Sufficient conditions for a game in strategic form with an aggregative structure to have a unique Nash equilibrium are presented. The setting presupposes that each player has ℝ+ as strategy set, makes smoothness assumptions, but allows for a possible discontinuity at the origin. The conditions, in terms of marginal payoffs, are in general easy to check. In particular they imply the needed pseudo-concavity. The results are proved by means of the Selten-Szidarovszky technique using virtual backward reply functions. Their power is illustrated by reproducing and improving various results in the literature concerning Cournot oligopolies.
We study two-player one-dimensional discrete Hotelling pure location games assuming that demand f(d) as a function of distance d is constant or strictly decreasing. We show that this game admits a best-reply potential. This result holds in particular for f(d) = wd with 0 < w ≤ 1. For this case special attention will be given to the structure of the equilibrium set and a conjecture about the increasingness of best reply correspondences will be made.
Wir geben einen sehr elementaren Beweis für das Newton-Polygon Verfahren für Polynome die in Linearfaktoren zerfallen. Der Beweis beruht hauptsächlich auf allgemeinen Eigenschaften der unteren konvexen Hülle. Aus der Algebra benutzen wir eigentlich nur den Wurzelsatz von Vieta.
The Hotelling pure location game is revisited. It is assumed that there there are two identical players, strategy sets are one-dimensional and demand as a function of distance is constant or strictly decreasing. Besides qualitative properties of conditional payoff functions, attention is given to the structure of the equilibrium set, best-response correspondences and existence of potentials.
The paper examines the claim that a virtuous cycle of more secure land rights, more land-saving investments, and denser populations requires the development of institutions that regulate competition over land. We construct a contest model that links the tenure security-investment relationship to the efforts of land users to enhance land rights themselves and the role of institutional protection. We study the impact of population growth on a close-to-subsistence economy, including the possibility that it weakens institutional protection. We derive sufficient conditions for a positive effect on land investment, but also show that population growth can push the economy into a low-productivity trap.
Coalition formation is often analysed in an almost non-cooperative way as a two-stage game that consists of a first stage comprising membership actions and a second stage with physical actions, such as the provision of a public good. We formalise this widely used approach for the case where in each stage actions are simultaneous. There is special attention to the case of a symmetric physical game. Various theoretical results, in particular for cartel games, are provided. As they are crucial, also recent results on uniqueness of coalitional equilibria of Cournot-like physical games are reconsidered. Various concrete examples are included. Finally, we discuss research strategies to obtain results about equilibrium coalition structures with abstract physical games in terms of qualitative properties of their primitives.
We consider the equilibrium uniqueness problem for a large class of Cournot oligopolies with convex cost functions and a proper price function p̃ with decreasing price flexibility. This class allows for (at 0) discontinuous industry revenue and in particular for p̃(y) = y- α. The paper illustrates in an exemplary way the Selten-Szidarovszky technique based on virtual backward reply correspondences. An algorithm for the calculation of the unique equilibrium is provided.
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For binary action games we present three properties which have in common that they are defined by conditions on marginal payoffs. The first two properties guarantee the existence of a special type of Nash equilibrium called semi-strict Nash equilibrium, for which we also show an algorithm to locate. The third one guarantees the existence of an exact potential, and can realize the aforementioned two properties in a class of exact potential games. The first one guarantees the existence of a generalized ordinal potential. Each symmetric binary action game possesses all the three properties. The results are illustrated by three applications.
We study a rather simplified game model of competition for status. Each player chooses a scalar variable (say, the level of conspicuous consumption), and then those who chose the highest level obtain the "high" status, while everybody else remains with the "low" status. Each player strictly prefers the high status, but they also have intrinsic preferences over their choices. The set of all feasible choices may be continuous or discrete, whereas the strategy sets of different players can only differ in their upper and lower bounds. The resulting strategic game with discontinuous utilities does not satisfy the assumptions of any general theorem known as of today. Nonetheless, the existence of a (pure strategy) Nash equilibrium, as well as the "finite best response improvement property," are established.
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We study the pure equilibrium set for a specific symmetric finite game in strategic form, referred to as the Hotelling bi-matrix game. General results that guarantee non-emptiness of this set (for all parametric values) do not seem to exist. We prove its non-emptiness by determining the pure equilibrium set. In this proof so-called demi-modality properties of the conditional payoff functions play an important role.
A family of oligopolies that possess a unique equilibrium was identified in the second authors doctoral dissertation. For such a family, it is therein specified a class of functions--economically related to the price function of a Cournot oligopoly---that satisfy a particular type of quasi-concavity. The first part of the present article (i) conceptualizes that type of quasi-concavity by introducing the notion of demi-concavity, (ii) considers two possible variants and (iii) provides some calculus properties. The second part, by relying on the results on demi-concavity, proves a Cournot equilibrium uniqueness theorem which is new for the journal literature and subsumes various results thereof. A third part shows an example that illustrates the novelty of the result.
We revisit two-person one-dimensional pure location games à la Anderson et al. (1992) and show that they admit continuous best-response potential functions (Voorneveld, 2000) if demand is sufficiently elastic (to the extent that the Principle of Minimum Differentiation fails); if demand is not that elastic (or is completely inelastic) they still admit continuous quasi-potential functions (Schipper, 2004). We also show that, even if a continuous best-response potential function exists, a generalized ordinal potential function (Monderer and Shapley, 1996) need not exist.
We study two-person one-dimensional non-price pure location games à la Anderson et al. (1992) under the setting that the strategy set is either a compact real interval or a finite integer interval and demand as a function of distance is constant (inelastic) or strictly decreasing (elastic). We show that on a finite integer interval, the game is a best- reply potential game (Voorneveld, 2000); on a compact real interval, it is a best-reply potential game if the demand is a sufficiently decreasing, strictly decreasing continuous function of distance; otherwise a quasi-potential game (Schipper, 2004). We also show that, even if a best-reply potential exists, a generalized ordinal potential (Monderer and Shapley, 1996) need not exist. Thus, on a finite integer interval, the game generally lacks the finite improvement property (Monderer and Shapley, 1996) but has the finite best-reply property (Milchtaich, 1996); on a compact real interval, the existence of a pure Nash equilibrium is secured by the existence of some continuous potential function, which, as we shall show, is indeed the case.
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In this article we take a close look at a specific type of behavioural experiment that Antonides conducted to study the endowment effect. We argue that if such experiments ignore to test for the presence of persons in the sample who are indifferent between alternatives, the identification procedure for establishing an endowment effect is fallible.
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A technique due to Selten and Szidarovszky for the analysis of Nash equilibria of games with an aggregative structure is reconsidered. Among other things it is shown that the transformation part of this technique can be extended to abstract games with co-strategy mappings and that this part allows for a purely algebraic setting.
This state-of-the-art collection of papers on the theory of Cournotian competition focuses on two main subjects: oligopolistic Cournot competition and contests. The contributors present various applications of the Cournotian Equilibrium Theory, addressing topics such as equilibrium existence and uniqueness, equilibrium structure, dynamic processes, coalitional behavior and welfare. Special emphasis is placed on the aggregative nature of the games that are relevant to such theory. This contributed volume was written to celebrate the 80th birthday of Prof. Koji Okuguchi, a pioneer in oligopoly theory.
The recent results in [14] on equilibrium (semi-)-uniqueness for homogeneous Cournot oligopolies with concave industry revenue and convex costs are refined and conceptualised. For this class of oligopolies also new results concerning the geometric structure of the equilibrium set E are provided. In particular, a subclass is identified for which E is a non-empty polytope on which the equilibrium aggregator is constant and a subclass for which E is a 1-dimensional polytope on which the equilibrium aggregator is injective.
[Verbesserung von: 1996 (mit W. Heijman). Floquet Theory and Economic Dynamics (Extended Version). Wageningen Economic Papers, 1996-5. Und Verbesserung von: 1993 (mit W. Heijman). Floquet Theory and Economic Dynamics. Wageningen Economic Papers, 1993-3.]
Floquet theory is an appropriate tool for studying ordinary linear recurrence and differential equations with periodic coefficients, and is a generalization of the theory for constant coefficients. Floquet theory has still not found its way into economics, although it seems to be relevant for economic dynamics. As well as a discussion of this relevance and an illustration of it in the context of the Samuelson-Hicks multiplier-accelerator model, this article contains an appendix that provides a quite complete exposition of Floquet theory for recurrence equations.
[Verbesserung von: 2015 (mit W. Heijman, P. Heringa und N. Abudaldah). Return of the Icecream Men. A Discrete Hotelling Game. WASS Working Paper No. 11. Kopie; Pfl
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We consider a finite symmetric game in strategic form with two players which can be interpreted as a discrete variant of the Hotelling game in a one or two-dimensional space. As the analytical investigation of this game is tedious, we simulate with Maple and formulate some conjectures. In addition we present a short literature overview.
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We reconsider the Nash equilibrium existence and uniqueness problem for transboundary pollution games. There is special attention for the equilibrium set E for effective compact transboundary pollution games with continuous strictly concave production functions, continuous convex damage cost functions and uniformly distributed transboundary pollution. For this case we show that E is a non-empty polytope and that for each country all equilibrium deposition levels are equal. If in addition each damage cost function is differentiable, then there is a unique equilibrium. The results are obtained by exploiting the aggregative structure of transboundary pollution games.
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We provide a simple proof of an equilibrium uniqueness result by Murphy, Sherali and Soyster for homogeneous Cournot oligopolies with concave industry revenue function and convex cost functions. We show how to adapt this proof to obtain substantial improvements of this result concerning capacity constraints, non-differentiable cost functions and industry revenue functions that are discontinuous at 0.
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We identify sufficient and necessary conditions for an aggregative game to have a unique Nash equilibrium. In particular, an improvement of a result of Gaudet and Salant (1991) for Cournot oligopolies is obtained. The results are obtained by exploiting the general relation between Nash equilibria and fixed points of the (virtual) aggregate cumulative best reply correspondence.
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[Verbesserung von: 2005 (mit M. Finus und B. Rundshagen) Uniqueness of Coalitional Equilibria. Fondazione Eni Enrico Mattei, Nota de Lavaro 23.2005. Kopie; Pfl
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In the so-called `new approach' of coalition formation it is important that coalitional equilibria are unique. Uniqueness comes down to existence and to semi-uniqueness, i.e. that there exists at most one equilibrium. Although conditions for existence are not problematic, conditions for semi-uniqueness are. We provide semi-uniqueness conditions by deriving a new equilibrium semi-uniqueness result for games in strategic form with higher dimensional strategy sets. The result applies in particular to Cournot-like games.
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We reconsider the recent work by Okuguchi (2010) on (possibly asymmetric) Cournotian firms with two production factors, one being inferior for each firm. It is shown there that an increase in the price of the inferior factor does raise the equilibrium industry output. In addition of providing a simpler and more rigorous proof of that result, we generalize it to the case of technologies with s ≥ 2 factors and also allow some firms not to use the inferior one.
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We consider a class of homogeneous Cournot oligopolies with concave integrated price flexibility and convex cost functions. We provide new results about the semi-uniqueness and uniqueness of (Cournot) equilibria for the oligopolies that satisfy these conditions. The condition of concave integrated price flexibility is implied by (but does not imply) the log-concavity of a continuous decreasing price function. So, our results generalize previous results for decreasing log-concave price functions and convex cost functions. We also discuss the particular type of quasi-concavity that characterizes the conditional revenue and profit functions of the firms in these oligopolies and we point out an error of the literature on the equilibrium uniqueness in oligopolies with log-concave price functions. Finally, we explain how the condition of concave integrated price flexibility relates to other conditions on the price and revenue functions usually considered in the literature, e.g., the concavity of the price function or the concavity of the aggregate revenue associated to a price function.
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[Verbesserung von: 2007 (mit H. Folmer) Linking of Repeated Games: When does it Lead to More Cooperation and Pareto Improvements? Fondazione Eni Enrico Mattei, Nota de Lavaro 60.2007. Kopie; Pfl
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We deal with the linkage of infinitely repeated games. Results are obtained by analysing the relations of the feasible individually rational payoff regions of the isolated games and the linked game. In fact we have to handle two geometric problems related to Minkowski sums, intersections and Pareto boundaries of convex sets.
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We show how the fixed point problem for a special type of correspondence R which satisfies a factorisation property can be handled by considering an associated more simple fixed point problem for a correspondence B with domain typically a subset of ℝ. In addition we analyse the fixed point problem for B under additional conditions on R that guarantee that B is at most singleton-valued. In fact we generalize, improve and make more conceptual a game theoretic technique developed by Selten and Szidarovszky.
