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A Class of N-Player Colonel Blotto Games with Multidimensional Private Information
University of Zurich Department of Economics Working Paper Series ISSN 1664-7041 (print) ISSN 1664-705X (online) Working Paper No. 336 A Class of N-Player Colonel Blotto Games with Multidimensional Private Information Christian Ewerhart and Dan Kovenock Revised version, February 2021 A Class of N-Player Colonel Blotto Games With Multidimensional Private Information Christian Ewerhart Dan Kovenocky Department of Economics Economic Science Institute University of Zurich Chapman University Revised version: February 1, 2021 Abstract. In this paper, we study N-player Colonel Blotto games with incomplete information about battlefield valuations. Such games arise in job markets, research and development, electoral competition, security analysis, and conflict resolution. For M N + 1 battlefields, we identify a Bayes-Nash equilibrium in which the resource ≥ allocation to a given battlefield is strictly monotone in the valuation of that battlefield. We also explore extensions such as heterogeneous budgets, the case M N, full-support ≤ type distributions, and network games. Keywords. Colonel Blotto games Private information Bayes-Nash equilibrium · · · Generalized Dirichlet distributions Networks · *) Corresponding author. Postal address: Schönberggasse 1, 8001 Zurich, Switzerland. E-mail address: [email protected]. ) E-mail address: [email protected]. y 1 Introduction In a Colonel Blotto game, players simultaneously and independently allocate their en- dowments of a resource across a set of battlefields. The player that deploys the largest amount of the resource to a given battlefield scores a win and enjoys a gain in utility equivalent to her valuation of that battlefield. Thus, a player’s utility corresponds to the sum of the valuations of all battlefields won by the player. -
Campaigning Is Hard but Approximately Manageable
Computational Analyses of the Electoral College: Campaigning Is Hard But Approximately Manageable Sina Dehghani,1 Hamed Saleh,2 Saeed Seddighin,3 Shang-Hua Teng4 1 Institute for Research in Fundamental Sciences 2 University of Maryland 3 TTIC 4 University of Southern California [email protected], [email protected], [email protected], [email protected] Abstract Electoral College, Political Campaigns, and In the classical discrete Colonel Blotto game—introduced Multi-Battleground Resource Allocation by Borel in 1921—two colonels simultaneously distribute their troops across multiple battlefields. The winner of each The president of the United States is elected by the Electoral battlefield is determined by a winner-take-all rule, indepen- College, which consists of electors selected based on 51 con- dently of other battlefields. In the original formulation, each current elections. The number of electors each of the 50 colonel’s goal is to win as many battlefields as possible. The states and D. C. can select is determined every 10 years by Blotto game and its extensions have been used in a wide the United States Census. All but two states1 use a winner- range of applications from political campaign—exemplified take-all system, and in typical election without any third- by the U.S presidential election—to marketing campaign, party candidate, the ticket that wins the majority votes in the from (innovative) technology competition to sports compe- state earns the right to have their slate of electors from that tition. Despite persistent efforts, efficient methods for find- state chosen to vote in the Electoral College. Thus, in prin- ing the optimal strategies in Blotto games have been elu- sive for almost a century—due to exponential explosion in ciple, the team that receives the majority of electoral votes the organic solution space—until Ahmadinejad, Dehghani, wins the race and its main candidate is elected as president. -
Ten Years of Economic Reforms in Russia: Windows in a Wall
Ten Years of Economic Reforms in Russia: Windows in a Wall CHRISTINE LANG* SHLOMO WEBER** I. Introduction The August crisis of 1998 had a devastating effect on financial markets in Russia. Russia announced a default on foreign debt payments for the three-month period (and is still having a difficult time meeting them), while the short-term debt was fully and unilaterally restructured towards long-term debt. In one year's time, the market volume of the Russian Trading System (RTS) stock market exchange has plunged from $70 billion to a mere $5 billion. The 1998 financial crisis is reminiscent of another crisis in 1989-1990 when Russia was still a part of the Soviet Union. Ten years of reform, first towards a socialist market economy and then to a market economy, have focused on an irreversible change in Russian economy and policy. In this paper we try to assess some of the gains and losses of the first decade of transition. We argue that, while democracy is still at risk in Russia, a Russian citizen has matured as a voter, a consumer, and an investor. Political and economic progress, however, are still hampered by the poor and criminalized state of business and political spheres. Russia badly needs a president, parliament, and government having the will and power to face the country's tremendous challenges. The maturation of the electorate provides for a better "Christine Lang is Visiting Scholar, Central Economic Mathematical Institute, Russian Academy of Sciences, Moscow, Russia, and Assistant Professor at the University of Geneva, Switzerland. She acknowledges the support of the Swiss National Research Foundation grant 8210-053488. -
Conference on Culture, Diversity and Development
The 6th CAS Annual Academic Conference Conference on Culture, Diversity and Development organized by the Center for the Study of Diversity and Social Interactions at the New Economic School and the Higher School of Economics in conjunction with the Centre for Advanced Studies, a joint project of the Higher School of Economics and the New Economic School October 18‐19, 2013 Higher School of Economics, room #311, 20, Myasnitskaya str., Moscow, Russia Organizing Committee Co‐Chairs: Paul Castañeda Dower (New Economic School) Martin Gilman (Higher School of Economics, CAS) Program Committee Co‐Chairs: Shlomo Weber (SMU and NES Center for the Study of Diversity and Social Interactions) Leonid Polishchuk (Higher School of Economics) The conference covers the following areas of research: theoretical challenges of the analysis of the community heterogeneity; the influence of institutions, religion and culture on the development and integration of communities and nations; issues of social interaction and social integration, including the processes of generation and maintenance of stable social groups; the influence of community heterogeneity on social mobility, preferences for redistribution of wealth, the emergence of civil conflict, the manifestation of separatism; issues of building trust and social capital in heterogeneous groups and communities, and their impact on the socio‐economic development. Conference Program Higher School of Economics, room #311, 20, Myasnitskaya str., Moscow, Russia Friday, October 18 8.30 – 9.00 Registration and Coffee 9.00 – 9.20 Introduction: Martin Gilman Greetings: Lev Yakobson (Higher School of Economics) Shlomo Weber (Southern Methodist University and NES Center for the Study of Diversity and Social Interactions) 9.20 – 10.00 Session I. -
Optimal Power Allocation Strategy Against Jamming Attacks Using the Colonel Blotto Game
Optimal Power Allocation Strategy Against Jamming Attacks Using the Colonel Blotto Game Yongle Wu, Beibei Wang, and K. J. Ray Liu Department of Electrical and Computer Engineering, University of Maryland, College Park, MD 20742, USA. {wuyl, bebewang, kjrliu}@umd.edu Abstract—Cognitive radio technologies have become a promis- adversaries. So far, there have been only a few papers regard- ing approach to increase the efficiency of spectrum utilization. ing security issues. For instance, the primary user emulation Although cognitive radio has been intensively studied in recent attack was described and a transmitter verification scheme was years, only a few works have discussed security aspects. In this paper, we focus on the jamming attack, one of major threats to proposed to test whether the given signal came from a primary cognitive radio networks, where a malicious user wants to jam user in [8]; [9] employed a Hammer model to identify, analyze the communications of secondary users by injecting interference. and assess denial of service attacks; in [10], security issues Aware of the absence of several primary users and the presence of the IEEE 802.22 standard were addressed; [11] discussed of a malicious user, a secondary user can allocate power to those an attack that malicious users attempted to affect an artificial- fallow bands with a randomized strategy, in hope of alleviating the damage caused by the malicious user. We model this scenario intelligence-based secondary user’s behavior by misleading the into a two-player zero-sum game, and derive its unique Nash learning process. Equilibrium under certain conditions using the Colonel Blotto In this paper, we mainly focus on one kind of malicious at- game approach, which provides a minimax strategy that the tacks in cognitive radio networks, namely, the jamming attack secondary user should adopt in order to minimize the worst- [12], where a malicious user, or the “jammer”, wants to prevent case damage caused by the malicious user. -
Nber Working Paper Series on the Joint Evolution Of
NBER WORKING PAPER SERIES ON THE JOINT EVOLUTION OF CULTURE AND INSTITUTIONS Alberto Bisin Thierry Verdier Working Paper 23375 http://www.nber.org/papers/w23375 NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambridge, MA 02138 April 2017 Thanks to Alberto Alesina, Gary Becker, Georgy Egorov, Luigi Guiso, David Levine, Massimo Morelli, Nicola Persico, Louis Putterman, Kenneth Shepsle, Shlomo Weber, and the many others we spoke with about this paper. We have presented the paper widely before writing it and we thank all the audiences for help, suggestions, and support. This research has been undertaken under financial support from the ERC Advanced Grant TECTACOM n324004. The views expressed herein are those of the authors and do not necessarily reflect the views of the National Bureau of Economic Research. NBER working papers are circulated for discussion and comment purposes. They have not been peer-reviewed or been subject to the review by the NBER Board of Directors that accompanies official NBER publications. © 2017 by Alberto Bisin and Thierry Verdier. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given to the source. On the Joint Evolution of Culture and Institutions Alberto Bisin and Thierry Verdier NBER Working Paper No. 23375 April 2017 JEL No. O10,P16,P48 ABSTRACT Explanations of economic growth and prosperity commonly identify a unique causal effect, e.g., institutions, culture, human capital, geography. In this paper we provide instead a theoretical modeling of the interaction between culture and institutions and their effects on economic activity. -
Generalizations on the Colonel Blotto Game
Generalizations on the Colonel Blotto Game Dan Kovenock∗and Brian Robersony Abstract In this paper we characterize the equilibrium in a relaxed version of the Colonel Blotto game in which battlefield valuations maybe heterogeneous across battlefields and asymmetric across players and the, possibly asymmetric, budget constraints hold only on average, and then show how this characterization partially extends to the full Colonel Blotto game, in which the budget constraints must be satisfied with probability one. For the non-constant-sum generalization of the Colonel Blotto game examined, we find that there exist non-pathological parameter configurations with multiple, payoff inequivalent, equilibria. ∗Dan Kovenock, Economic Science Institute, Argyros School of Business and Economics, Chapman Uni- versity, One University Drive, Orange, CA 92866 USA t:714-628-7226 E-mail: [email protected] yBrian Roberson, Purdue University, Department of Economics, Krannert School of Management, 403 W. State Street, West Lafayette, IN 47907 USA t: 765-494-4531 E-mail: [email protected] (Correspondent) 1 1 Introduction The Colonel Blotto game is a two-player resource allocation game in which each player is endowed with a level of resources to allocate across a set of battlefields, within each battlefield the player that allocates the higher level of resources wins the battlefield, and each player's payoff is the sum of the valuations of the battlefields won. This simple game, that originates with Borel (1921), illustrates the fundamental strategic considerations that arise in multi- dimensional resource allocation competition such as: political campaign resource allocation, research and development competition where innovation involves obtaining a collection of interrelated patents, attack and defense of a collection of targets, etc. -
Faster and Simpler Algorithm for Optimal Strategies of Blotto Game
Faster and Simpler Algorithm for Optimal Strategies of Blotto Game Soheil Behnezhad∗, Sina Dehghani∗, Mahsa Derakhshan∗, MohammadTaghi HajiAghayi∗, Saeed Seddighin∗ Department of Computer Science, University of Maryland fsoheil, dehghani, mahsaa, hajiagha, [email protected] Abstract In the Colonel Blotto game, which was initially introduced by Borel in 1921, two colonels simultaneously distribute their troops across different battlefields. The winner of each battlefield is determined independently by a winner-take-all rule. The ultimate payoff of each colonel is the number of battlefields he wins. The Colonel Blotto game is commonly used for analyzing a wide range of applications from the U.S presidential election, to innovative technology competitions, to advertisement, to sports, and to politics. There has been persistent efforts for finding the optimal strategies for the Colonel Blotto game. After almost a century Ahmadinejad, Dehghani, Hajiaghayi, Lucier, Mahini, and Seddighin [2] provided an algorithm for finding the optimal strategies in polynomial time. Ahmadinejad et al. [2] first model the problem by a Linear Program (LP) with both an expo- nential number of variables and an exponential number of constraints which makes the problem intractable. Then they project their solution to another space to obtain another exponential-size LP, for which they can use Ellipsoid method. However, despite the theoretical importance of their algorithm, it is highly impractical. In general, even Simplex method (despite its exponen- tial running time in practice) performs better than Ellipsoid method in practice. In this paper, we provide the first polynomial-size LP formulation of the optimal strategies for the Colonel Blotto game. We use linear extension techniques. -
Political Game Theory Nolan Mccarty Adam Meirowitz
Political Game Theory Nolan McCarty Adam Meirowitz To Liz, Janis, Lachlan, and Delaney. Contents Acknowledgements vii Chapter 1. Introduction 1 1. Organization of the Book 2 Chapter 2. The Theory of Choice 5 1. Finite Sets of Actions and Outcomes 6 2. Continuous Outcome Spaces* 10 3. Utility Theory 17 4. Utility representations on Continuous Outcome Spaces* 18 5. Spatial Preferences 19 6. Exercises 21 Chapter 3. Choice Under Uncertainty 23 1. TheFiniteCase 23 2. Risk Preferences 32 3. Learning 37 4. Critiques of Expected Utility Theory 41 5. Time Preferences 46 6. Exercises 50 Chapter 4. Social Choice Theory 53 1. The Open Search 53 2. Preference Aggregation Rules 55 3. Collective Choice 61 4. Manipulation of Choice Functions 66 5. Exercises 69 Chapter 5. Games in the Normal Form 71 1. The Normal Form 73 2. Solutions to Normal Form Games 76 3. Application: The Hotelling Model of Political Competition 83 4. Existence of Nash Equilibria 86 5. Pure Strategy Nash Equilibria in Non-Finite Games* 93 6. Application: Interest Group Contributions 95 7. Application: International Externalities 96 iii iv CONTENTS 8. Computing Equilibria with Constrained Optimization* 97 9. Proving the Existence of Nash Equilibria** 98 10. Strategic Complementarity 102 11. Supermodularity and Monotone Comparative Statics* 103 12. Refining Nash Equilibria 108 13. Application: Private Provision of Public Goods 109 14. Exercises 113 Chapter 6. Bayesian Games in the Normal Form 115 1. Formal Definitions 117 2. Application: Trade restrictions 119 3. Application: Jury Voting 121 4. Application: Jury Voting with a Continuum of Signals* 123 5. Application: Public Goods and Incomplete Information 126 6. -
Introduction to the Special Issue
Introduction to the special issue Giacomo Bonanno Andreas Herzig Department of Economics IRIT-CNRS University of California Davis (USA) Toulouse University (France) [email protected] [email protected] Wiebe van der Hoek J´er^omeLang Department of Computer Science LAMSADE University of Liverpool (UK) University Paris-Dauphine (France) [email protected] [email protected] January 6, 2012 This special issue of the International Journal of Game Theory contains a selection of papers presented at the 9th Conference on Logic and the Foun- dations of the Theory of Games and Decisions (LOFT9), which took place in Toulouse July 5{7, 2010. While this special issue collects papers with a stronger game-theoretic content, a second set of papers that are more logic-oriented can be found in a special issue of the Journal of Applied Non- Classical Logics. The LOFT conferences, which have been a regular biannual event since 1994, are interdisciplinary events that bring together researchers from a variety of fields: cognitive psychology, computer science and artifi- cial intelligence, economics, game theory, linguistics, logic, mind sciences, philosophy, social choice and statistics.1 In its original conception, LOFT had as its central theme the applica- tion of logic, in particular modal epistemic logic, to foundational issues in 1The first conference was hosted by the Centre International de Recherches Mathe- matiques in Marseille (France), the next four took place at the International Centre for Economic Research in Torino (Italy), the sixth was hosted by the Graduate School of Management in Leipzig (Germany), the seventh took place at the University of Liverpool (United Kingdom) and the eighth at the University of Amsterdam (The Netherlands). -
Ten Years of Economic Reforms in Russia: Windows in a Wall
Ten Years of Economic Reforms in Russia: Windows in a Wall CHRISTINE LANG* SHLOMO WEBER** I. Introduction The August crisis of 1998 had a devastating effect on financial markets in Russia. Russia announced a default on foreign debt payments for the three-month period (and is still having a difficult time meeting them), while the short-term debt was fully and unilaterally restructured towards long-term debt. In one year's time, the market volume of the Russian Trading System (RTS) stock market exchange has plunged from $70 billion to a mere $5 billion. The 1998 financial crisis is reminiscent of another crisis in 1989-1990 when Russia was still a part of the Soviet Union. Ten years of reform, first towards a socialist market economy and then to a market economy, have focused on an irreversible change in Russian economy and policy. In this paper we try to assess some of the gains and losses of the first decade of transition. We argue that, while democracy is still at risk in Russia, a Russian citizen has matured as a voter, a consumer, and an investor. Political and economic progress, however, are still hampered by the poor and criminalized state of business and political spheres. Russia badly needs a president, parliament, and government having the will and power to face the country's tremendous challenges. The maturation of the electorate provides for a better "Christine Lang is Visiting Scholar, Central Economic Mathematical Institute, Russian Academy of Sciences, Moscow, Russia, and Assistant Professor at the University of Geneva, Switzerland. She acknowledges the support of the Swiss National Research Foundation grant 8210-053488. -
A Class of N-Player Colonel Blotto Games with Multidimensional Private Information
A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Ewerhart, Christian; Kovenock, Daniel J. Working Paper A class of N-player Colonel Blotto games with multidimensional private information Working Paper, No. 336 Provided in Cooperation with: Department of Economics, University of Zurich Suggested Citation: Ewerhart, Christian; Kovenock, Daniel J. (2019) : A class of N-player Colonel Blotto games with multidimensional private information, Working Paper, No. 336, University of Zurich, Department of Economics, Zurich, http://dx.doi.org/10.5167/uzh-176995 This Version is available at: http://hdl.handle.net/10419/206852 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 University of Zurich Department of Economics Working Paper Series ISSN 1664-7041 (print) ISSN 1664-705X (online) Working Paper No.