Bargaining and Collusion in a Regulatory Model
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Frequently Asked Questions in Mathematics
Frequently Asked Questions in Mathematics The Sci.Math FAQ Team. Editor: Alex L´opez-Ortiz e-mail: [email protected] Contents 1 Introduction 4 1.1 Why a list of Frequently Asked Questions? . 4 1.2 Frequently Asked Questions in Mathematics? . 4 2 Fundamentals 5 2.1 Algebraic structures . 5 2.1.1 Monoids and Groups . 6 2.1.2 Rings . 7 2.1.3 Fields . 7 2.1.4 Ordering . 8 2.2 What are numbers? . 9 2.2.1 Introduction . 9 2.2.2 Construction of the Number System . 9 2.2.3 Construction of N ............................... 10 2.2.4 Construction of Z ................................ 10 2.2.5 Construction of Q ............................... 11 2.2.6 Construction of R ............................... 11 2.2.7 Construction of C ............................... 12 2.2.8 Rounding things up . 12 2.2.9 What’s next? . 12 3 Number Theory 14 3.1 Fermat’s Last Theorem . 14 3.1.1 History of Fermat’s Last Theorem . 14 3.1.2 What is the current status of FLT? . 14 3.1.3 Related Conjectures . 15 3.1.4 Did Fermat prove this theorem? . 16 3.2 Prime Numbers . 17 3.2.1 Largest known Mersenne prime . 17 3.2.2 Largest known prime . 17 3.2.3 Largest known twin primes . 18 3.2.4 Largest Fermat number with known factorization . 18 3.2.5 Algorithms to factor integer numbers . 18 3.2.6 Primality Testing . 19 3.2.7 List of record numbers . 20 3.2.8 What is the current status on Mersenne primes? . -
The Three Types of Collusion: Fixing Prices, Rivals, and Rules Robert H
University of Baltimore Law ScholarWorks@University of Baltimore School of Law All Faculty Scholarship Faculty Scholarship 2000 The Three Types of Collusion: Fixing Prices, Rivals, and Rules Robert H. Lande University of Baltimore School of Law, [email protected] Howard P. Marvel Ohio State University, [email protected] Follow this and additional works at: http://scholarworks.law.ubalt.edu/all_fac Part of the Antitrust and Trade Regulation Commons, and the Law and Economics Commons Recommended Citation The Three Types of Collusion: Fixing Prices, Rivals, and Rules, 2000 Wis. L. Rev. 941 (2000) This Article is brought to you for free and open access by the Faculty Scholarship at ScholarWorks@University of Baltimore School of Law. It has been accepted for inclusion in All Faculty Scholarship by an authorized administrator of ScholarWorks@University of Baltimore School of Law. For more information, please contact [email protected]. ARTICLES THE THREE TYPES OF COLLUSION: FIXING PRICES, RIVALS, AND RULES ROBERTH. LANDE * & HOWARDP. MARVEL** Antitrust law has long held collusion to be paramount among the offenses that it is charged with prohibiting. The reason for this prohibition is simple----collusion typically leads to monopoly-like outcomes, including monopoly profits that are shared by the colluding parties. Most collusion cases can be classified into two established general categories.) Classic, or "Type I" collusion involves collective action to raise price directly? Firms can also collude to disadvantage rivals in a manner that causes the rivals' output to diminish or causes their behavior to become chastened. This "Type 11" collusion in turn allows the colluding firms to raise prices.3 Many important collusion cases, however, do not fit into either of these categories. -
Collusion Constrained Equilibrium
Theoretical Economics 13 (2018), 307–340 1555-7561/20180307 Collusion constrained equilibrium Rohan Dutta Department of Economics, McGill University David K. Levine Department of Economics, European University Institute and Department of Economics, Washington University in Saint Louis Salvatore Modica Department of Economics, Università di Palermo We study collusion within groups in noncooperative games. The primitives are the preferences of the players, their assignment to nonoverlapping groups, and the goals of the groups. Our notion of collusion is that a group coordinates the play of its members among different incentive compatible plans to best achieve its goals. Unfortunately, equilibria that meet this requirement need not exist. We instead introduce the weaker notion of collusion constrained equilibrium. This al- lows groups to put positive probability on alternatives that are suboptimal for the group in certain razor’s edge cases where the set of incentive compatible plans changes discontinuously. These collusion constrained equilibria exist and are a subset of the correlated equilibria of the underlying game. We examine four per- turbations of the underlying game. In each case,we show that equilibria in which groups choose the best alternative exist and that limits of these equilibria lead to collusion constrained equilibria. We also show that for a sufficiently broad class of perturbations, every collusion constrained equilibrium arises as such a limit. We give an application to a voter participation game that shows how collusion constraints may be socially costly. Keywords. Collusion, organization, group. JEL classification. C72, D70. 1. Introduction As the literature on collective action (for example, Olson 1965) emphasizes, groups often behave collusively while the preferences of individual group members limit the possi- Rohan Dutta: [email protected] David K. -
Subgame-Perfect Ε-Equilibria in Perfect Information Games With
Subgame-Perfect -Equilibria in Perfect Information Games with Common Preferences at the Limit Citation for published version (APA): Flesch, J., & Predtetchinski, A. (2016). Subgame-Perfect -Equilibria in Perfect Information Games with Common Preferences at the Limit. Mathematics of Operations Research, 41(4), 1208-1221. https://doi.org/10.1287/moor.2015.0774 Document status and date: Published: 01/11/2016 DOI: 10.1287/moor.2015.0774 Document Version: Publisher's PDF, also known as Version of record Document license: Taverne Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. -
Interim Correlated Rationalizability
Interim Correlated Rationalizability The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Dekel, Eddie, Drew Fudenberg, and Stephen Morris. 2007. Interim correlated rationalizability. Theoretical Economics 2, no. 1: 15-40. Published Version http://econtheory.org/ojs/index.php/te/article/view/20070015 Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:3196333 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA Theoretical Economics 2 (2007), 15–40 1555-7561/20070015 Interim correlated rationalizability EDDIE DEKEL Department of Economics, Northwestern University, and School of Economics, Tel Aviv University DREW FUDENBERG Department of Economics, Harvard University STEPHEN MORRIS Department of Economics, Princeton University This paper proposes the solution concept of interim correlated rationalizability, and shows that all types that have the same hierarchies of beliefs have the same set of interim-correlated-rationalizable outcomes. This solution concept charac- terizes common certainty of rationality in the universal type space. KEYWORDS. Rationalizability, incomplete information, common certainty, com- mon knowledge, universal type space. JEL CLASSIFICATION. C70, C72. 1. INTRODUCTION Harsanyi (1967–68) proposes solving games of incomplete -
Tacit Collusion in Oligopoly"
Tacit Collusion in Oligopoly Edward J. Green,y Robert C. Marshall,z and Leslie M. Marxx February 12, 2013 Abstract We examine the economics literature on tacit collusion in oligopoly markets and take steps toward clarifying the relation between econo- mists’analysis of tacit collusion and those in the legal literature. We provide examples of when the economic environment might be such that collusive pro…ts can be achieved without communication and, thus, when tacit coordination is su¢ cient to elevate pro…ts versus when com- munication would be required. The authors thank the Human Capital Foundation (http://www.hcfoundation.ru/en/), and especially Andrey Vavilov, for …nancial support. The paper bene…tted from discussions with Roger Blair, Louis Kaplow, Bill Kovacic, Vijay Krishna, Steven Schulenberg, and Danny Sokol. We thank Gustavo Gudiño for valuable research assistance. [email protected], Department of Economics, Penn State University [email protected], Department of Economics, Penn State University [email protected], Fuqua School of Business, Duke University 1 Introduction In this chapter, we examine the economics literature on tacit collusion in oligopoly markets and take steps toward clarifying the relation between tacit collusion in the economics and legal literature. Economists distinguish between tacit and explicit collusion. Lawyers, using a slightly di¤erent vocabulary, distinguish between tacit coordination, tacit agreement, and explicit collusion. In hopes of facilitating clearer communication between economists and lawyers, in this chapter, we attempt to provide a coherent resolution of the vernaculars used in the economics and legal literature regarding collusion.1 Perhaps the easiest place to begin is to de…ne explicit collusion. -
Imbalanced Collusive Security Games
Imbalanced Collusive Security Games Han-Ching Ou, Milind Tambe, Bistra Dilkina, and Phebe Vayanos Computer Science Department, University of Southern California {hanchino,tambe,dilkina,phebe.vayanos }@usc.edu Abstract. Colluding adversaries is a crucial challenge for defenders in many real-world applications. Previous literature has provided Collusive Security Games (COSG) to model colluding adversaries, and provided models and algorithms to generate defender strategies to counter colluding adversaries, often by devising strategies that inhibit collusion [6]. Unfortunately, this previous work focused exclusively on situations with perfectly matched adversaries, i.e., where their rewards were symmetrically distributed. In the real world, however, defenders often face adversaries where their rewards are asymmetrically distributed. Such inherent asymmetry raises a question as to whether human adversaries would at- tempt to collude in such situations, and whether defender strategies to counter such collusion should focus on inhibiting collusion. To address these open ques- tions, this paper: (i) explores and theoretically analyzes Imbalanced Collusive Security Games (ICOSG) where defenders face adversaries with asymmetrically distributed rewards; (ii) conducts extensive experiments of three different adver- sary models involving 1800 real human subjects and (iii) derives novel analysis of the reason behind why bounded rational attackers models outperform perfectly rational attackers models. The key principle discovered as the result of our -
Econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible
A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Stähler, Frank Working Paper — Digitized Version Markov perfection and cooperation in repeated games Kiel Working Paper, No. 760 Provided in Cooperation with: Kiel Institute for the World Economy (IfW) Suggested Citation: Stähler, Frank (1996) : Markov perfection and cooperation in repeated games, Kiel Working Paper, No. 760, Kiel Institute of World Economics (IfW), Kiel This Version is available at: http://hdl.handle.net/10419/46946 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 Commons Licences), you genannten Lizenz gewährten Nutzungsrechte. may exercise further usage rights as specified in the indicated licence. www.econstor.eu Kieler Arbeitspapiere Kiel Working Papers Kiel Working Paper No. 760 Markov perfection and cooperation in repeated games by Frank Stahler August 1996 Institut fur Weltwirtschaft an der Universitat Kiel The Kiel Institute of World Economics ISSN 0342 - 0787 The Kiel Institute of World Economics Diisternbrooker Weg 120 D-24105Kiel, FRG Kiel Working Paper No. -
Tic-Tac-Toe Is Not Very Interesting to Play, Because If Both Players Are Familiar with the Game the Result Is Always a Draw
CMPSCI 250: Introduction to Computation Lecture #27: Games and Adversary Search David Mix Barrington 4 November 2013 Games and Adversary Search • Review: A* Search • Modeling Two-Player Games • When There is a Game Tree • The Determinacy Theorem • Searching a Game Tree • Examples of Games Review: A* Search • The A* Search depends on a heuristic function, which h(y) = 6 2 is a lower bound on the 23 s distance to the goal. x 4 If x is a node, and g is the • h(z) = 3 nearest goal node to x, the admissibility condition p(y) = 23 + 2 + 6 = 31 on h is that 0 ≤ h(x) ≤ d(x, g). p(z) = 23 + 4 + 3 = 30 Review: A* Search • Suppose we have taken y off of the open list. The best-path distance from the start s to the goal g through y is d(s, y) + d(y, h(y) = 6 g), and this cannot be less than 2 23 d(s, y) + h(y). s x 4 • Thus when we find a path of length k from s to y, we put y h(z) = 3 onto the open list with priority k p(y) = 23 + 2 + 6 = 31 + h(y). We still record the p(z) = 23 + 4 + 3 = 30 distance d(s, y) when we take y off of the open list. Review: A* Search • The advantage of A* over uniform-cost search is that we do not consider entries x in the closed list for which d(s, x) + h(x) h(y) = 6 is greater than the actual best- 2 23 path distance from s to g. -
Collusion and Heterogeneity of Firms∗
Collusion and Heterogeneity of Firms∗ Ichiro Obaray Federico Zincenko z June 13, 2016 Abstract We examine the impact of heterogeneous discounting on collusion in dynamic Bertrand competition. We show exactly when collusion can be sustained and how collusion would be organized efficiently with heterogeneous discounting. First we show that collusion is possible if and only if the average discount factor exceeds a certain threshold, with or without capacity constraints. Next we identify a dynamic pattern of market share that characterizes efficient collusion and obtain the unique long-run prediction despite the presence of multiple equilibria. In the long run, the most patient firm and the most impatient firm tend to dominate the market. JEL Classification: C72, C73, D43. Keywords: Bertrand Competition, Capacity Constraint, Collusion, Repeated Game, Subgame Perfect Equilibrium. Unequal Discounting. ∗This article was developed from the second chapter (co-authored with I. Obara) of Zincenko's doctoral dissertation at UCLA. We would like to thank the editor and anonymous referees for their useful comments. We also like to thank the participants at many seminars at various universities, 2011 Asian Meeting of the Econometric Society at Korea University, 2012 Southwest Economic Theory Conference at UC San Diego, and 2013 North American Summer Meeting of the Econometric Society at USC. All remaining errors are ours. yUniversity of California, Los Angeles; [email protected]. zUniversity of Pittsburgh; [email protected]. 1 1 Introduction To understand when and how collusion arises with heterogeneous discounting, we study a dynamic Bertrand competition model, in which firms discount future profits at different discount rates, and examine the impact of heterogeneous discounting on collusion. -
Game Playing
Game Playing Philipp Koehn 29 September 2015 Philipp Koehn Artificial Intelligence: Game Playing 29 September 2015 Outline 1 ● Games ● Perfect play – minimax decisions – α–β pruning ● Resource limits and approximate evaluation ● Games of chance ● Games of imperfect information Philipp Koehn Artificial Intelligence: Game Playing 29 September 2015 2 games Philipp Koehn Artificial Intelligence: Game Playing 29 September 2015 Games vs. Search Problems 3 ● “Unpredictable” opponent ⇒ solution is a strategy specifying a move for every possible opponent reply ● Time limits ⇒ unlikely to find goal, must approximate ● Plan of attack: – computer considers possible lines of play (Babbage, 1846) – algorithm for perfect play (Zermelo, 1912; Von Neumann, 1944) – finite horizon, approximate evaluation (Zuse, 1945; Wiener, 1948; Shannon, 1950) – first Chess program (Turing, 1951) – machine learning to improve evaluation accuracy (Samuel, 1952–57) – pruning to allow deeper search (McCarthy, 1956) Philipp Koehn Artificial Intelligence: Game Playing 29 September 2015 Types of Games 4 deterministic chance perfect Chess Backgammon information Checkers Monopoly Go Othello imperfect battleships Bridge information Blind Tic Tac Toe Poker Scrabble Philipp Koehn Artificial Intelligence: Game Playing 29 September 2015 Game Tree (2-player, Deterministic, Turns) 5 Philipp Koehn Artificial Intelligence: Game Playing 29 September 2015 Simple Game Tree 6 ● 2 player game ● Each player has one move ● You move first ● Goal: optimize your payoff (utility) Start Your move Opponent -
Constructing Stationary Sunspot Equilibria in a Continuous Time Model*
A Note on Woodford's Conjecture: Constructing Stationary Title Sunspot Equilibria in a Continuous Time Model(Nonlinear Analysis and Mathematical Economics) Author(s) Shigoka, Tadashi Citation 数理解析研究所講究録 (1994), 861: 51-66 Issue Date 1994-03 URL http://hdl.handle.net/2433/83844 Right Type Departmental Bulletin Paper Textversion publisher Kyoto University 数理解析研究所講究録 第 861 巻 1994 年 51-66 51 A Note on Woodford’s Conjecture: Constructing Stationary Sunspot Equilibria in a Continuous Time Model* Tadashi Shigoka Kyoto Institute of Economic Research, Kyoto University, Yoshidamachi Sakyoku Kyoto 606 Japan 京都大学 経済研究所 新後閑 禎 Abstract We show how to construct stationary sunspot equilibria in a continuous time model, where equilibrium is indeterninate near either a steady state or a closed orbit. Woodford’s conjecture that the indeterminacy of equilibrium implies the existence of stationary sunspot equilibria remains valid in a continuous time model. 52 Introduction If for given equilibrium dynamics there exist a continuum of non-stationary perfect foresight equilibria all converging asymptoticaUy to a steady state (a deterministic cycle resp.), we say the equilibrium dynamics is indeterminate near the steady state (the deterministic cycle resp.). Suppose that the fundamental characteristics of an economy are deterministic, but that economic agents believe nevertheless that equihibrium dynamics is affected by random factors apparently irrelevant to the fundamental characteristics (sunspots). This prophecy could be self-fulfilling, and one will get a sunspot equilibrium, if the resulting equilibrium dynamics is subject to a nontrivial stochastic process and confirns the agents’ belief. See Shell [19], and Cass-Shell [3]. Woodford [23] suggested that there exists a close relation between the indetenninacy of equilibrium near a deterministic steady state and the existence of stationary sunspot equilibria in the immediate vicinity of it.