Claude d'Aspremont

ON HOTELLING'S "STABILITY IN COMPETITION"

The purpose of this note is to show that the so-called Principle of Minimum Differentiation, as based on Hotelling’s 1929 paper “Stability in Competition” is invalid. The purpose of this note is to show that the so-called Principle of Minimum Differentiation, as based on Hotelling’s 1929 celebrated paper (Hotelling [3]), is invalid. Firstly, we assert that, contrary to the statement formulated by Hotelling in his model, nothing can be said about the tendency of both sellers to agglomerate at the center of the market. The reason is that no equilibrium price solution will exist when both sellers are not far enough from each other. Secondly, we consider a slightly modified version of Hotelling’s example, for which there exists a price equilibrium solution everywhere. We show however that, for this version, there is a tendency for both sellers to maximize their differentiation. This example thus constitutes a counterexample to Hotelling’s conclusions. We shall first recall Hotelling’s model and notations. On a line of length `, two sellers A and B of a homogeneous product, with zero production cost, are located at respective distances a and b from the ends of this line (a+ b ≤ `; a ≥ 0, b ≥ 0). Customers are evenly distributed along the line, and each customer consumes exactly a single unit of this commodity per unit of time, irrespective of its price. Since the product is homogeneous, a customer will buy from the seller Econometrica, 47(5), 1145–1150, September 1979. Center for Operations Research and Econometrics

Equity and the informational basis of collective choice

We consider the problem of a planner or ethical observer who wants to derive a collective preference ordering over a set of feasible alternatives from the knowledge of individual utility functions. By assumption, he is concerned with social welfare judgements, not with committee decisions. As a tool of analysis, we use the concept of social welfare functional (SWFL), which was developed by Sen [9] on foundations originally laid down by Arrow [1]. Rather than to compare SWFL’s directly, we treat them somewhat like composite goods and we compare sets of axioms which characterize them. We select five such sets, which differ mainly with respect to the planner’s informational basis. This term refers to an “invariance” axiom which defines in each case the measurability and comparability properties of individual utility functions. Taking up a suggestion of Sen’s [10], we focus our attention on the implications of each informational basis for the equity content of collective choice. Our study does not treat all possible invariance axioms; it does not even exhaust all the most relevant ones. However, we think that it brings about some logical clarification. Among other things, we characterize utilitarianism and the leximin (or lexical maximin) principle by means of two sets of axioms which differ only in one respect, viz. the invariance axiom. The paper is divided into three sections. In Section 1 we describe our problem formally, we discuss our invariance axioms, and we show that some of them are equivalent, in the light of the Review of Economic Studies, 44(2), 199-209, 1977. On this distinction, see Sen [11].

Bayesian incentive compatible beliefs

Abstract The problem of incentives for correct revelation is studied as a game with incomplete information where players have individual beliefs concerning others types. General conditions on the beliefs are given which are shown to be sufficient for the existence of a Pareto-efficient mechanism for which truth-telling is a Bayesian equilibrium.

Stackelberg-solvable games and pre-play communication

Abstract The concept of Stackelberg-solvable games is introduced and analyzed as a generalization of 2-person zero-sum games. Then, the problem of sincere pre-play communication is examined and the incentive compatibility of the Nash-equilibrium selection is shown to correspond to the Stackelberg-equilibrium property.

Balanced Bayesian mechanisms

We present a new condition on beliefs that guarantee the Bayesian implementability of all efficient social decision rules. We show that this condition is easy to verify and is both more interpretable and more general than the conditions that are found in the literature. We also study conditions guaranteeing the Bayesian implementability of all social decision rules with balanced budget mechanisms.

Competition for market share or for market size: oligopolistic equilibria with varying competitive toughness

For an industry producing a composite commodity, we propose a comprehensive concept of oligopolistic equilibrium, allowing for a parameterized continuum of regimes varying in competitive toughness. Each firm sets simultaneously its price and its quantity under two constraints, relative to its market share and to market size. The price and the quantity equilibrium outcomes always belong to the set of oligopolistic equilibria. When firms are identical and we let their number increase, any sequence of symmetric oligopolistic equilibria converges to the monopolistic competition outcome. Further results are derived in the symmetric CES case, concerning in particular the collusive solution enforceability.

Ordinal Bayesian Incentive Compatible Representations of Committees

We consider the problem of strategic manipulation for decision schemes that provide an adequate representation (in some sense) of the distribution of power within a committee. “Strategy-proof representation” is very restrictive: it implies that the committee contains exactly one minimal winning coalition. So we introduce the weaker concept of “Ordinally Bayesian Incentive Compatible representation” and prove the existence of such representations for weak games under some conditions. Finally, constructing examples, we show first how necessary these conditions are—including the use of chance in the voting procedure — and second that we cannot avoid Condorcets paradox.

Price–quantity competition with varying toughness☆

For an industry producing a single homogeneous good, we define and characterize the concept of oligopolistic equilibrium, allowing for a parameterized continuum of regimes with varying competitive toughness. This parameterization will appear to be equivalent to the one used in the empirical literature. The Cournot and the competitive outcomes coincide, respectively, with the softest and the toughest oligopolistic equilibrium outcome. The concept offers an alternative to the conjectural variations approach with better foundations. It can be viewed as a canonical description of oligopolistic behavior which can receive different theoretical justifications and provide a convenient tool for modeling purposes. Two illustrative cases (linear and isoelastic demands) are developed and the possibility of endogenizing (strategically) the choice of competitive toughness by the firms is examined.

Incentives and the existence of Pareto-optimal revelation mechanisms☆

Abstract From the characterization of strongly and Bayesian incentive compatible Paretooptimal mechanisms with transferable utilities, we derive the following results. If there are only two types per individual then a strongly incentive compatible Paretooptimal mechanism exists. If there are only two individuals (with more than three types) then there are sets of beliefs (open in the class of all beliefs) for which no Bayesian incentive compatible Pareto-optimal mechanism exists. If there are more than two individuals then the class of beliefs for which such mechanisms exist is open and dense in the class of all beliefs.

Oligopolistic Competition as a Common Agency Game

In applying the common agency framework to the context of an oligopolistic industry, we want to go beyond the classical dichotomy between Cournot and Bertrand competition. We define two games, the oligopolistic game and the corresponding concept of oligopolistic equilibrium, and an associated auxiliary game that can be interpreted as a common agency game and that has the same set of equilibria. The parameterization of the set of (potential) equilibria in terms of competitive toughness is derived from the first order conditions of this auxiliary game. The enforceability of monopolistic competition, of price and quantity competition, and of collusion is examined in this framework. We then describe the (reduced) set of equilibria one would obtain, first in the nonintrinsic case and then in the case where a global approach would be adopted instead of partial equilibrium approach. Finally, we illustrate the use of the concept of oligopolistic equilibrium and of the corresponding parameterization by referring to the standard case of symmetric quadratic utility.