Walter Trockel | Bielefeld University (original) (raw)
Papers by Walter Trockel
Springer eBooks, Sep 5, 2005
In this chapter we develop the basic theory of stochastic optimization. After some introductory r... more In this chapter we develop the basic theory of stochastic optimization. After some introductory remarks about the mathematical representation of uncertainty, we investigate the key ingredients of general stochastic programs, i.e. decision strategies, constraints, and objective functions. Subsequently, the static and dynamic versions of a stochastic optimization problem are formulated, and some elementary regularity conditions are discussed. Under these regularity conditions, the static and dynamic versions of the stochastic program at hand can be shown to be well-defined, solvable, and equivalent. Finally, we discuss two useful indicators, which enjoy wide popularity in literature: the expected value of perfect information (EVPI) represents the maximum amount to be paid in return for complete and accurate information about the future, whereas the value of the stochastic solution (VSS) quantifies the cost of ignoring uncertainty in choosing a decision.
Lecture Notes in Economics and Mathematical Systems
Lecture Notes in Economics and Mathematical Systems, 1984
Social Science Research Network, 2012
These notes consist of two parts. In the first one I present a counter example to Proposition 3 o... more These notes consist of two parts. In the first one I present a counter example to Proposition 3 of Anbarci & Sun (2013). In the second part I propose a correction based on an axiom similar to one used by Salonen (1988) in the first axiomatization of the Discrete Raiffa solution.
RePEc: Research Papers in Economics, 2017
The paper presents and discusses an alternative approach to bargaining games. N-person bargaining... more The paper presents and discusses an alternative approach to bargaining games. N-person bargaining games with complete information are shown to induce in a canonical way an Arrow-Debreu economy with production and private ownership. The unique Walras stable competitive equilibrium of this economy is shown to coincide with an asymmetric Nash bargaining solution of the underlying game with weights corresponding to the shares in production. In the case of an economy with equal shares in production, the unique competitive equilibrium coincides with the symmetric Nash bargaining solution. As this in turn represents the unique Shapley nontransferable utility (NTU) value our paper solves a problem posed by Shubik, namely to find a model in which the Shapley NTU value is a Walrasian equilibrium. "There has been some controversy about the interpretation of the A-transfer-value... no consensus has yet emerged on the significance of these concerns, which had been addressed.., in numerous explorations of the A-transfer-value as a tool for analysing games and markets..." (Roth, 1985).
RePEc: Research Papers in Economics, 1991
RePEc: Research Papers in Economics, 1986
RePEc: Research Papers in Economics, 1987
RePEc: Research Papers in Economics, 2017
Studies in economic design, 2019
Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch ge... more Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence.
Journal of Mathematical Economics, Dec 1, 1980
We sugges! in this paper to treat the problem of smoothing demand by aggregation in a two-step pr... more We sugges! in this paper to treat the problem of smoothing demand by aggregation in a two-step procedure, corresponding to the two different constituents of consumption characteristics. wealth and preferences. Instead of imposing a manifold structure on preferences we exploit the nice structure of wealth-space. The first step of this procedure, aggregation with respect to wealth. is carried out. It is shown th!at, for any preference, aggregation with respect to wealth yields a mean demand which is almost everywhere Ct. Moreover, it is shown that for an important class of preferences, vanishing Gaussian curvature of indifference surfaces does not destroy differentiab!!ity of the mean demand function. *The authors have benefitted from conversations with several colleagues and guests at the Sonderforschungsbereich 21 in Bonn. Financial support of the Deutsche Forschungsgemeinschaft is gratefully acknowledged. 'For prices in this exceptional set demand need not be unique. In another paper [Dicrker Dierker Trockel (3 980)] we show that for most preferences aggregation with respect to ~vealth yelds a continuous mean demand function.
Economics Letters, Jun 1, 1996
Economic Theory Bulletin, Jun 12, 2014
This article provides an axiomatic characterization of the discrete Raiffa solution for two-perso... more This article provides an axiomatic characterization of the discrete Raiffa solution for two-person bargaining games. The extension to n>2$$n>2 players is straightforward. This solution had been introduced as one of four “arbitration schemes” by Raiffa (Arbitration schemes for generalized-two person games, 1951; Ann Math Stud 28:361–387, 1953). The axiomatization expresses a consistency property by which the standard midpoint solution for TU-bargaining games can be extended to general NTU-bargaining games. The underlying linear approximation from inside captures a dual view to the linear approximation from outside that underlies Nash’s (Econometrica 18:155–162, 1950) axiomatization of his Nash solution that is also embodied in Shapley’s (Utility comparison and the theory of games. In La Décision, pp. 251–263, 1969) \lambda λ—transfer principle and, even earlier, in a lemma by Harsanyi (Contributions to the theory of games IV, pp 325–355, 1959). Finally, the present axiomatization is compared with other ones in the literature that are motivated by Kalai (Econometrica 45:1623–1630, 1977) axiom of step-by-step negotiation.
International Game Theory Review, Sep 1, 1999
Journal of Mathematical Economics, 2011
Economic Theory, Sep 1, 2000
The present paper provides three different support results for the Nash bargaining solution of n-... more The present paper provides three different support results for the Nash bargaining solution of n-person bargaining games. First, for any bargaining game there is defined a non-cooperative game in strategic form, whose unique Nash equilibrium induces a payoff vector that coincides with the Nash solution of the bargaining game. Next this game is modified in such a way that the unique Nash equilibrium that supports the Nash solution is even in dominant strategies. After that an n-stage game in extensive form is presented whose unique subgame perfect equilibrium supports the Nash solution of the bargaining game. Finally, the support results are shown to induce implementation results in the sense of mechanism theory.
RePEc: Research Papers in Economics, 2020
RePEc: Research Papers in Economics, Aug 15, 2011
Howard (1992) argues that the Nash bargaining solution is not Nash implementable, as it does not ... more Howard (1992) argues that the Nash bargaining solution is not Nash implementable, as it does not satisfy Maskin monotonicity. His arguments can be extended to other bargaining solutions as well. However, by defining a social choice correspondence that is based on the solution rather than on its realizations, one can overcome this shortcoming. We even show that such correspondences satisfy a stronger version of monotonicity that is even sufficient for Nash implementability.
Springer eBooks, Sep 5, 2005
In this chapter we develop the basic theory of stochastic optimization. After some introductory r... more In this chapter we develop the basic theory of stochastic optimization. After some introductory remarks about the mathematical representation of uncertainty, we investigate the key ingredients of general stochastic programs, i.e. decision strategies, constraints, and objective functions. Subsequently, the static and dynamic versions of a stochastic optimization problem are formulated, and some elementary regularity conditions are discussed. Under these regularity conditions, the static and dynamic versions of the stochastic program at hand can be shown to be well-defined, solvable, and equivalent. Finally, we discuss two useful indicators, which enjoy wide popularity in literature: the expected value of perfect information (EVPI) represents the maximum amount to be paid in return for complete and accurate information about the future, whereas the value of the stochastic solution (VSS) quantifies the cost of ignoring uncertainty in choosing a decision.
Lecture Notes in Economics and Mathematical Systems
Lecture Notes in Economics and Mathematical Systems, 1984
Social Science Research Network, 2012
These notes consist of two parts. In the first one I present a counter example to Proposition 3 o... more These notes consist of two parts. In the first one I present a counter example to Proposition 3 of Anbarci & Sun (2013). In the second part I propose a correction based on an axiom similar to one used by Salonen (1988) in the first axiomatization of the Discrete Raiffa solution.
RePEc: Research Papers in Economics, 2017
The paper presents and discusses an alternative approach to bargaining games. N-person bargaining... more The paper presents and discusses an alternative approach to bargaining games. N-person bargaining games with complete information are shown to induce in a canonical way an Arrow-Debreu economy with production and private ownership. The unique Walras stable competitive equilibrium of this economy is shown to coincide with an asymmetric Nash bargaining solution of the underlying game with weights corresponding to the shares in production. In the case of an economy with equal shares in production, the unique competitive equilibrium coincides with the symmetric Nash bargaining solution. As this in turn represents the unique Shapley nontransferable utility (NTU) value our paper solves a problem posed by Shubik, namely to find a model in which the Shapley NTU value is a Walrasian equilibrium. "There has been some controversy about the interpretation of the A-transfer-value... no consensus has yet emerged on the significance of these concerns, which had been addressed.., in numerous explorations of the A-transfer-value as a tool for analysing games and markets..." (Roth, 1985).
RePEc: Research Papers in Economics, 1991
RePEc: Research Papers in Economics, 1986
RePEc: Research Papers in Economics, 1987
RePEc: Research Papers in Economics, 2017
Studies in economic design, 2019
Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch ge... more Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence.
Journal of Mathematical Economics, Dec 1, 1980
We sugges! in this paper to treat the problem of smoothing demand by aggregation in a two-step pr... more We sugges! in this paper to treat the problem of smoothing demand by aggregation in a two-step procedure, corresponding to the two different constituents of consumption characteristics. wealth and preferences. Instead of imposing a manifold structure on preferences we exploit the nice structure of wealth-space. The first step of this procedure, aggregation with respect to wealth. is carried out. It is shown th!at, for any preference, aggregation with respect to wealth yields a mean demand which is almost everywhere Ct. Moreover, it is shown that for an important class of preferences, vanishing Gaussian curvature of indifference surfaces does not destroy differentiab!!ity of the mean demand function. *The authors have benefitted from conversations with several colleagues and guests at the Sonderforschungsbereich 21 in Bonn. Financial support of the Deutsche Forschungsgemeinschaft is gratefully acknowledged. 'For prices in this exceptional set demand need not be unique. In another paper [Dicrker Dierker Trockel (3 980)] we show that for most preferences aggregation with respect to ~vealth yelds a continuous mean demand function.
Economics Letters, Jun 1, 1996
Economic Theory Bulletin, Jun 12, 2014
This article provides an axiomatic characterization of the discrete Raiffa solution for two-perso... more This article provides an axiomatic characterization of the discrete Raiffa solution for two-person bargaining games. The extension to n>2$$n>2 players is straightforward. This solution had been introduced as one of four “arbitration schemes” by Raiffa (Arbitration schemes for generalized-two person games, 1951; Ann Math Stud 28:361–387, 1953). The axiomatization expresses a consistency property by which the standard midpoint solution for TU-bargaining games can be extended to general NTU-bargaining games. The underlying linear approximation from inside captures a dual view to the linear approximation from outside that underlies Nash’s (Econometrica 18:155–162, 1950) axiomatization of his Nash solution that is also embodied in Shapley’s (Utility comparison and the theory of games. In La Décision, pp. 251–263, 1969) \lambda λ—transfer principle and, even earlier, in a lemma by Harsanyi (Contributions to the theory of games IV, pp 325–355, 1959). Finally, the present axiomatization is compared with other ones in the literature that are motivated by Kalai (Econometrica 45:1623–1630, 1977) axiom of step-by-step negotiation.
International Game Theory Review, Sep 1, 1999
Journal of Mathematical Economics, 2011
Economic Theory, Sep 1, 2000
The present paper provides three different support results for the Nash bargaining solution of n-... more The present paper provides three different support results for the Nash bargaining solution of n-person bargaining games. First, for any bargaining game there is defined a non-cooperative game in strategic form, whose unique Nash equilibrium induces a payoff vector that coincides with the Nash solution of the bargaining game. Next this game is modified in such a way that the unique Nash equilibrium that supports the Nash solution is even in dominant strategies. After that an n-stage game in extensive form is presented whose unique subgame perfect equilibrium supports the Nash solution of the bargaining game. Finally, the support results are shown to induce implementation results in the sense of mechanism theory.
RePEc: Research Papers in Economics, 2020
RePEc: Research Papers in Economics, Aug 15, 2011
Howard (1992) argues that the Nash bargaining solution is not Nash implementable, as it does not ... more Howard (1992) argues that the Nash bargaining solution is not Nash implementable, as it does not satisfy Maskin monotonicity. His arguments can be extended to other bargaining solutions as well. However, by defining a social choice correspondence that is based on the solution rather than on its realizations, one can overcome this shortcoming. We even show that such correspondences satisfy a stronger version of monotonicity that is even sufficient for Nash implementability.