NASH EQUILIBRIA IN A MODEL OF MULTIPRODUCT PRICE COMPETITION: AN ASSIGNMENT PROBLEM

We study the market interaction of a finite number of single-product firms and a representative buyer, where the buyer consumes bundles of these goods. The buyers' value function determines their willingness to pay for subsets of goods. We show that subgame perfect Nash-equilibrium outcomes are solutions of the linear relaxation of an integer programming assignment problem and that they always exits. The (subgame perfect) Nash-equilibrium price set is characterized by the Pareto frontier of the associated dual problem's projection on the firms' price vectors. We identify the Nash-equilibrium prices for monotonic buyers' value functions and, more importantly, we show that some central solution concepts in cooperative game theory are (subgame perfect) equilibrium prices of our strategic game.Multiproduct price competition, interger programming, subgame perfect nash equilibria

PSEUDORANDOM PROCESSES: ENTROPY AND AUTOMATA

This paper studies implementation of cooperative payoffs in finitely repeated games when players implement their strategies by finite automata of big sizes. Specifically, we analyze how much we have to depart from fully rational behavior to achieve the Folk Theorem payoffs, i.e., which are the maximum bounds on automata complexity which yield cooperative behavior in long but not infinite interactions. To this end we present a new approach to the implementation of the mixed strategy equilibrium paths leading to cooperation. The novelty is to offer a new construction of the set of the pure strategies which belong to the mixed strategy equilibrium. Thus, we consider the subset of strategies which is characterized by both the complexity of the finite automata and the entropy associated to the underlying coordination process. The equilibrium play consists of both a communication phase and the play of a cycle which depends on the chosen message. The communication set is designed by tools of Information Theory. Moreover, the characterization of this set is given by the complexity of the weaker player that implements the equilibrium play. We offer a domain of definition of the smallest automaton which includes previous domains in the literature.Complexity; Cooperation; Entropy; Automata; Repeated Games.

The Value for Actions-Set Games

Action-Set games are transferable utility games where the set of players is finite, every player has a finite set of actions, and the worth of the game is a function of the actions taken by the players. In this setting a rule has to determine individual payoffs at each combinations of actions. Following an axiomatic approach, we define the set of Consistent Bargaining Equilibria.Action-set games, Shapley value, Prekernel, Consistent Bargaining Equilibria

REPEATED GAMES WITH PROBABILISTIC HORIZON

Repeated games with probabilistic horizon are defined as those games where players have a common probability structure over the length of the game's repetition, T. In particular, for each t, they assign a probability pt to the event that "the game ends in period t". In this framework we analyze Generalized Prisoners' Dilemma games in both finite stage and differentiable stage games. Our construction shows that it is possible to reach cooperative equilibria under some conditions on the distribution of the discrete random variable T even if the expected length of the game is finite. More precisely, we completely characterize the existence of sub-game perfect cooperative equilibria in finite stage games by the (first order) convergence speed: the behavior in the limit of the ratio between the ending probabilities of two consecutive periods. Cooperation in differentiable stage games is determined by the second order convergence speed, which gives a finer analysis of the probability convergence process when the first convergence speed is zero.Leptokurtic distributions are defined as those distributions for which the (first order) convergence speed is zero and they preclude cooperation in finite stage games with probabilistic horizon. However, this negative result is obtained in differential stage games only for a subset of these distributions.repeated games, probabilistic horizon, cooperation

MIXED BUNDLING STRATEGIES AND MULTIPRODUCT PRICE COMPETITION

This paper deals with price competition among multiproduct firms. We consider a model with n firms and one representative buyer. Each firm produces a set of products that can be different or identical to the other firms' products. The buyer is characterized by her willingness to pay -in monetary terms- for every subset of products. To handle the combinatorial complexity of this general setting we use the linear relaxation of an integer programming package assignment problem. This approach allows to characterize all the equilibrium outcomes. We look for subgame perfect Nash equilibrium prices in mixed bundling strategies, i.e., when firms offer consumers the option of buying goods separately or else packages of them at a discount over the single good prices. We find that a mixed bundling subgame perfect Nash equilibrium price vector always exists. Also, the associated equilibrium outcome is always efficient, in the sense that it maximizes the social surplus. We extend the analysis to a model with m buyers and offer the conditions under which the equilibrium outcome set is non-empty.Multiproduct price competition, Integer Programming, Mixed Bundling Strategies, Subgame Perfect Nash Equilibria.

Decisiones de consumo: un ejemplo usando una solución dinámica "Cournot- Nash"

In recent years there have been numerous studies about the formation of the savings rate. However, little attention has been paid to the growing incorporation of women to the labour market on the concumption decision making of households. This decision making changes when both partners have utility functions which are combined inside the household. In this situation, duopolistic behaviours can take place and the final result -the stock of wealth or savings- will depend on the strategy, co-operative or not, followed by the participants. Our model has two basic features: the underlying stock of wealth changes as a result of the actions of both participants, and each participants takes in account of the other's actions. This strategic aspect is studied, for example, by using the concept of a Cournot- Nash equilibrium in which each partner's reaction depends on the stock of wealth and not on previous behaviour. This, the model is a discreet-time alogue of a differential game. The paper examines the dynamic and steady- state proporties of savings level that result from the participants' interactions.Durante los últimos años se ha prestado gran atención al estudio de la formación del nivel de ahorro. Sin embargo, hay un factor que no ha sido suficientemente elaborado: la repercusión de la creciente incorporación de la mano de obra femenina al mundo laboral sobre el nivel de decisiones de consumo o ahorro de las economías domésticas. Dicha toma tradicional de decisiones se ve alterada cuando los dos cónyuges tienen funciones de utilidad separadas que combinan dentro del "hogar". Esto puede dar lugar a situaciones duopolísticas en las que el resultado final -el nivel de riqueza o ahorro- dependerá de la estrategia -colusiva o no - seguida por los participantes. El modelo -un ejercicio simple en la línea comentada- presenta dos características básicas: el nivel de ahorros cambia como resultado de la acción de ambos cónyuges y cada uno de ellos toma en mente las acciones del otro. El aspecto estratégico se estudia usando el concepto de equilibrio Cournot-Nash en un contexto dinámico -la reacción de cada participante en el juego dependerá del nivel de ahorro (variable de estado) y no de comportamientos anteriores-. Así. el modelo es análogo a un juego diferencial en tiempo discreto. El artículo examina las propiedades dinámicas y steady state del nivel de ahorro que resultan de la interacción de los participantes

Nash Equilibrium and information transmission coding and decoding rules

The design of equilibrium protocols in sender-receiver games where communication is noisy occupies an important place in the Economic literature. This paper shows that the common way of constructing a noisy channel communication protocol in Information Theory does not necessarily lead to a Nash equilibrium. Given the decoding scheme, it may happen that, given some state, it is better for the sender to transmit a message that is different from that prescribed by the codebook. Similarly, when the sender uses the codebook as prescribed, the receiver may sometimes prefer to deviate from the decoding scheme when receiving a message.Noisy channel, Shannon's Theorem, sender-receiver games, Nash equilibrium

Pragmatic Languages with Universal Grammars

This paper shows the existence of an equilibrium pragmatic Language with a universal grammar as a coordination device under communication misunderstandings. Such a language plays a key role in achieving efficient outcomes in those Sender-Receiver games where there may exist noisy information transmission. The Language is pragmatic in the sense that the Receiver’ best response depends on the context, i.e, on the payoffs and on the initial probability distribution of the states of nature of the underlying game. The Language has a universal grammar because the coding rule does not depend on such specific parameters and can then be applied to any Sender-Receiver game with noisy communication.grammar, pragmatic language, prototypes, separating equilibria

Asymmetric Information and Endogenous Signalling the Case of Unknown Intercept and Random Output

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PRODUCT QUALITY AND DISTRIBUTION CHANNELS

We introduce strategic behaviour in assigning a certain distribution channel to a product of a particular quality. We propose a variety of models to analyze and study some of the determinants of the choice of distribution channels. Taking the Gabszewicz and Thisse's (1979) model as a benchmark, we first study whether there exist strategic incentives for delegation of sales in a vertically differentiated duopoly. Secondly, product quality is associated with a particular distribution channel. Finally, the model is extended to account for multi-quality production. The resulting equilibria of every game depend on the relative market profitability, the degree of vertical differentiation (i.e. the relative marginal utility of income for quality and the non-buying option), and hence on the intensity of inter-quality and intra-quality competition. In all of the games analyzed, delegation appears as an equilibrium action. In the first game it is a dominant action for both manufacturers. In the second game, at least one of the manufacturers delegates sales. Whether it is one or both crucially depends on market profitability for each quality and the intensity of inter-quality competition. In the third of the games, the single-product manufacturer delegates sales at equilibrium whereas the multi-product manufacturer delegates only one of the qualities. The multi-product manufacturer employs wholesale prices together with the decision of not delegating both qualities to optimally combine the trade-off between the intensity of intra-quality competition and intra-firm competition.vertical differentiation, distribution channels, multi-quality production.