between the players. Each feasible coalition, including the set of all players, contains a nonempty subset called the choice set of the coalition. Only players that are in the choice set of a coalition are able to join to feasible subcoalitions to form the coalition and to obtain a marginal contribution. We demonstrate that all restricted communication systems that have been studied in the literature take the form of a quasi-building set for an appropriate set system and choice set. Every quasi-building set determines a nonempty collection of maximal strictly nested sets and each such set induces a rooted tree satisfying that every node of the tree is a player that is in the choice set of the feasible coalition that consists of himself and all his successors in the tree. Each tree corresponds to a marginal vector of the underlying game at which each player gets as payoff his marginal contribution when he joins his successors in the tree. As solution concept of a quasi-building set game we propose the average marginal vector (AMV) value, being the average of the marginal vectors that correspond to the trees induced by all maximal strictly nested sets of the quasi-building set. Properties of this solution are also studied. To establish core stability we introduce appropriate convexity conditions of the game with respect to the underlying quasi-building set. For some specifications of quasi-building sets, the AMV-value coincides with solutions known in the literature, for example, for building set games the solution coincides with the gravity center solution and the Shapley value recently defined for this class. For graph games it therefore differs from the well-known Myerson value. For a full communication system the solution coincides with the classical Shapley value.

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References (43)

  1. Albizuri M.J. (2008) Axiomatizations of the Owen value without effi- ciency. Mathematical Social Sciencies 55, 78-89.
  2. Albizuri M.J. and Zarzuelo J.M. (2004) On coalitional semivalues. Games and Economic Behavior 49, 221-243.
  3. Alonso-Meijide J.M., Carreras F. and Puente M.A. (2007) Axiomatic characterizations of the symmetric coalitional binomial semivalues. Dis- crete Applied Mathematics 155(17), 2282-2293.
  4. Alonso-Meijide J.M. and Fiestras-Janeiro M.G. (2002) Modification of the Banzhaf value for games with a coalition structure. Annals of Oper- ations Research 109, 213-227.
  5. Amer R. and Carreras F. (1995) Cooperation indices and coalitional value. TOP 3(1), 117-135.
  6. Amer R., Carreras F. and Giménez J.M. (2002) The modified Banzhaf value for games with a coalition structure: an axiomatic characteriza- tion. Mathematical Social Sciencies 43, 45-54.
  7. Aumann R.J. and Drèze J.H. (1974) Cooperative games with coalition structure. International Journal of Game Theory 3, 217-237.
  8. Banzhaf J.F. (1965) Weighted voting doesn't work: A mathematical anal- ysis. Rutgers Law Review 19, 317-343.
  9. Bergantiños G., Casas B., Fiestras-Janeiro G. and Vidal-Puga J. (2007) A focal-point solution for bargaining problems with coalition structure. Mathematical Social Sciences 54(1), 35-58.
  10. Bergantiños G. and Vidal-Puga J. (2005) The consistent coalitional value. Mathematics of Operations Research 30(4), 832-851.
  11. Carreras F. and Puente M.A. (2006) A parametric family of mixed coali- tional values. Recent Advances in Optimization, Lecture Notes in Eco- nomics and Mathematical Systems. Ed. by A. Seeger. Springer, 323-339.
  12. Calvo E., Lasaga J. and Winter E. (1996) The principle of balanced con- tributions and hierarchies of cooperation. Mathematical Social Sciences 31, 171-182.
  13. Calvo E. and Santos J.C. (2000) A value for multichoice games. Math- ematical Social Sciences 40, 341-354.
  14. Calvo E. and Santos J.C. (2006) The serial property and restricted bal- anced contributions in discrete cost sharing problems. Top 14(2), 343- 353.
  15. Chae S. and Heidhues P. (2004) A group bargaining solution. Mathemat- ical Social Sciences 48(1), 37-53.
  16. Deegan J. and Packel E.W. (1978) A new index of power for simple n-person games. International Journal of Game Theory 7, 113-123.
  17. Hamiache G. (1999) A new axiomatization of the Owen value for games with coalition structures. Mathematical Social Sciencies 37, 281-305.
  18. Hamiache G. (2001) The Owen value values friendship. International Journal of Game Theory 29, 517-532.
  19. Harsanyi J.C. (1977) Rational behavior and bargaining equilibrium in games and social situations. Cambridge University Press.
  20. Hart S. and Kurz M. (1983) Endogenous formation of coalitions. Econo- metrica 51(4), 1047-1064.
  21. Kalai E. and Samet D. (1987) On weighted Shapley values. International Journal of Game Theory 16, 205-222.
  22. Kalai E. and Samet D. (1988) Weighted Shapley values. The Shapley value: Essays in honor of L. Shapley. Ed. by A. Roth. Cambridge. Cam- bridge University Press, 83-99.
  23. Kalandrakis (2006) Proposal rights and political power. American Jour- nal of Political Science 50(2), 441-448.
  24. Levy A. and McLean R.P. (1989) Weighted coalition structure values. Games and Economic Behavior 1, 234-249.
  25. Malawski M. (2004) "Counting" power indices for games with a priori unions. Theory and Decision 56, 125-140.
  26. Młodak A. (2003) Three additive solutions of cooperative games with a priori unions. Applicationes Mathematicae 30, 69-87.
  27. Moulin H. (1995) On Additive Methods to Share Joint Costs. The Japanese Economic Review 46, 303-332.
  28. Myerson R.B. (1977) Graphs and cooperation in games. Mathematics of Operations Research 2, 225-229.
  29. Myerson R.B. (1980) Conference structures and fair allocation rules. International Journal of Game Theory 9, 169-182.
  30. Owen G. (1975) Multilinear extensions and the Banzhaf value. Naval Research Logistics Quarterly 22, 741-750.
  31. Owen G. (1977) Values of games with a priori unions. Essays in math- ematical economics and game theory. Ed. by R. Henn R. and O. Moeschlin. Berlin. Springer-Verlag, 76-88.
  32. Owen G. (1978) Characterization of the Banzhaf-Coleman index. SIAM Journal on Applied Mathematics 35, 315-327.
  33. Peleg B. (1989) Introduction to the theory of cooperative games (Chapter 8). The Shapley value. RM 88 Center for Research in Mathematical Economics and Game Theory. The Hebrew University. Jerusalem. Israel.
  34. Puente M.A. (2000) Aportaciones a la representabilidad de juegos sim- ples y al cálculo de soluciones de esta clase de juegos (in Spanish). Ph.D. Thesis. Polytechnic University of Catalonia, Spain.
  35. Ruiz L.M., Valenciano F. and Zarzuelo J.M. (1996) The least square prenucleolus and the least square nucleolus. Two values for TU games based on the excess vector. International Journal of Game Theory 25, 113-134.
  36. Shapley L.S. (1953a) Additive and non-additive set functions. Ph.D. Thesis. Princeton University.
  37. Shapley L.S. (1953b) A value for n-person games. Contributions to the Theory of Games II. Ed. by H.W. Kuhn and A.W. Tucker. Princeton NJ. Princeton University Press, 307-317.
  38. Vázquez-Brage M., Van den Nouweland A. and García-Jurado I. (1997) Owen's coalitional value and aircraft landing fees. Mathematical Social Sciences 34, 273-286.
  39. Vidal-Puga J. (2006) The Harsanyi paradox and the "right to talk" in bargaining among coalitions. EconWPA Working paper number 0501005. Available at http://ideas.repec.org/p/wpa/wuwpga/0501005.html.
  40. Winter E. (1989) A value for cooperative games with level structure of cooperation. International Journal or Game Theory 18, 227-240.
  41. Winter E. (1992) The consistency and potential for values of games with coalition structure. Games and Economic Behavior 4, 132-144.
  42. Young H.P. (1985) Monotonic solutions of cooperative games. Interna- tional Journal of Game Theory 14, 65-72.
  43. Zhang X. (1995) The pure bargaining problem among coalitions. Asia- Pacific Journal of Operational Research 12, 1-15.

On non-transferable utility games with coalition structure

International Journal of Game Theory, 1991

We introduce a solution function for Non-transferable Utility (NTU) games when prior coalition structure isgiven. This solution function generalizes both the Harsanyi solution function for NTU games and the Owen solution for TU games with coalition structure.

Solidarity induced by group contributions: the MI$$^k$$-value for transferable utility games

Operational Research

The most popular values in cooperative games with transferable utilities are perhaps the Shapley and the Shapley like values which are based on the notion of players' marginal productivity. The equal division rule on the other hand, is based on egalitarianism where resource is equally divided among players, no matter how productive they are. However none of these values explicitly discuss players' multilateral interactions with peers in deciding to form coalitions and generate worths. In this paper we study the effect of multilateral interactions of a player that accounts for her contributions with her peers not only at an individual level but also on a group level. Based on this idea, we propose a value called the MI k-value and characterize it by the axioms of linearity, anonymity, efficiency and a new axiom: the axiom of MN k-player. An MN kplayer is one whose average marginal contribution due to her multilateral interactions upto level k is zero and can be seen as a generalization of the standard null player axiom of the Shapley value. We have shown that the MI k-value on a variable player set is asymptotically close to the equal division rule. Thus our value realizes solidarity among players by incorporating both their individual and group contributions.

An Axiomatization of the Non-Transferable Utility Value

Econometrica, 1985

M. Maschler, B. Peleg, and M. Perles are gratefully acknowledged. 2. I.e., games with side payments, representable by a coalitional worth (''characteristic'') function. 3. See the references of [2]. 4. The random order expected contribution formula for the TU value, and the maximum product formula for the Nash Bargaining Problem.

On some decisive players for linear efficient and symmetric values in cooperative games with transferable utility

2018

The main goal of the paper is to shed light on economic allocations issues, in particular by focusing on individuals who receive nothing (that is an amount of zero allocation or payoff). It is worth noting that such individuals may be considered, in some contexts, as poor or socially excluded. To this end, our study relies on the notion of cooperative games with transferable utility and the Linear Efficient and Symmetric values (called LES values) are considered as allocation rules. Null players in Shapley sense are extensively studied ; two broader classes of null players are introduced. The analysis is facilitated by the help of a parametric representation of LES values. It is clearly shown that the control of what a LES value assigns as payoffs to null players gives significant information about the characterization of the value. Several axiomatic characterizations of subclasses of LES values are provided using our approach.

A comparative axiomatic characterization of the Banzhaf–Owen coalitional value

Decision Support Systems, 2007

A compact axiomatic characterization of the modified Banzhaf value for games with a coalition structure (Banzhaf-Owen value, for short) is provided. The axiomatic system used here can be compared with parallel axiomatizations of other coalitional values such as the Owen value or the Alonso-Fiestras value, thus giving arguments to defend the use of one of them that will depend on the context where they are to be applied. proposed by Penrose [36] and Coleman [17]) that gave rise to a Banzhaf value β on all cooperative games first defined by Owen [30]. Many axiomatic characterizations of one, the other or both values may be found in the literature (see, e.g., Owen [32], Dubey and Shapley [18], Young [45], Lehrer [26], Straffin [40], Amer and Carreras [7], Nowak [28] or Laruelle and Valenciano [24]). A most interesting one was stated by Feltkamp [20], who gave parallel characterizations of the Shapley and Banzhaf values that enhance the similarities and differences between them. Indeed, only one property distinguishes these values: efficiency for the Shapley value versus total power for the Banzhaf value.

The value in games with restricted cooperation

2015

We consider cooperative games in which the cooperation among players is restricted by a set system, which outlines the set of feasible coalitions that actually can be formed by players in the game. In our setting, the structure of this set system is completely free, and the only restriction is that the empty set belongs to it. An extension of the Shapley value is provided as the sum of the dividends that players obtain in the game. In this general setting, we offer two axiomatic characterizations for the value: one by means of component efficiency and fairness, and the other one with efficiency and balanced contributions.

The consistent coalitional value for hyperplane games

We introduce a value for hyperplane games with coalition structure. This value coincides with the consistent value for trivial coalition structures, and with the Owen value for transferable utility games with coalition structure. Furthermore, we present two characterizations: the …rst one using a consistency property and the second one using properties of balanced contributions. We thank David Perez-Castrillo for helpful comments. Financial support from the Ministerio de Ciencia y Tecnologia and FEDER through grant BEC2002-04102-C02-01 and Xunta de Galicia through grant PGIDIT03PXIC30002PN is gratefully aknowledged.

An Almost Ideal Sharing Scheme for Coalition Games with Externalities

SSRN Electronic Journal, 2000

We propose a class of sharing schemes for the distribution of the gains from cooperation for coalition games with externalities. In the context of the partition function, it is shown that any member of this class of sharing schemes leads to the same set of stable coalitions in the sense of d' Aspremont et al. (1983). These schemes are "almost ideal" in that they stabilize these coalitions which generate the highest global welfare among the set of "potentially stable coalitions". Our sharing scheme is particularly powerful for economic problems that are characterized by positive externalities from coalition formation and which therefore are likely to suffer from severe free-riding.

Multilateral Value for TU Cooperative Games

A value of a TU Cooperative game represents an assessment by a player of her gains for participating in a coalition. One of the most important values in the literature of TU games is the Shapley value [18]. It is indeed an aggregation of the marginal contributions of a player based on her bilateral interactions. In this paper we introduce a new value for TU Cooperative games. The notion of multilateral interaction of a player is proposed that accounts not only for the player's own inclusion or exclusion in a coalition as considered in computing the Shapley value but also for her influence on her peers in their decision of joining or leaving the coalition together. We characterize this value by the axioms of linearity, anonymity, efficiency and a new axiom: the axiom of parasite player. A parasite player extracts worths of other players. Our model makes her role less significant in presence of multilateral interactions.

E ciency in Coalition Games with Externalities

2006

A natural,extension,of superadditivity,is not su¢ cient to imply that the grand,coalition is e¢ cient when,externalities are present. We provide a con- dition –analogous,to convexity–that,is su¢ cient for the grand coalition to be e¢ cient and show,that this also implies that the (appropriately de…ned) core is nonempty. Moreover, we propose a mechanism which implements the most e¢ cient partition for all coalition formation,games,and characterize the payo¤ division of the mechanism. JEL Classi…cation Numbers: C71, C72, D62 Keywords: Coalition formation, externalities, partition function games, Shapley value, implementation.

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THE SHAPLEY-SOLIDARITY VALUE FOR GAMES WITH A COALITION STRUCTURE

International Game Theory Review, 2013

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.