On some geometry and equivalence classes of normal form games

Equivalence classes of normal form games are defined using the geometryof correspondences of standard equilibiurm concepts like correlated, Nash,and robust equilibrium or risk dominance and rationalizability. Resultingequivalence classes are fully characterized and compared across differentequilibrium concepts for 2 x 2 games. It is argued that the procedure canlead to broad and game-theoretically meaningful distinctions of games aswell as to alternative ways of viewing and testing equilibrium concepts.Larger games are also briefly considered.

English grammar games in primary education

The purpose of this study was to analyse pupils’ English grammar acquisition from competitive and cooperative approaches. After searching a wide range of authors’ contribution to English language learning, grammar acquisition, classroom environment and language games. A hundred and twenty pupils from three schools; two from Ripoll and one from Campdevànol were enrolled in a specific grammar games intervention. This was imparted in three different phases: first of all, I interviewed the three teachers from the three schools, then I put into practice my competitive and cooperative games which I designed especially for this study (all the sessions were carried, assessed and registered by myself); finally, all pupils answered a questionnaire related to their experiences in my grammar games intervention. Analysis of teaching interventions showed that, in terms of English language acquisition, pupils used different strategies to show up understanding and achieve the objective of the game such as: recalling their background knowledge, expressing sentences influenced by their internal language and their mother tongue. Data collected revealed that most difficulties were founded in team work, even more in competitive games. The results also showed that team work is something which has to be developed step by step in order to achieve language learning and all pupils’ active participation successfully.

Axiomatization of the nucl eolus of assignment games

On the domain of general assignment games (with possible reservation prices) the core is axiomatized as the unique solution satisfying two consistency principles: projection consistency and derived consistency. Also, an axiomatic characterization of the nucleolus is given as the unique solution that satisfies derived consistency and equal maximum complaint between groups. As a consequence, we obtain a geometric characterization of the nucleolus. Maschler et al. (1979) provide a geometrical characterization for the intersection of the kernel and the core of a coalitional game, showing that those allocations that lie in both sets are always the midpoint of certain bargaining range between each pair of players. In the case of the assignment game, this means that the kernel can be determined as those core allocations where the maximum amount, that can be transferred without getting outside the core, from one agent to his / her optimally matched partner equals the maximum amount that he / she can receive from this partner, also remaining inside the core. We now prove that the nucleolus of the assignment game can be characterized by requiring this bisection property be satisfied not only for optimally matched pairs but also for optimally matched coalitions. Key words: cooperative games, assignment game, core, nucleolus

Payoff-dependent balancedness and cores (revised version)

We prove the non-emptiness of the core of an NTU game satisfying a condition of payoff-dependent balancedness, based on transfer rate mappings. We also define a new equilibrium condition on transfer rates and we prove the existence of core payoff vectors satisfying this condition. The additional requirement of transfer rate equilibrium refines the core concept and allows the selection of specific core payoff vectors. Lastly, the class of parametrized cooperative games is introduced. This new setting and its associated equilibrium-core solution extend the usual cooperative game framework and core solution to situations depending on an exogenous environment. A non-emptiness result for the equilibrium-core is also provided in the context of a parametrized cooperative game. Our proofs borrow mathematical tools and geometric constructions from general equilibrium theory with non convexities. Applications to extant results taken from game theory and economic theory are given.

Free Mobility and Taste-Homogeneity of Jurisdiction Structures

We consider a population of agents distributed on the unit interval. Agents form jurisdictions in order to provide a public facility and share its costs equally. This creates an incentive to form large entities. Individuals also incur a transportation cost depending on their location and that of the facility which makes small jurisdictions advantageous. We consider a fairly general class of distributions of agents and generalize previous versions of this model by allowing for non-linear transportation costs. We show that, in general, jurisdictions are not necessarily homogeneous. However, they are if facilities are always intraterritory and transportation costs are superadditive. Superadditivity can be weakened to strictly increasing and strictly concave when agents are uniformly distributed. Keywords: Consecutiveness, stratification, local public goods, coalition formation, country formation. JEL Classification: C71 (Cooperative Games), D71 (Social Choice; Clubs; Committees; Associations), H73 (Interjurisdictional Differentials and Their Effects).

Simultaneous Nash bargaining with consistent beliefs

We propose and analyze a new solution concept, the R solution, for three-person, transferable utility, cooperative games. In the spirit of the Nash Bargaining Solution, our concept is founded on the predicted outcomes of simultaneous, two-party negotiations that would be the alternative to the grand coalition. These possibly probabilistic predictions are based on consistent beliefs. We analyze the properties of the R solution and compare it with the Shapley value and other concepts. The R solution exists and is unique. It belongs to the bargaining set and to the core whenever the latter is not empty. In fact, when the grand coalition can simply execute one of the three possible bilateral trades, the R solution is the most egalitarian selection of the bargaining set. Finally, we discuss how the R solution changes important conclusions of several well known Industrial Organization models.

A comment on the Nash program and the theory of implementation

The mechanisms in the Nash program for cooperative games are madecompatible with the framework of the theory of implementation. This is donethrough a reinterpretation of the characteristic function that avoids feasibilityproblems, thereby allowing an analysis that focuses exclusively on the payoff space. In this framework, we show that the core is the only majorcooperative solution that is Maskin monotonic. Thus, implementation of mostcooperative solutions must rely on refinements of the Nash equilibrium concept(like most papers in the Nash program do). Finally, the mechanisms in theNash program are adapted into the model.

Finite horizon bargaining and the consistent field

This paper explores the relationships between noncooperative bargaining games and the consistent value for non-transferable utility (NTU) cooperative games. A dynamic approach to the consistent value for NTU games is introduced: the consistent vector field. The main contribution of the paper is to show that the consistent field is intimately related to the concept of subgame perfection for finite horizon noncooperative bargaining games, as the horizon goes to infinity and the cost of delay goes to zero. The solutions of the dynamic system associated to the consistent field characterize the subgame perfect equilibrium payoffs of the noncooperative bargaining games. We show that for transferable utility, hyperplane and pure bargaining games, the dynamics of the consistent fields converge globally to the unique consistent value. However, in the general NTU case, the dynamics of the consistent field can be complex. An example is constructed where the consistent field has cyclic solutions; moreover, the finite horizon subgame perfect equilibria do not approach the consistent value.

The effect of intergroup competition on group coordination: An experimental study

We report an experiment on the effect of intergroup competition on group coordination in the minimal-effort game (Van Huyck et al., 1990). The competition was between two 7-person groups. Each player in each group independently chose an integer from 1 to 7. The group with the higher minimum won the competition and each of its members was paid according to the game s original payoff matrix. Members of the losing group were paid nothing. In case of a tie, each player was paid half the payoff in the original matrix. This treatment was contrasted with two control treatments where each of the two groups played an independent coordination game, either with or without information about the minimum chosen by the outgroup. Although the intergroup competition does not change the set of strict equilibria, we found that it improved collective rationality by moving group members in the direction of higher-payoff equilibria. Merely providing group members with information about the minimal-effort level in the other group was not sufficient to generate this effect.

Some game-theoretic grounds for meeting people half-way

It is well known that, in distributions problems, fairness rarely leads to a single viewpoint (see, for instance, Young (1994)). In this context, this paper provides interesting bases that support the simple and commonly observed behavior of reaching intermediate agreements when two prominent distribution proposals highlight a discrepancy in sharing resources. Specifi cally, we formalize such a conflicting situation by associating it with a `natural' cooperative game, called bifocal distribution game, to show that both the Nucleolus (Schmeidler (1969)) and the Shapley value (Shapley (1953a)) agree on recommending the average of the two focal proposals. Furthermore, we analyze the interpretation of the previous result by means of axiomatic arguments. Keywords: Distribution problems, Cooperative games, Axiomatic analysis, Nucleolus, Shapley value. JEL Classi fication Numbers: C71, D63, D71.

On the coincidence of the Mas-Colell bargaining set and the core

In this paper we prove that the Mas-Colell bargaining set coincides with the core for three-player balanced and superadditive cooperative games. This is no longer true without the superadditivity condition or for games with more than three-players. Furthermore, under the same assumptions, the coincidence between the Mas-Collel and the individual rational bargaining set (Vohra (1991)) is revealed. Keywords: Cooperative game, Mas-Colell bargaining set, balancedness, individual rational bargaining set. JEL classi fication: C71, D63, D71.

Els jocs cooperatius desenvolupats a l'àrea d'educació física milloren les relaciones que s'estableixen entre alumnes de cicle superior de primària

Introducció: Teodorico (2004) ens diu que amb el joc cooperatiu els alumnes poden valorar, compartir i reflexionar sobre la relació que estableixen amb els altres companys. Per això, l’objectiu de la meva investigació és avaluar l’efecte que pot tenir l’aplicació d’una unitat de programació, centrada en el joc cooperatiu, en la relació que s’estableix entre els alumnes de 5è de primària, focalitzant l’atenció amb el líder i el menys acceptat del grup-classe. Mètodes: La intervenció s’avalua amb un disseny quasi-experimental pre-post amb grup control. I des del punt de vista dels dissenys observacionals podem parlar d’un disseny de seguiment, idiogràfic i multidimensional. El total de la mostra d’aquesta investigació era de 48 alumnes, dividit en el grup experimental (25) i en el grup control (23). La durada de la intervenció va ser de 8 setmanes i es va utilitzar un qüestionari ad hoc per la confecció dels sociograma i un instrument observacional ad hoc de les relacions de grup-classe. Resultats: El grup experimental estava format per 4 grups d’alumnes, al final de la unitat només se’n formaven 2, en canvi, el grup control, tan abans com després, estava format pels mateixos grups d’alumnes. Un dels menys acceptats del grup experimental ha millorat la relació amb alguns dels companys, en canvi els menys acceptat del grup control no ha millorat cap tipus de relació. Conclusions: Crec que els jocs cooperatius són una bona eina educativa per intentar millorar les relacions, ja que amb més temps crec que els resultats d’aquesta investigació haguessin sigut més positius.

Les relacions entre alumnes a la classe d’Educació Física: la consciència de les actituds i valors en el treball cooperatiu en l’etapa de primària

La meva investigació de metodologia qualitativa es base en observar el grau de consciència que tenen els alumnes de tercer de primària, de l’escola El Gegant del Rec, dels valors, les actituds i les normes que es treballen a través de jocs cooperatius, els quals estan presentats a partir d’una unitat de programació. El motiu del meu estudi és que l’educació en valors ha d’estar present a l’àrea d’Educació Física, ja que és un escenari molt potent i és molt interessant observar i potenciar als discents diferents actituds i valors a través de jocs i normes.

Characterization of Reward Functions in Networks with Costs

In this paper we study network structures in which the possibilities for cooperation are restricted and can not be described by a cooperative game. The benefits of a group of players depend on how these players are internally connected. One way to represent this type of situations is the so-called reward function, which represents the profits obtainable by the total coalition if links can be used to coordinate agents' actions. The starting point of this paper is the work of Vilaseca et al. where they characterized the reward function. We concentrate on those situations where there exist costs for establishing communication links. Given a reward function and a costs function, our aim is to analyze under what conditions it is possible to associate a cooperative game to it. We characterize the reward function in networks structures with costs for establishing links by means of two conditions, component permanence and component additivity. Finally, an economic application is developed to illustrate the main theoretical result.

A Sequential Allocation Problem: The Asymptotic Distribution of Resources

In this paper we consider a sequential allocation problem with n individuals. The first individual can consume any amount of some endowment leaving the remaining for the second individual, and so on. Motivated by the limitations associated with the cooperative or non-cooperative solutions we propose a new approach. We establish some axioms that should be satisfied, representativeness, impartiality, etc. The result is a unique asymptotic allocation rule. It is shown for n = 2; 3; 4; and a claim is made for general n. We show that it satisfies a set of desirable properties. Key words: Sequential allocation rule, River sharing problem, Cooperative and non-cooperative games, Dictator and ultimatum games. JEL classification: C79, D63, D74.