Time Preference and Economic Progress
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Approximate Nash Equilibria in Large Nonconvex Aggregative Games
Approximate Nash equilibria in large nonconvex aggregative games Kang Liu,∗ Nadia Oudjane,† Cheng Wan‡ November 26, 2020 Abstract 1 This paper shows the existence of O( nγ )-Nash equilibria in n-player noncooperative aggregative games where the players' cost functions depend only on their own action and the average of all the players' actions, and is lower semicontinuous in the former while γ-H¨oldercontinuous in the latter. Neither the action sets nor the cost functions need to be convex. For an important class of aggregative games which includes con- gestion games with γ being 1, a proximal best-reply algorithm is used to construct an 1 3 O( n )-Nash equilibria with at most O(n ) iterations. These results are applied in a nu- merical example of demand-side management of the electricity system. The asymptotic performance of the algorithm is illustrated when n tends to infinity. Keywords. Shapley-Folkman lemma, aggregative games, nonconvex game, large finite game, -Nash equilibrium, proximal best-reply algorithm, congestion game MSC Class Primary: 91A06; secondary: 90C26 1 Introduction In this paper, players minimize their costs so that the definition of equilibria and equilibrium conditions are in the opposite sense of the usual usage where players maximize their payoffs. Players have actions in Euclidean spaces. If it is not precised, then \Nash equilibrium" means a pure-strategy Nash equilibrium. This paper studies the approximation of pure-strategy Nash equilibria (PNE for short) in a specific class of finite-player noncooperative games, referred to as large nonconvex aggregative games. Recall that the cost functions of players in an aggregative game depend on their own action (i.e. -
A Stated Preference Case Study on Traffic Noise in Lisbon
The Valuation of Environmental Externalities: A stated preference case study on traffic noise in Lisbon by Elisabete M. M. Arsenio Submitted in accordance with the requirements for the degree of Doctor of Philosophy University of Leeds Institute for Transport Studies August 2002 The candidate confirms that the work submitted is his own and that appropriate credit has been given where reference has been made to the work of others. Acknowledgments To pursue a PhD under a part-time scheme is only possible through a strong will and effective support. I thank my supervisors at the ITS, Dr. Abigail Bristow and Dr. Mark Wardman for all the support and useful comments throughout this challenging topic. I would also like to thank the ITS Directors during my research study Prof. Chris Nash and Prof A. D. May for having supported my attendance in useful Courses and Conferences. I would like to thank Mr. Stephen Clark for the invaluable help on the computer survey, as well as to Dr. John Preston for the earlier research motivation. I would like to thank Dr. Hazel Briggs, Ms Anna Kruk, Ms Julie Whitham, Mr. F. Saremi, Mr. T. Horrobin and Dr. R. Batley for the facilities’ support. Thanks also due to Prof. P. Mackie, Prof. P. Bonsall, Mr. F. Montgomery, Ms. Frances Hodgson and Dr. J. Toner. Thanks for all joy and friendship to Bill Lythgoe, Eric Moreno, Jiao Wang, Shojiro and Mauricio. Special thanks are also due to Dr. Paul Firmin for the friendship and precious comments towards the presentation of this thesis. Without Dr. -
Lecture Notes General Equilibrium Theory: Ss205
LECTURE NOTES GENERAL EQUILIBRIUM THEORY: SS205 FEDERICO ECHENIQUE CALTECH 1 2 Contents 0. Disclaimer 4 1. Preliminary definitions 5 1.1. Binary relations 5 1.2. Preferences in Euclidean space 5 2. Consumer Theory 6 2.1. Digression: upper hemi continuity 7 2.2. Properties of demand 7 3. Economies 8 3.1. Exchange economies 8 3.2. Economies with production 11 4. Welfare Theorems 13 4.1. First Welfare Theorem 13 4.2. Second Welfare Theorem 14 5. Scitovsky Contours and cost-benefit analysis 20 6. Excess demand functions 22 6.1. Notation 22 6.2. Aggregate excess demand in an exchange economy 22 6.3. Aggregate excess demand 25 7. Existence of competitive equilibria 26 7.1. The Negishi approach 28 8. Uniqueness 32 9. Representative Consumer 34 9.1. Samuelsonian Aggregation 37 9.2. Eisenberg's Theorem 39 10. Determinacy 39 GENERAL EQUILIBRIUM THEORY 3 10.1. Digression: Implicit Function Theorem 40 10.2. Regular and Critical Economies 41 10.3. Digression: Measure Zero Sets and Transversality 44 10.4. Genericity of regular economies 45 11. Observable Consequences of Competitive Equilibrium 46 11.1. Digression on Afriat's Theorem 46 11.2. Sonnenschein-Mantel-Debreu Theorem: Anything goes 47 11.3. Brown and Matzkin: Testable Restrictions On Competitve Equilibrium 48 12. The Core 49 12.1. Pareto Optimality, The Core and Walrasian Equiilbria 51 12.2. Debreu-Scarf Core Convergence Theorem 51 13. Partial equilibrium 58 13.1. Aggregate demand and welfare 60 13.2. Production 61 13.3. Public goods 62 13.4. Lindahl equilibrium 63 14. -
The Role of Time Preference on Wealth
THE ROLE OF TIME PREFERENCE ON WEALTH _______________________________________ A Thesis presented to the Faculty of the Graduate School at the University of Missouri-Columbia _______________________________________________________ In Partial Fulfillment of the Requirements for the Degree Master of Science _____________________________________________________ by ANDREW MARK ZUMWALT Dr. Deanna L. Sharpe, Thesis Co-Supervisor Dr. Sandra J. Huston, Thesis Co-Supervisor December 2008 The undersigned, appointed by the dean of the Graduate School, have examined the thesis entitled THE ROLE OF TIME PREFERENCE ON WEALTH presented by Andrew Mark Zumwalt, a candidate for the degree of master of science, and hereby certify that, in their opinion, it is worthy of acceptance. Professor Deanna L. Sharpe Professor Sandra J. Huston Professor Thomas Johnson ad maiorem dei gloriam ACKNOWLEDGEMENTS All throughout my life, I have been supported by strong individuals. These people have been the pillars of my success, and without their influence and guidance, I would be nowhere near where I am now. I can only hope that I have, in turn, enriched their lives. I aspire to fulfill a similar role for the young men and women I meet. While I believe that my success has been caused by the cumulative efforts of many individuals, I would like to take this limited space to thank those that have helped with this part of my life. First, I would like to thank the members of my committee. Dr Thomas Johnson’s experience from agricultural economics provided insight into overlooked ideas in my research when viewed in a broader context. He has also been extremely understanding with regard to interesting timetables, and I’ve thoroughly enjoyed learning from him in both inside and outside the classroom. -
Game Theory: Preferences and Expected Utility
Game Theory: Preferences and Expected Utility Branislav L. Slantchev Department of Political Science, University of California – San Diego April 19, 2005 Contents. 1 Preferences 2 2 Utility Representation 4 3 Choice Under Uncertainty 5 3.1Lotteries............................................. 5 3.2PreferencesOverLotteries.................................. 7 3.3vNMExpectedUtilityFunctions............................... 9 3.4TheExpectedUtilityTheorem................................ 11 4 Risk Aversion 16 1 Preferences We want to examine the behavior of an individual, called a player, who must choose from among a set of outcomes. Begin by formalizing the set of outcomes from which this choice is to be made. Let X be the (finite) set of outcomes with common elements x,y,z. The elements of this set are mutually exclusive (choice of one implies rejection of the others). For example, X can represent the set of candidates in an election and the player needs to chose for whom to vote. Or it can represent a set of diplomatic and military actions—bombing, land invasion, sanctions—among which a player must choose one for implementation. The standard way to model the player is with his preference relation, sometimes called a binary relation. The relation on X represents the relative merits of any two outcomes for the player with respect to some criterion. For example, in mathematics the familiar weak inequality relation, ’≥’, defined on the set of integers, is interpreted as “integer x is at least as big as integer y” whenever we write x ≥ y. Similarly, a relation “is more liberal than,” denoted by ’P’, can be defined on the set of candidates, and interpreted as “candidate x is more liberal than candidate y” whenever we write xPy. -
Nine Lives of Neoliberalism
A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Plehwe, Dieter (Ed.); Slobodian, Quinn (Ed.); Mirowski, Philip (Ed.) Book — Published Version Nine Lives of Neoliberalism Provided in Cooperation with: WZB Berlin Social Science Center Suggested Citation: Plehwe, Dieter (Ed.); Slobodian, Quinn (Ed.); Mirowski, Philip (Ed.) (2020) : Nine Lives of Neoliberalism, ISBN 978-1-78873-255-0, Verso, London, New York, NY, https://www.versobooks.com/books/3075-nine-lives-of-neoliberalism This Version is available at: http://hdl.handle.net/10419/215796 Standard-Nutzungsbedingungen: Terms of use: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Documents in EconStor may be saved and copied for your Zwecken und zum Privatgebrauch gespeichert und kopiert werden. personal and scholarly purposes. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle You are not to copy documents for public or commercial Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich purposes, to exhibit the documents publicly, to make them machen, vertreiben oder anderweitig nutzen. publicly available on the internet, or to distribute or otherwise use the documents in public. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, If the documents have been made available under an Open gelten abweichend von diesen Nutzungsbedingungen die in der dort Content Licence (especially Creative -
The Geographies of Soft Paternalism in the UK: the Rise of the Avuncular State and Changing Behaviour After Neoliberalism
The Geographies of Soft Paternalism in the UK: the rise of the avuncular state and changing behaviour after neoliberalism Rhys Jones, Jessica Pykett and Mark Whitehead, Aberystwyth University Forthcoming, Geography Compass Abstract Soft paternalism or libertarian paternalism has emerged as a new rationality of governing in the UK under New Labour, denoting a style of governing which is aimed at both increasing choice and ensuring welfare. Popularised in Thaler and Sunstein’s best-selling title, Nudge, the approach of libertarian paternalism poses new questions for critical human geographers interested in how the philosophies and practices of the Third Way are being adapted and developed in a range of public policy spheres, including the environment, personal finance and health policy. Focusing specifically on the case of the UK, this article charts how soft paternalism appeals to the intellectual influences of behavioural economics, psychology and the neurosciences, amongst others, to justify government interventions based on the ‘non-ideal’ or irrational citizen. By identifying the distinctive mechanisms associated with this ‘behaviour change’ agenda, such as ‘choice architecture’, we explore the contribution of behavioural geographies and political geographies of the state to further understanding of the techniques and rationalities of governing. 1 The Geographies of Soft Paternalism: the rise of the avuncular state and changing behaviour after neoliberalism 1. Introduction: The Chicago School comes to Britain Mark II. On March 24 2009, David Cameron stood next to the eminent behavioural economist Richard Thaler at the London Stock Exchange. David Cameron was in the City of London to deliver his keynote speech on banking reform and the Tory’s response to the credit crunch. -
Variable-Time-Preference-He-Et-Al.Pdf
Cognitive Psychology 111 (2019) 53–79 Contents lists available at ScienceDirect Cognitive Psychology journal homepage: www.elsevier.com/locate/cogpsych Variable time preference ⁎ T Lisheng Hea, , Russell Golmanb, Sudeep Bhatiaa,1 a University of Pennsylvania, United States b Carnegie Mellon University, United States ARTICLE INFO ABSTRACT Keywords: We re-examine behavioral patterns of intertemporal choice with recognition that time pre- Decision making ferences may be inherently variable, focusing in particular on the explanatory power of an ex- Intertemporal choice ponential discounting model with variable discount factors – the variable exponential model. We Stochastic choice provide analytical results showing that this model can generate systematically different choice Preference variability patterns from an exponential discounting model with a fixed discount factor. The variable ex- Computational modeling ponential model accounts for the common behavioral pattern of decreasing impatience, which is typically attributed to hyperbolic discounting. The variable exponential model also generates violations of strong stochastic transitivity in choices involving intertemporal dominance. We present the results of two experiments designed to evaluate the variable exponential model in terms of quantitative fit to individual-level choice data. Data from these experiments reveal that allowing for a variable discount factor significantly improves the fit of the exponential model, and that a variable exponential model provides a better account of individual-level choice probabilities than hyperbolic discounting models. In a third experiment we find evidence of strong stochastic transitivity violations when intertemporal dominance is involved, in accordance with the variable exponential model. Overall, our analytical and experimental results indicate that exponential discounting can explain intertemporal choice behavior that was supposed to be beyond its descriptive scope if the discount factor is permitted to vary at random. -
Time Preference and Its Relationship with Age, Health, and Survival Probability
Judgment and Decision Making, Vol. 4, No. 1, February 2009, pp. 1–19 Time preference and its relationship with age, health, and survival probability Li-Wei Chao∗1, Helena Szrek2, Nuno Sousa Pereira2,3 and Mark V. Pauly4 1 Population Studies Center, University of Pennsylvania 2 Center for Economics and Finance, University of Porto 3 Faculty of Economics, University of Porto 4 Health Care Systems Department, The Wharton School, University of Pennsylvania Abstract Although theories from economics and evolutionary biology predict that one’s age, health, and survival probability should be associated with one’s subjective discount rate (SDR), few studies have empirically tested for these links. Our study analyzes in detail how the SDR is related to age, health, and survival probability, by surveying a sample of individuals in townships around Durban, South Africa. In contrast to previous studies, we find that age is not significantly related to the SDR, but both physical health and survival expectations have a U-shaped relationship with the SDR. Individuals in very poor health have high discount rates, and those in very good health also have high discount rates. Similarly, those with expected survival probability on the extremes have high discount rates. Therefore, health and survival probability, and not age, seem to be predictors of one’s SDR in an area of the world with high morbidity and mortality. Keywords: subjective discount rate; delay discounting; expected survival probability; health; age; South Africa. 1 Introduction diately, and the degree to which an individual discounts the future reward will be measured as the subjective dis- People generally prefer to receive a reward sooner rather count rate or SDR (which we define formally below). -
Revealed Preference with a Subset of Goods
JOURNAL OF ECONOMIC THEORY 46, 179-185 (1988) Revealed Preference with a Subset of Goods HAL R. VARIAN* Department of Economics, Unioersity of Michigan, Ann Arbor, Michigan 48109 Received November 4. 1985; revised August 14. 1987 Suppose that you observe n choices of k goods and prices when the consumer is actually choosing from a set of k + 1 goods. Then revealed preference theory puts essentially no restrictions on the behavior of the data. This is true even if you also observe the quantity demanded of good k+ 1, or its price. The proofs of these statements are not difficult. Journal cf Economic Literature Classification Number: 022. ii? 1988 Academic Press. Inc. Suppose that we are given n observations on a consumer’s choices of k goods, (p,, x,), where pi and xi are nonnegative k-dimensional vectors. Under what conditions can we find a utility function u: Rk -+ R that rationalizes these observations? That is, when can we find a utility function that achieves its constrained maximum at the observed choices? This is, of course, a classical question of consumer theory. It has been addressed from two distinct viewpoints, the first known as integrabilitv theory and the second known as revealed preference theory. Integrability theory is appropriate when one is given an entire demand function while revealed preference theory is more suited when one is given a finite set of demand observations, the case described above. In the revealed preference case, it is well known that some variant of the Strong Axiom of Revealed Preference (SARP) is a necessary and sufficient condition for the data (pi, xi) to be consistent with utility maximization. -
Revealed Preference in Game Theory
Revealed Preference in Game Theory Ad´am´ Galambos 1 Department of Managerial Economics and Decision Sciences, Kellogg School of Management, Northwestern University, 2001 Sheridan Road, Evanston, IL, 60208 Abstract I characterize joint choice behavior generated by the pure strategy Nash equilib- rium solution concept by an extension of the Congruence Axiom of Richter(1966) to multiple agents. At the same time, I relax the “complete domain” assumption of Yanovskaya(1980) and Sprumont(2000) to “closed domain.” Without any restric- tions on the domain of the choice correspondence, determining pure strategy Nash rationalizability is computationally very complex. Specifically, it is NP–complete even if there are only two players. In contrast, the analogous problem with a single decision maker can be determined in polynomial time. Key words: Nash equilibrium; Revealed preference; Complexity Email address: [email protected] (Ad´am´ Galambos). 1 I am grateful to Professor Marcel K. Richter for many inspiring and stimulating discussions on these topics, as well as many suggestions. I wish to thank Profes- sors Beth Allen, Andrew McLennan and Jan Werner, and participants of the Mi- cro/Finance and Micro/Game theory workshops at the University of Minnesota for their comments. This paper is based on my doctoral dissertation at the University of Minnesota. I gratefully acknowledge the financial support of the NSF through grant SES-0099206 (principal investigator: Professor Jan Werner). 27 September 2005 1 Introduction What are the testable implications of the Nash equilibrium solution? If we observed a group of agents play different games, could we tell, without knowing their preferences, whether they are playing according to Nash equilibrium? Such questions could be of interest to a regulatory agency, wanting to know if some firms they observe in the market are behaving in a competitive or in a collusive way. -
14.12 Game Theory Lecture Notes Theory of Choice
14.12 Game Theory Lecture Notes Theory of Choice Muhamet Yildiz (Lecture 2) 1 The basic theory of choice We consider a set X of alternatives. Alternatives are mutually exclusive in the sense that one cannot choose two distinct alternatives at the same time. We also take the set of feasible alternatives exhaustive so that a player’s choices will always be defined. Note that this is a matter of modeling. For instance, if we have options Coffee and Tea, we define alternatives as C = Coffee but no Tea, T = Tea but no Coffee, CT = Coffee and Tea, and NT = no Coffee and no Tea. Take a relation on X. Note that a relation on X is a subset of X X.Arelation º × is said to be complete if and only if, given any x, y X,eitherx y or y x.A º ∈ º º relation is said to be transitive if and only if, given any x, y, z X, º ∈ [x y and y z] x z. º º ⇒ º Arelationisapreference relation if and only if it is complete and transitive. Given any preference relation ,wecandefine strict preference by º Â x y [x y and y x], Â ⇐⇒ º 6º and the indifference ∼ by x ∼ y [x y and y x]. ⇐⇒ º º Apreferencerelationcanberepresented by a utility function u : X R in the → following sense: x y u(x) u(y) x, y X. º ⇐⇒ ≥ ∀ ∈ 1 The following theorem states further that a relation needs to be a preference relation in order to be represented by a utility function.