Research in Game theory                            



   Working Papers

 The coalitional value in finite-type continuum games  (2009)

  Abstract: 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 defined for the class of continuos games with a finite 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 finite set of players.

   PDF document

Weighted solidarity values (2013) (with Esther Gutiérrez-López)

 Abstract: We present a noncooperative bargaining protocol among n players, applied to the setting of cooperative games in coalitional form with transferable utility. In this model, players are chosen randomly to make proposals until one is accepted unanimously, and after each proposal rejection, the probability that players leave the game increases. If after a rejection, some players withdraw the bargaining, the remaining players continue the process. We define a new family of values, called the weighted solidarity values, and we show that these values arise as the associated equilibrium payoffs of this bargaining protocol. In these values players have an altruistic behavior between them as the null player property is not satisfied.

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Published Papers 


 Redistribution of tax resources: a cooperative game theory approach (2021)

 Abstract: We consider the problem of how to distribute public expenditure among the different regions of an economic entity after all taxes have been collected. Typical examples are: the regions that make up a country, the states of a federal country, or the countries of a confederation of countries. We model the problem as a cooperative game in coalitional form, called the tax game. This game estimates the fiscal resources collected in each region, or coalition of regions, by differentiating between what comes from economic activity within each region and what comes from trade with the other regions. This methodology provides a measure of the disagreement within a region, or coalitions of regions, with respect to the budget received. Similarly, the stability of a budget allocation can be inferred by its situation within the core of the corresponding tax game. We consider the Spanish case as an example and show that the current regional financial system has a moderate degree of instability. We introduce two budget allocation rules, both borrowed from the cooperative games literature: the balanced allocation, which coincides with the nucleolus and with the Shapley value of the tax game, and the weighted balanced allocation, which coincides with the weighted Shapley value. We compare both budget allocation rules with the current Spanish financial system.

    SERIEs Vol. 12, p. 633-686. 

TaxFederalism.xlsx (2019) Excel file associated to the analysis of Spanish Fiscal Balances (SCPT). Years 20111-2014  Excel Document  

The Equal Collective Gains Value in Cooperative Games (2022) (with Esther Gutiérrez López)

    Abstract: The property of equal collective gains means that each player should obtain the same benefit from the contributions of the other players in the game. We show that this property jointly with efficiency characterize a new value, called the equal collective gains value. 

    International Journal of Game Theory  Vol. 51 (1), p. 249-278.

 Recursive and Bargaining values  (2021) (with Esther Gutiérrez-López)

 Abstract: We introduce two families of values for TU-games: the recursive and bargaining values. Bargaining values are obtained as the equilibrium payoffs of the symmetric non-cooperative bargaining game proposed by Hart and Mas-Colell (1996). We show that bargaining values have a recursive structure in their definition, and we call this property recursiveness. All efficient, linear, and symmetric values that satisfy recursiveness are called recursive values. We generalize the notions of potential, and balanced contributions property, to characterize the family of recursive values. Finally, we show that if a time discount factor is considered into the bargaining model, every bargaining value has its corresponding discounted bargaining value.

     Mathematical Social Sciences Vol. 113, p. 97-106. 

 A strategic approach for the discounted Shapley values  (2016) (with Esther Gutiérrez-López)

 Abstract: The family of discounted Shapley values is analyzed for cooperative games in coalitional form. We consider the bargaining protocol of the alternating random proposer introduced in Hart and Mas-Colell (Econometrica 64:357-380, 1996). We demonstrate that the discounted Shapley values arise as the expected payoffs associated with the bargaining equilibria when a time discount factor is considered. In a second model, we replace the time cost with the probability that the game ends without agreements. This model also implements these values in transferable utility games, moreover, the model implements the α-consistent values in the nontransferable utility setting.

 Theory and Decision Vol. 80 (2), p. 271-293. PDF Document

 Axiomatic characterization of the weighted solidarity values (2014) (with Esther Gutiérrez-López)

Abstract: We define and characterize the class of all weighted solidarity values. Our first characterization employs the classical axioms determining the solidarity value (except symmetry), that is, efficiency, additivity and the A-null player axiom, and two new axioms called proportionality and strong individual rationality. In our second axiomatization, the additivity and the A-null player axioms are replaced by a new axiom called average marginality.

 Mathematical Social Sciences Vol 71 (2014), p. 6-11. PDF Document

The Shapley-Solidarity value for games with a coalition structure  (2013) (with Esther Gutiérrez-López) 

 Abstract: 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.

 International Game Theory Review Vol. 15 (1) (2013). PDF Document

 Dynamic Models of International Environmental Agreements: A Differential Game Approach  (2012) (with Santiago Rubio)

Abstract: This article provides a survey of dynamic models of international environmental agreements (IEAs). The focus is on environmental problems that are caused by a stock pollutant as are the cases of the acid rain and climate change. For this reason, the survey only reviews the literature that utilizes dynamic state-space games to analyze the formation of international agreements to control pollution. The survey considers both the cooperative approach and the noncooperative approach. In the case of the latter, the survey distinguishes between the models that assume binding agreements and those that assume the contrary. An evaluation of the state of the art is presented in the conclusions along with suggestions for future research.

 International Review of Environmental and Resource Economics Vol. 6 (4) (2012), p. 289-339. PDF Document

 Solidarity in games with a coalition structure (with Esther Gutiérrez)

 Abstract: A new axiomatic characterization of the two-step Shapley value (Kamijo (2007)) is presented based in a solidarity principle for the members of a union: when the game changes due to the addition or deletion of players outside the union, all members of the union will share the same gains/losses

 Mathematical Social Sciences Vol. 60 (2010), p. 196-203. PDF_document

 Random Marginal and Random Removal Values 

 Abstract: We propose two variations of the non-cooperative bargaining model for games in coalitional form, introduced by Hart and Mas-Colell (1996a). These strategic games implement, in the limit, two new NTU-values: The random marginal and the random removal values. The main characteristic of these proposals is that they always select a unique payoff allocation in NTU-games. The random marginal value coincides with the Consistent NTU-value (Maschler and Owen, 1989) for hyperplane games, and with the Shapley value for TU games (Shapley, 1953). The random removal coincides with the solidarity value (Novak and Radzik, 1994) in TU-games. In large games we show that, in the special class of market games, the random marginal value coincides with the Shapley NTU-value (Shapley,1969), and that the random removal value coincides with the equal split value.

  International Journal of Game Theory Vol. 37 (4). p. 533-564. 2008   PDF Document  Erratum PDFDocument

The Serial Principle and Restricted Balanced Contributions in Discrete Cost Sharing Problems (with Juan Carlos Santos)

 Abstract: We show that the axioms of Efficiency, Serial Principle and Restricted Balanced Contributions, characterize the Moulin's rule (Moulin, 1995) in discrete cost allocation problems.

     TOP Vol. 14 (2). p. 343-353. April 2006. PDF Document


Bargaining with ordinal and cardinal players (with Hans Peters)

 Abstract: We consider bargaining problems with at least one cardinal player and with ordinal players, and provide a complete description of utility invariant solutions of such problems for two players. For the n-person case we provide a procedure that: (i) returns a given cardinal solution if there are only cardinal players; (ii) is based on the ordinal solution for gradual bargaining problems, introduced by O'Neill et al. (2001), for the ordinal players. Finally, we introduce the so-called cardinal concession solution as another example of a utility invariant solution.

 Games & Economic Behavior Vol. 52 (1). p 20-33. July 2005.

 A Value for Mixed Action-Set Games (with Juan Carlos Santos)

 Abstract: 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 finite number of activity levels in which they can join a coalition, and continuous players that possess a continuum of levels.

 International Journal of Game Theory Vol. 30 (1). p 61-78. 2001.

 Prices in Mixed Cost Allocation Problems (with Juan Carlos Santos)

Abstract: We consider mixed cost allocation problems, i.e., joint cost problems that involve two types of heterogeneous outputs, divisible and indivisible. The Aumann-Shapley price mechanism is extended to this setting. We also present a set of properties which characterize this cost sharing rule.

 Games & Economic Behavior Vol. 37 (2). p 243-58. November 2001.

 Dynamics and Axiomatics of the Equal Area Bargaining Solution (with Hans Peters)

Abstract: We present an alternative formulation of the two-person equal area bargaining solution based on a dynamical process describing the disagreement point set. This alternative formulation provides an interpretation of the idea of equal concessions. Furthermore, it leads to an axiomatic characterization of the solution.

 International Journal of Game Theory Vol. 29 (1). p 81-92. February 2000.

 Replication Invariance on NTU Games ( with Inaki Garcia, Jose M  Zarzuelo)

 Abstract: Two concepts of replication (conflictual and non-conflictual) are extended from the class of pure bargaining games to the class of NTU games. The behavior of the Harsanyi, Shapley NTU, Egalitarian and Maschler-Owen solutions of the replica games is compared with that of the Nash and Egalitarian solutions in pure bargaining games.

 International Journal of Game Theory Vol. 29 (4). p 473-86. 2000.

 Weighted Weak Semivalues (with Juan Carlos Santos)

 Abstract: We introduce two new value solutions: weak semivalues and weighted weak semivalues. They are subfamilies of probabilistic values, and they appear by adding the axioms of balanced contributions and weighted balanced contributions respectively. We show that the effect of the introduction of these axioms is the appearance of consistency in the beliefs of players about the game.

 International Journal of Game Theory Vol. 29 (1). p 1-9. February 2000.

 The Multichoice Consistent Value (with Esther Gutierrez, Juan Carlos Santos)

 Abstract: We consider multichoice NTU games, i.e., cooperative NTU games in which players can participate in the game with several levels of activity. For these games, we define and characterize axiomatically the multichoice consistent value, which is a generalization of the consistent NTU value for NTU games and of the multichoice value for multichoice TU games. Moreover, we show that this value coincides with the consistent NTU value of a replicated NTU game and we provide a probabilistic interpretation.

 International Journal of Game Theory Vol. 29 (2). p 177-88. July 2000.

 A Value for Multichoice Games (with Juan Carlos Santos)

Abstract: A multichoice game is a generalization of a cooperative TU game in which each player has several activity levels. We study the solution for these games proposed by Van Den Nouweland et al. (1995). We show that this solution applied to the discrete cost sharing model coincides with the Aumann-Shapley method proposed by Moulin (1995). Also, we show that the Aumann-Shapley value for continuum games can be obtained as the limit of multichoice values for admissible convergence sequences of multichoice games. Finally, we characterize this solution by using the axioms of balanced contributions and efficiency.

 Mathematical Social Sciences Vol. 40 (3). p 341-54. November 2000.

 Scoring Rules: A Cooperative Game-Theoretic Approach (with Esther Gutierrez, Inaki Garcia)

 Abstract: In this work we define the game of the alternatives for each preference profile, and establish relations between scoring rules and cooperative solution concepts for that game, such as the family of semivalues and the family of least square values.

 Social Choice & Welfare Vol. 16 (3). p 491-512. May 1999.

 Values of games with probabilistic graphs (vith J Javier Lasaga , Anne van den Nouweland )

 Abstract: In this paper we consider games with probabilistic graphs. The model we develop is an extension of the model of games with communication restrictions by Myerson (1977). In the Myerson model each pair of players is joined by a link in the graph if and only if these two players can communicate directly. The current paper considers a more general setting in which each pair of players has some probability of direct communication. The value is defined and characterized in this context. It is a natural extension of the Myerson value and it turns out to be the Shapley value of a modified game.

 Mathematical Social Sciences Vol. 37 (1): p 79-95, Jan 1999.

 Potentials in Cooperative TU-Games (with Juan Carlos Santos)

 Abstract: This paper is devoted to the study of solutions for cooperative TU-games which admit a potential function, such as the potential associated with the Shapley value (introduced by Hart and Mas-Colell). We consider the finite case and the finite type continuum. Several characterizations of this family are offered and, as a main result, it is shown that each of these solutions can be obtained by applying the Shapley value to an appropriately modified game. We also study the relationship of the potential with the noncooperative potential games, introduced by Monderer and Shapley, for the multilinear case in the continuum finite type setting.

 Mathematical Social Sciences Vol. 34 (2). p 175-90. October 1997.

 Probabilistic Graphs and Power Indices: An Application to the Spanish Parliament (with J Javier Lasaga)

 Abstract: In this paper we consider games with probabilistic graphs. The model we develop is an extension of the model of games with communication restrictions by Myerson (1977). In the Myerson model each pair of players is joined by a link in the graph if and only if these two players can communicate directly. The current paper considers a more general setting in which each pair of players has some probability of direct communication. The value is defined and characterized in this context. It is a natural extension of the Myerson value and it turns out to be the Shapley value of a modified game.

 Journal of Theoretical Politics Vol. 9 (4): p 477-502, October 1997.

 A Prekernel Characterization by Means of Stability Properties (with Esther Gutierrez)

 Abstract: We define two new properties for payoff allocations in the set of pre-imputations of a TU-cooperative game: Strong stability and balanced surplus property. By means of these properties we give a new characterization of the Prekernel of a game. Also a characterization of the Least-core of a game is done in terms of strong stability.

 TOP Vol 4 (2). p 257-267. December 1996.

 The Principle of Balanced Contributions and Hierarchies of Cooperation (vith J Javier Lasaga, Eyal Winter)

Abstract: The principle of balanced contributions has appeared repeatedly in the literature on the Shapley value. This principle is akin to the reciprocity properties shared by almost all cooperative solution concepts. We provide a new axiomatization for the level structure value. This axiomatization has the advantage that it can be applied to many important subdomains of TU games. We use the Hart-Mas-Colell potential function as a tool to prove our main result, and establish another interesting characterization for the value as a by-product.

 Mathematical Social Sciences Vol. 31 (3). p 171-82. June 1996.

 On the Axiomatization of the Tau-Value (vith S.H. Tijs, F. Valenciano, J.M. Zarzuelo)

 Abstract: The Tau-value is a solution concept for a subclass of games with transferable utility introduced and axiomatized by Tijs. In this note an alternative characterization of the Tau-value by means of five axioms is offered. Two of them are well-known: efficiency and translation equivalence; the other three relate the solution of a game with the minimal and maximal aspiration vectors involved in the definition of the value.

 TOP Vol 3(1). p 35-46. June 1995.

  Extension of the Perles-Maschler Solution to N-Person Bargaining Games (with Esther Gutiérrez)

 Abstract: The superadditive solution for 2-person Nash bargaining games was axiomatically defined in Perles/Maschler (1981). In Perles (1982) it was shown that the axioms are incompatible even for 3-person bargaining games. In this paper we offer a generalization method of this solution concept for n-person games. In this method, the Kalai-Smorodinsky solution (1975) is revealed as the rule followed to determine the movements along the path of intermediate agreements.

International Journal of Game Theory Vol. 23 (4). p 325-346, 1994.