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Recently a framework was developed for aggregative variational inequalities by means of the Selten-Szidarovszky technique. By referring to this framework, a powerful Nash equilibrium uniqueness theorem for sum-aggregative games is derived. Payoff functions are strictly quasi-concave in own strategies but may be discontinuous at the origin. Its power is illustrated by reproducing and generalising in a few lines an equilibrium uniqueness result in Corchon and Torregrosa (2020) for Cournot oligopolies with the Bulow-Pfleiderer price function. Another illustration addresses an asymmetric contest with endogenous valuations in Hirai and Szidarovszky (2013).
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We present a game theoretic model of media bias based on the Pure Location Hotelling Game. Contrary to the usual restrictive assumption of inelastic demand, we allow that demand is elastic and introduce in this context the concepts of quite inelastic and quite elastic demand. The real world interpretation of the media bias model is explained in detail for its discrete variant for unlimited media providers and arbitrary distribution of individuals.
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We enrich the theory of variational inequalities in the case of an aggregative structure by implementing recent results obtained by using the Selten-Szidarovszky technique. We derive existence, semi-uniqueness and uniqueness results for solutions and provide a computational method. As an application we derive very powerful practical equilibrium results for Nash equilibria of sum-aggregative games and illustrate with Cournot oligopolies.
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Various Nash equilibrium results for a broad class of aggregative games are presented. The main ones concern equilibrium uniqueness. The setting presupposes that each player has ℝ+ as strategy set, makes smoothness assumptions but allows for a discontinuity of stand-alone payoff functions at 0; this possibility is especially important for various contest and oligopolistic games. Conditions are completely in terms of marginal reductions which may be considered as primitives of the game. For many games in the literature they can easily be checked. They automatically imply that conditional payoff functions are strictly quasi-concave. The results are proved by means of the Szidarovszky variant of the Selten-Szidarovszky technique. Their power is illustrated by reproducing quickly and improving upon various results for economic games.
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-response correspondences will be made.
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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 cspondences. 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.
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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.
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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.
The article corrects, improves and formalises the procedure in Heijman and Goossen (2009) to determine an optimal landscape and its monetary value, assuming a Cobb-Douglas benefit function.
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Economic development generally implies that traditional egalitarian norms and beliefs are replaced by modern individualistic values. Particularly when opportunities for advancement are unequally presented to people, this transformation may be accompanied by polarization and violent conflict. We illustrate this point by describing the processes of land privatization in Sub-Saharan Africa and then present two models that capture some salient aspects of this transformation in rural communities, including the possibility of polarization. We find that the support of egalitarian norms is notably strong when new opportunities are available for only a few people or when the community is socially unstable. Moreover, in unstable communities, polarization is strongest when the group with the most lucrative opportunities comprises half the population.
We consider the Nash equilibrium existence problem for Cournotian games and we provide two results for it. The first is compatible with utility functions that are discontinuous at the origin, but requires the nonemptiness of best-replies at the origin (and in some sense is known); the second requires the discontinuity of utility functions at the origin and the emptiness of best-replies at the origin (and to the best of our knowledge is not known). Both results are proved by reducing the Nash equilibrium existence problem to a simple fixpoint existence problem for a real function defined on a real interval. The results are then applied to Cournot oligopolies and winner-take-all contest games; various examples illustrate the novelty of the results obtained.
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We revisit and compare equilibrium uniqueness results for homogeneous Cournot oligopolies. In doing this we provide various useful and interesting results for which it is difficult to give appropriate reference in the literature. We also propose problems for future research.
One might expect that after their identification in the 19th century, all aspects of Giffen goods would have been studied by now. This appears not to be the case. This book contains the latest insights into the theory of Giffen goods. In the past, surprisingly few goods could be categorized as ''Giffen''. This may be because of a lack of understanding the character of these goods. Therefore, the theories explained in this book may also produce a solid basis for further empirical research in the field. Experts throughout the world have contributed to this book, which predominantly pursues a mathematically rigorous approach. It may be used by researchers in the field of fundamental economics and in graduate-level courses in advanced microeconomics.
[ Verbesserung von: 2009 (mit W. Heijman). On Simple Concrete Giffen Utility Functions: Old and New Results. Mansholt Graduate School, Discussion Paper 45. Kopie; Pfl
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We present a theoretical review of the literature on concrete utility functions for Giffen and inferior goods within the context of the utility maximisation problem under a budget restriction and provide new functional forms. The presentation is organised around the specific properties such utility functions have. These properties include strict increasingness, quasi-concavity, and the applicability of Gossen's second law.
We consider Giffen goods for the very general setting of an upper semi-continuous utility function. Two not equivalent definitions for this notions are discussed. Concerning this we show that there exists a (well-defined) non-decreasing demand function ℝ++ → ℝ that is nowhere strictly increasing. Also we reconsider the relation between Giffen and inferior goods.
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A class of games in strategic form with the following property is identified: for every n ∈ E, i.e. Nash equilibrium, the (Nash) sum ∑_l nl is constant. For such a game sufficient conditions for E to be polyhedral and semi-uniqueness (i.e. # E ≤ 1) are given. The abstract results are illustrated by applying them to a class of games that covers various types of Cournot oligopoly and transboundary pollution games. The way of obtaining the results is by analysing so-called left and right marginal reductions.
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We analyze a game in strategic form, where each player′s payoff depends on his action and his social status, which is given by his rank in the actions distribution. Our focus is on the relation between the degree of heterogeneity among status-seeking players and the distribution of their Nash equilibrium actions. We find that if among players intrinsic concerns are sufficiently important relative to status concerns, individual equilibrium actions diverge, but if status concerns are relatively important, individual equilibrium actions are the same. Another key result of the analysis is that, in contrast to what is usually claimed, status seeking need not always be socially inefficient. If players are sufficiently heterogeneous, there exists a Nash equilibrium that is unique, separating, and Pareto efficient.
Given non-degenerate intervals Xi of ℝ and an increasing ordered mapping Φ: X1 x … x XN → ℝN, games in strategic form between N players with the Xi as action sets with the following three properties are studied: the set of Nash equilibria E is convex, Φ is constant on E and in each Nash equilibrium at least one payoff function is not partially differentiable w.r.t. its own action. The results are illustrated for a special class of aggregative games that include the formal transboundary pollution games with global transboundary pollution.
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The formal mathematical structure of direct sum games is studied, in particular their relation with multiobjective games. A canonical analysis is performed and the notion of equilibrated direct sum game is introduced. For games in the latter class, and even for not too large deviations of them, there is special attention for strictly individually rational full cooperative payoff vectors.
As far as we know, the problem of existence of a maximiser of the welfare function in traffic congestion models à la Mohring and Harwitz has never been posed. In this paper we present a mathematically rigorous existence result and take the opportunity to provide a similar variant of their self-financing result.
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We generalize a less known Nash equilibrium uniqueness result for games in strategic form. Its power is illustrated by applying it to a Public Goods Game, a Homogeneous Cournot Oligopoly Game and a Formal Transboundary Pollution Game.
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Transboundary Pollution and International Cooperation. In: The International Yearbook of Environmental and Resource Economics 2000/2001, 231-166. Edward Elgar, Cheltenham. Editors: T. Tietenberg, H. Folmer. Kopie; Pfl
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Asymptotics, for coupling constant A tending to zero and to infinity, for the bands and gaps of the Schrödinger operator (HA g)(n) = g(n+1) + Abng(n) + g(n-1) on l2(ℤ) with periodic potential b are studied. In this context special attention is paid to `asymptotic spectral powers'. Three different perturbation techniques are used to obtain results. In this first paper the emphasis lies on results that can be obtained by a first-order degenerate or arbitrary-order non-degenerate Rayleigh-Schrödinger perturbation calculus.
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Some features of international environmental problems are considered. A basic problem is to induce countries to adopt a cooperative approach. One of the instruments to induce countries to cooperate is an exchange of concessions in fields of relative strengths, such as swapping trade concessions for cooperation on international environmental problems. This instrument will be modeled in this paper with tensor games. Both tradeoff and non-tradeoff tensor games will be addressed with emphasis on tradeoff tensor games with linear strict weights. The relationship between the Pareto-equilibria of a non-tradeoff tensor game and the Nash-equilibria of the associated tradeoff tensor games will be studied. Due to structural similarities between tensor games and repeated multiple objective games, some attention will be also paid to the latter. Relationships between objects related to Folk theorems for the tradeoff tensor game with completely additive weights and the corresponding objects for its constituting isolated games will be studied. As many international environmental problems have prisoners' dilemma characteristics, it is analyzed how interconnection may enhance cooperation in prisoners' dilemma games.
Dieser Artikel betrifft eine ausfürliche Kurzfassung von ``1994 (mit H. Folmer) Interconnected Games and International Environmental Problems. II, Annals of Operations Research, 54, 97-117''.
The purpose of this paper is to introduce the concept of interconnected games and to show its relevance for modeling international environmental problems. It is argued that an interconnected game approach to international environmental problems may enhance cooperation and provide an alternative to the use of financial side payments to induce countries to cooperate. Two types of interconnected games are distinguished in this paper, i.e. direct sum games and tensor games. In the former all the constituting isolated games are games in strategic form and in the latter they are repeated games. In both cases the interconnected game can be interpreted as a multiple objective game, but only the setting where a trade-off is made for the vector-payoffs is considered. In addition to the formal definition of these types of interconnected games, some elementary results concerning Nash equilibria of such games are derived.
This essay deals with two extensions of classic business cycle models in the context of resource economics. The first is based on the traditional post-Keynesian multiplier accelerator model and is a typical demand side dynamic macroeconomic model. It explains the business cycle for resource use out of changes in the effective demand. The second, a dynamic microeconomic model, based on the traditional cobweb model, explains the variation in the price level of resources. Both models are described by second order linear difference equations. Special attention is given to the problem of asymptotic stability in the case of Floquetian coefficients. Furthermore each model is used to analyse a specific economic policy problem.
We obtain partial result on the conjecture that for the almost Mathieu operator at irrational frequency, α, the measure of the spectrum, S(α, λ, θ) = | 4 - 2 | λ | |. For |λ| ≠ 2 we show that if αn is rational and αn → α irrational, then S+(αn, λ) → | 4 - 2 | λ| |.
We solve the coexistence problem for the periodic discrete Mathieu operator in all parametric cases. The main tool in the proof will be Bezout's theorem for projective plane curves. As an additional result we obtain the gap opening and gap growth powers for this operator.
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In some models of solid state physics, mysterious phenomena can be observed: The clustering and nesting of energy spectra. This happens in particular with models leading to the discrete Mathieu equation g(n+1) + (2 A cos(2 π n α - ν) - ε) g(n) + g(n-1) =0 (n ∈ ℤ). Hofstadter (1976) gave an empirical description of the combinatorics of both phenomena. In this paper we shall give a natural explanation of the combinatorics by introducing and applying the concept of infinitesimal clustering.
En partant de l'équation de Schrödinger pour un électron qui se déplace dans un potentiel périodique, soumis en même temps s à un champ magnétique extérieur, nous arrivons à l'équation presque-Mathieu, étant une équation aux différences. Inspiré par les résultats théoriques pour l'équation différentielle de Mathieu, nous obtenons des résultats semblables pour l'équation presque Mathieu. En outre nous calculons a l'aide d'un ordinateur puissant le spectre d'énergie de cette équation. En particulier nous obtenons des variantes de papillons de Hofstadter.
