Mein Doktorvater war J. Duistermaat. Jede Person in der unterstehenden Liste hat als Doktorvater der nachfolgenden Person; Platz und Jahreszahl beziehen sich auf die Doktorarbeit. Zur Erstellung dieser Liste habe ich Mathematics Genealogy Project benutzt.

- Pierre von Mouche (1959--), Utrecht 1988.
- Johannes Duistermaat (1942--2010), Utrecht 1968.
- Hans Freudenthal (1905--1990), Berlin 1930.
- Heinz Hopf (1894--1971) und Ludwig Bieberbach, Berlin 1925.
- Erhard Schmidt (1876--1959) und Ludwig Bieberbach. Göttingen 1905.
- David Hilbert (1862--1943), Königsberg 1885.
- Ferdinand von Lindemann (1852--1939), Erlangen-Nürnberg 1873.
- Felix Klein (1849--1925) und Julius Plücker (1801-1868), Bonn 1868.
- Rudolf Lipschitz (1832-1903), Berlin 1853.
- Gustav Dirichlet (1805-1859) und Martin Ohm, Bonn 1827.
- Siméon Poisson (1781-1840) und Jean-Baptiste Fourier, Paris 1800.
- Joseph Lagrange (1736--1813) und Pierre-Simon Laplace,
- Leonard Euler(1707--1783).
- Johann Bernoulli (1667--1748).

- 2019 (mit W. Pijnappel).
*Das Newton-Polygon Verfahren für Polynome die in Linearfaktoren zerfallen.*

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. -
2019 (mit J.-I. Itaya).
*Equilibrium Uniqueness in Aggregative Games: Very Practical Conditions.*Zur Publikation angeboten.

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. - 2019 (mit A. Hagen und H.-P. Weikard).
*The Two-Stage Game Approach to Coalition Formation: Where we Stand and Ways to Go.*Zur Publikation akzeptiert (Games).

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. - 2019 (mit R. Haagsma).
*Securing Land Rights under Rapid Population Growth: the Feasibility of Institutional Land Rights Protection in Africa.*Zur Publikation akzeptiert (Journal of Institutional and Theoretical Economics).

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 for 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. -
2019 (mit T. Sato).
*Cournot Equilibrium Uniqueness: at 0 Discontinuous Industry Revenue and Decreasing Price Flexibility.*International Game Theory Review, 21, 2, 1--19. Kopie.

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. - 2018 (mit T. Iimura und T. Watanabe).
*Binary Action Games: Deviation Properties, Semi-Strict Equilibria and Potentials*. Discrete Applied Mathematics, 251, 57--68. Kopie; Pflege.

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. - 2018 (mit N. Kukushkin).
*Cournot Tatonnement and Nash Equilibrium in Binary Status Games.*Economics Bulletin, 38, 2, 1038--1044. Kopie.

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. - 2018
(mit W. Pijnappel).
*The Hotelling Bi-matrix Game.*Optimization Letters, 12, 1, 187--202, Kopie; Pflege.

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. - 2018 (mit F. Quartieri).
*Cournot Equilibrium Uniqueness via Demi-Concavity.*Optimization, 67, 4, 441--455, Kopie.

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. - 2017 (mit T. Iimura und T. Watanabe).
*Best-Response Potential for Hotelling Pure Location Games.*Economics Letters, Volume 160, 73--77, Kopie.

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. - 2017 (mit T. Iimura und T. Watanabe).
*Best-reply Potential for Two-Person One-Dimensional Pure Location Games.*Tokyo Metropolitan University Research Paper Series, No. 178. Kopie.

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. - 2016 (mit R. Haagsma).
*On the Endowment Effect in `Apple-Mars' Experiments.*Apstract, 10, 2-3, 47--50, Kopie; Pflege.

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. - 2016.
*The Selten-Szidarovszky Technique: the Transformation Part.*In: Recent Advances in Game Theory and Applications, 147--164. Editors: L. Petrosyan and V. Mazalov. Birkhauser. ISBN 9783319438382. Kopie; Pflege.

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. - 2016 (mit F. Quartieri).
*Equilibrium Theory for Cournot Oligopolies and Related Games: Essays in Honour of Koji Okuguchi.*Springer Series in Game Theory. Editors: P. v. Mouche and F. Quartieri. ISBN 978-3-319-29253-3. Kopie.

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. - 2016.
*On the Geometric Structure of the Cournot Equilibrium Set: the Case of Concave Industry Revenue and Convex Costs.*In:*Equilibrium Theory for Cournot Oligopolies and Related Games: Essays in Honour of Koji Okuguchi.*Springer Series in Game Theory, 63--88. Editors: P. v. Mouche and F. Quartieri. ISBN 978-3-319-29253-3. Kopie.

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. - 2015 (mit W. Heijman).
*Floquet Theory and Economic Dynamics II.*WASS Working Paper No. 15. DOI:10.13140/RG.2.1.1067.4009. Kopie.

[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. - 2015 (mit W. Heijman, P. Heringa und N. Abudaldah).
*Return of the Icecream Men. A Discrete Hotelling Game.*Romanian Journal of Regional Science, 9, 2, 39-48. Kopie.

[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; Pflege. ]

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. - 2015 (mit H. Folmer).
*Nash Equilibria of Transboundary Pollution Games.*In: Handbook of Research Methods and Applications in Environmental Studies, 504--524. Edward-Elgar. Editor: M. Ruth. Kopie; Pflege.

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. - 2015 (mit F. Quartieri).
*Cournot Equilibrium Uniqueness in Case of Concave Industry Revenue: a Simple Proof.*Economics Bulletin, 35, 2, 1299-1305. Kopie; Pflege.

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. - 2015 (mit T. Yamazaki).
*Sufficient and Necessary Conditions for Equilibrium Uniqueness in Aggregative Games.*Journal of Nonlinear and Convex Analysis, 16, 2, 353-364. Kopie; Pflege.

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. - 2014 (mit M. Finus und B. Rundshagen).
*On Uniqueness of Coalitional Equilibria.*In: Contributions to Game Theory and Management. Volume VII, 51-60. Editors: L. Petrosjan, N. Zenkevich. St. Petersburg State University. ISSN 2310-2608. Kopie. Pflege.

[Verbesserung von: 2005 (mit M. Finus und B. Rundshagen)*Uniqueness of Coalitional Equilibria.*Fondazione Eni Enrico Mattei, Nota de Lavaro 23.2005. Kopie; Pflege. ]

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. - 2014 (mit P. Bertoletti).
*Inferior Factor in Cournot Oligopoly Revisited.*Journal of Economics, 112, 85-90. Kopie; Pflege.

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. - 2013 (mit F. Quartieri).
*On the Uniqueness of Cournot Equilibrium in Case of Concave Integrated Price Flexibility.*Journal of Global Optimization, 57, 3, 707-718. Kopie. Pflege.

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. - 2013 (mit H. Folmer).
*Analysing the Folk Theorem for Linked Repeated Games.*In: Contributions to Game Theory and Management. Volume VI, 146-164. Editors: L. Petrosjan, N. Zenkevich. Graduate School of Management St. Petersburg. ISBN 978-5-9924-0080-9. Kopie. Pflege.

[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; Pflege. ]

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. - 2013 (mit F. Quartieri und F. Szidarovszky).
*On a Fixed Point Problem Transformation Method.*Proceedings of the 10th international conference on fixed point theory and its applications (ICFPTA), 179--190. ISBN 978-606-17-0420-0. ISSN 1661-7738. Kopie; Pflege.

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. - 2013 (mit W. Heijman).
*A Procedure for Determining an Optimal Landscape and its Monetary Value.*In: The Economic Value of Landscapes. Routledge Studies in Ecological Economics, Volume 26, 123--135. Editors C. van der Heide and W. Heijman. Kopie.

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. - 2013 (mit R. Haagsma).
*Egalitarian Norms, Economic Development, and Ethnic Polarization.*Journal of Comparative Economics, 41, 719-744. DOI. Pflege.

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.

Komplette Publikationsliste.

- (mit P. Moffatt und E. Smits).
*On Concrete Giffen Utility Functions with Very Nice Theoretical Properties.* -
*Multi Objective Games: Separability, Tensor Sums and Permutations.* -
*Biconcavity.* -
*Nash Equilibrium Uniqueness in Asymmetric Rent-seeking Contests: a short proof.* - (mit W. Pijnappel).
*On Fully Reducible Polynomials with Formal Power Series as Coefficients.* -
*The Selten-Szidarovszky Technique: Backward versus Virtual backward Reaction Correspondences.* -
*A Potpourri of Useful Results for Games in Strategic Form, Aggregative Games and Cournot Oligopolies.* -
*Shock-independency of the Periodic Samuelson-Hicks Model.* -
*Transboundary Pollution Games.* -
*On Equilibrium Existence for Ordinal Status Games with Finitely Many Players.* - (mit W. Pijnappel.)
*Pi in the Mandelbrot Set Revisited.* -
*The Monopoly Profit Function.* - (mit W. Pijnappel).
*Vers une Théorie de Perturbation de Rayleigh-Schrödinger Algébrique.* -
*Micro-economie (Buch).* -
*Spectral Asymptotics of Periodic Discrete Schrödinger Operators. II.* -
*The Selten-Szidarovszky Technique (Buch).* -
*Matrices de Jacobi Périodiques (Buch).* -
*Nombre de Bandes de l'Opérateur Discret de Harper Triangulaire.*

Meine Erdöszahl ist 3 (Erdös - Totik - Simon - von Mouche).