Characterization of the Walrasian equilibria of the assignment model

We study the assignment model where a collection of indivisible goods are sold to a set of buyers who want to buy at most one good. We characterize the extreme and interior points of the set of Walrasian equilibrium price vectors for this model. Our characterizations are in terms of demand sets of buyers. Using these characterizations, we also give a unique characterization of the minimum and the maximum Walrasian equilibrium price vectors. Also, necessary and sufficient conditions are given under which the interior of the set of Walrasian equilibrium price vectors is non-empty. Several of the results are derived by interpreting Walrasian equilibrium price vectors as potential functions of an appropriate directed graph.     

Pseudocontinuous functions and existence of Nash equilibria

In topological spaces, we introduce a new class of functions (pseudocontinuous functions) and we present some characterizations and properties. In particular, we show that any preference relation endowed of utility functions is continuous if and only if any utility is pseudocontinuous. A maximum theorem is proved for such a class of functions and connections with similar results are investigated. Finally, the existence of Nash equilibria for games with pseudocontinuous payoffs is obtained.     

Deriving Nash equilibria as the supercore for a relational system

In this paper, under a binary relation that refines the standard relation which only accounts for single profitable deviations, we obtain that the set of NE strategy profiles of every finite non-cooperative game in normal form coincides with the supercore (Roth, 1976) of its associated abstract system. Further, under the standard relation we show when these two solution concepts coincide.     

Imitation and price competition in a differentiated market

We study the stable market outcome that evolves in a spatially differentiated market when price-competing firms choose actions by imitation of the most profitable firm. We compare and contrast the stable outcomes under two imitation procedures: one, where each firm immediately imitates the most profitable firm, and the other when a firm imitates another firm only if it is more profitable while being “sufficiently similar” (in context of the market segment it operates in) or “sufficiently close”. In either case, the symmetric pure strategy Nash equilibrium is always a stable outcome. However, when imitation of the most profitable firm is immediate and market differentiation is ‘moderate’, states with prices lower than the Nash equilibrium are also stable. In contrast, when imitation of the most profitable firm is more gradual and market differentiation is below a threshold, states with prices above the Nash equilibrium are also stable. Thus, while competitive evolutionary pressure in this imitation based model does result in the Nash equilibrium always being stable, other outcomes may be stable as well. Interestingly, the states that are stable under gradual imitation give the firms a higher profit than the stable states under immediate imitation.     

Workers' enterprises in price competition

Extending earlier equivalence results for perfect competition and for (imperfect) quantity competition, here it is shown that in price competition, too, the behavior of a workers‘ enterprise (defined by the coincidence of its workers with its partners) is tamed by the worker-partnership market so as to be identical with that of a twin entrepreneurial firm. Thus, not only is Bertrand–Nash equilibrium unaltered when an entrepreneurial firm is replaced by its workers’ enterprise twin, but such a replacement also leaves Bertrand-von Stackelberg equilibrium intact whether we switch the leader or a follower from one form of ownership/management to the other.     

A genetic algorithm for the partial binary constraint satisfaction problem: an application to a frequency assignment problem

We describe a genetic algorithm for the partial constraint satisfaction problem. The typical elements of a genetic algorithm, selection, mutation and cross-over, are filled in with combinatorial ideas. For instance, cross-over of two solutions is performed by taking the one or two domain elements in the solutions of each of the variables as the complete domain of the variable. Then a branch-and-bound method is used for solving this small instance. When tested on a class of frequency assignment problems this genetic algorithm produced the best known solutions for all test problems. This feeds the idea that combinatorial ideas may well be useful in genetic algorithms.     

Supply-constrained fixed price equilibria in monetary economies

In this paper we consider an economy with restrictions on the relative prices of non-money commodities. The non-money commodities are partitioned into two groups, index makers and price following commodities. Then two cases are considered, namely that the relative prices of the index makers are fixed, respectively free. In both cases the money prices are flexible, whereas the relative prices of the price following commodities are tied to the prices of the index makers. The existence of a supply-constrained equilibrium is proved with (i) no rationing on the money commodities (stores of value), and (ii) at least one non-money commodity is not rationed. If prices of the index makers are fixed the result strengthens a theorem of Dehez and Drèze, if the prices are free a theorem of Kurz is strengthened. This paper is not only concerned with these existence results, but also with the question whether supply-constrained equilibria should appear more frequently than demand-constrained equilibria.     

Endogenous timing in a mixed duopoly: price competition with managerial delegation

We introduce a managerial delegation contract into the mixed duopoly model and examine its influence on price setting in a mixed duopoly in the context of the endogenous‐timing problem. We obtain the result that owners of a public and a private firm prefer to delay the setting of the prices of their products as much as possible. Thus, in equilibrium, the firms choose their prices simultaneously in the latter stage of the game. This is in contrast to the findings of the entrepreneurial case, according to which firms choose prices simultaneously in the former stage. Copyright © 2009 John Wiley & Sons, Ltd.     

Correlated equilibria in homogeneous good Bertrand competition

We show that there is a unique correlated equilibrium, identical to the unique Nash equilibrium, in the classic Bertrand oligopoly model with homogeneous goods and identical marginal costs. This provides a theoretical underpinning for the so-called “Bertrand paradox” as well as its most general formulation to date. Our proof generalizes to asymmetric marginal costs and arbitrarily many players in the following way: The market price cannot be higher than the second lowest marginal cost in any correlated equilibrium.     

Wholesale price discrimination: Innovation incentives and upstream competition

In intermediate good markets where there are alternative supply sources, wholesale price discrimination may enhance innovation incentives downstream. We consider a vertical chain where a dominant firm and a competitive fringe supply imperfect substitutes to duopoly retailers which carry both varieties. We show that a ban on price discrimination by the dominant supplier makes uniform pricing credible and reduces retailers’ incentives to decrease the cost of acquiring the competitively supplied variety, leading to higher upstream profits and lower downstream welfare. Our analysis complements existing results by identifying a novel channel through which wholesale price discrimination can improve dynamic market efficiency.     

Global analysis and economics III: Pareto Optima and price equilibria

Among other things a global version of the fundamental theorem of welfare economics is proved. One starts with a pure exchange economy with fixed total resources where hypotheses of differentiability, convexity, and monotonicity are made on the utility functions. Let ? be the set of price equilibria where the initial allocation coincides with the final one. Then the map which assigns to such a price equilibrium, the corresponding allocation is a diffeomorphism (a complete correspondence) between ? and the set of Pareto Optima.     

Stability of a Cournot-Nash equilibrium: The multiproduct case

This paper examines the stability conditions of Cournot-Nash equilibria in models in which each firm produces multiple outputs in m related markets. Using a discrete adjustment process, we establish sufficient and necessary conditions for stability. We derive further specialized stability conditions for three important cases, namely a multiproduct duopoly, a multiproduct oligopoly with homogeneous products, and a single-product oligopoly with product heterogeneity. The paper shows that the stability conditions of the m markets individually are in general sufficient nor necessary for the stability of the m markets as a group. In addition, an example is given of how the stability conditions can be applied to determine comparative statics for multiproduct oligopoly.     

The existence of rational expectations equilibria in a large economy with noisy price observations

Non-revealing rational expectations equilibria exist in microeconomic pure exchange economies in which a continuum of uninformed agents have suitably distributed noisy price observations. Slight dispersion in the prices observed by the subset of uninformed consumers is the key condition for continuity (and smoothness) of aggregate excess demand, although individual demands are discontinuous. It leads to equilibria in which markets approximately clear in a strong sense. The equilibria are obtained by applying a fixed point argument to state-dependent excess demand functions.     

Ex-post favoring ranks: a fairness notion for the random assignment problem

We introduce two notions of ex-post fairness, namely ex-post favoring ranks (EFR) and robust ex-post favoring ranks, which consider whether objects are received by those agents who have the highest rank for them. We examine their compatibility with standard properties of random assignments and state some impossibility theorems. We also propose and formalize a revised version of the Boston mechanism and prove that it provides an EFR random assignment.

     

Existence of equilibria in atomless production economies with price dependent preferences

The existence of a competitive equilibrium for an economy with a measure space of agents is proved by reducing the economy to a three-person game and then applying Debreu's lemma (1952). In addition to the relative simplicity and shortness of the proof, we generalize known results [Aumann (1966), Hildenbrand (1970), Schmeidler (1969)] by allowing both price dependent preferences and satiation.     

Purification of Bayesian–Nash equilibria in large games with compact type and action spaces

We present a purification result for incomplete information games with a large but finite number of players that allows compact metric spaces for both actions and types. We then compare our framework and findings to the early purification theorems of Rashid (1983. Equilibrium points of non-atomic games: asymptotic results. Economics Letters 12, 7–10), Cartwright and Wooders (2002 On equilibrium in pure strategies in games with many players. University of Warwick Working Paper 686 (revised 2005)), Kalai (2004. Large robust games. Econometrica 72, 1631–1665) and Wooders, Cartwright and Selten (2006. Behavioral conformity in games with many players. Games and Economic Behavior 57, 347–360). Our proofs are elementary and rely on the Shapley–Folkman theorem.