Cooperative Games: Core and Shapley Value
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Is Shapley Cost Sharing Optimal?∗ (For the Special Issue in Honor of Lloyd Shapley)
Is Shapley Cost Sharing Optimal?∗ (For the special issue in honor of Lloyd Shapley) Shahar Dobzinskiy Aranyak Mehtaz Tim Roughgardenx Mukund Sundararajan{ August 29, 2016 Abstract A general approach to the design of budget-balanced cost-sharing mechanisms is to use the Shapley value, applied to the given cost function, to define payments from the players to the mechanism. Is the corresponding Shapley value mechanism \optimal" in some sense? We consider the objective of minimizing worst-case inefficiency subject to a revenue constraint, and prove results in three different regimes. 1. For the public excludable good problem, the Shapley value mechanism minimizes the worst-case efficiency loss over all truthful, deterministic, and budget-balanced mechanisms that satisfy equal treatment. This result follows from a characteriza- tion of the Shapley value mechanism as the unique one that satisfies two additional technical conditions. 2. For the same problem, even over a much more general mechanism design space that allows for randomization and approximate budget-balance and does not im- pose equal treatment, the worst-case efficiency loss of the Shapley value mecha- nism is within a constant factor of the minimum possible. ∗The results in Sections 3{5 appeared, in preliminary form, in the extended abstract [8]. The results in Section 6 appeared in the PhD thesis of the fourth author [27]. yDepartment of Applied Math and Computer Science, the Weizmann Institute of Science, Rohovot 76100, Israel. This work was done while the author was visiting Stanford University, and supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities, and by grants from the Israel Science Foundation and the USA-Israel Bi-national Science Foundation. -
Nash-Type Bargaining with Coherent Acceptability Measures
Splitting a Random Pie: Nash-Type Bargaining with Coherent Acceptability Measures Walter J. Gutjahr University of Vienna, Vienna, Austria, [email protected] Raimund M. Kovacevic Vienna University of Technology, Vienna, Austria, [email protected] David Wozabal Technische Universit¨atM¨unchen, Munich, Germany, [email protected] We propose an axiomatic solution for cooperative stochastic games where risk averse players bargain for the allocation of profits from a joint project that depends on management decisions by the agents. We model risk preferences by coherent acceptability functionals and show that in this setting the axioms of Pareto optimality, symmetry, and strategy proofness fully characterize a bargaining solution, which can be efficiently computed by solving a stochastic optimization problem. Furthermore, we demonstrate that there is no conflict of interest between players about management decisions and characterize special cases where random payoffs of players are simple functions of overall project profit. In particular, we show that for players with distortion risk functionals, the optimal bargaining solution can be represented by an exchange of standard options contracts with the project profit as the underlying. We illustrate the concepts in the paper by a detailed example of risk averse households that jointly invest into a solar plant. Key words : Stochastic bargaining games; coherent risk measures; stochastic programming; photovoltaics History : 1. Introduction If a project is undertaken jointly by several agents and there is no efficient market for individual contributions, the question of how to distribute the project's profits arises. The case when profits are deterministic is well studied in the field of cooperative game theory. -
Game Theory Lecture Notes
Game Theory: Penn State Math 486 Lecture Notes Version 2.1.1 Christopher Griffin « 2010-2021 Licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License With Major Contributions By: James Fan George Kesidis and Other Contributions By: Arlan Stutler Sarthak Shah Contents List of Figuresv Preface xi 1. Using These Notes xi 2. An Overview of Game Theory xi Chapter 1. Probability Theory and Games Against the House1 1. Probability1 2. Random Variables and Expected Values6 3. Conditional Probability8 4. The Monty Hall Problem 11 Chapter 2. Game Trees and Extensive Form 15 1. Graphs and Trees 15 2. Game Trees with Complete Information and No Chance 18 3. Game Trees with Incomplete Information 22 4. Games of Chance 24 5. Pay-off Functions and Equilibria 26 Chapter 3. Normal and Strategic Form Games and Matrices 37 1. Normal and Strategic Form 37 2. Strategic Form Games 38 3. Review of Basic Matrix Properties 40 4. Special Matrices and Vectors 42 5. Strategy Vectors and Matrix Games 43 Chapter 4. Saddle Points, Mixed Strategies and the Minimax Theorem 45 1. Saddle Points 45 2. Zero-Sum Games without Saddle Points 48 3. Mixed Strategies 50 4. Mixed Strategies in Matrix Games 53 5. Dominated Strategies and Nash Equilibria 54 6. The Minimax Theorem 59 7. Finding Nash Equilibria in Simple Games 64 8. A Note on Nash Equilibria in General 66 Chapter 5. An Introduction to Optimization and the Karush-Kuhn-Tucker Conditions 69 1. A General Maximization Formulation 70 2. Some Geometry for Optimization 72 3. -
Stable Allocations and the Practice of Market Design
15 OCTOBER 2012 Scientific Background on the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 2012 STABLE ALLOCATIONS AND THE PRACTICE OF MARKET DESIGN compiled by the Economic Sciences Prize Committee of the Royal Swedish Academy of Sciences THE ROYAL SWEDISH ACADEMY OF SCIENCES has as its aim to promote the sciences and strengthen their influence in society. BOX 50005 (LILLA FRESCATIVÄGEN 4 A), SE-104 05 STOCKHOLM, SWEDEN TEL +46 8 673 95 00, FAX +46 8 15 56 70, [email protected] HTTP://KVA.SE 1 Introduction Economists study how societies allocate resources. Some allocation problems are solved by the price system: high wages attract workers into a particu- lar occupation, and high energy prices induce consumers to conserve energy. In many instances, however, using the price system would encounter legal and ethical objections. Consider, for instance, the allocation of public-school places to children, or the allocation of human organs to patients who need transplants. Furthermore, there are many markets where the price system operates but the traditional assumption of perfect competition is not even approximately satis…ed. In particular, many goods are indivisible and het- erogeneous, whereby the market for each type of good becomes very thin. How these thin markets allocate resources depends on the institutions that govern transactions. This year’s prizewinning work encompasses a theoretical framework for analyzing resource allocation, as well as empirical studies and actual redesign of real-world institutions such as labor-market clearinghouses and school ad- missions procedures. The foundations for the theoretical framework were laid in 1962, when David Gale and Lloyd Shapley published a mathematical inquiry into a certain class of allocation problems. -
Game Theoretic Interaction and Decision: a Quantum Analysis
games Article Game Theoretic Interaction and Decision: A Quantum Analysis Ulrich Faigle 1 and Michel Grabisch 2,* 1 Mathematisches Institut, Universität zu Köln, Weyertal 80, 50931 Köln, Germany; [email protected] 2 Paris School of Economics, University of Paris I, 106-112, Bd. de l’Hôpital, 75013 Paris, France * Correspondence: [email protected]; Tel.: +33-144-07-8744 Received: 12 July 2017; Accepted: 21 October 2017; Published: 6 November 2017 Abstract: An interaction system has a finite set of agents that interact pairwise, depending on the current state of the system. Symmetric decomposition of the matrix of interaction coefficients yields the representation of states by self-adjoint matrices and hence a spectral representation. As a result, cooperation systems, decision systems and quantum systems all become visible as manifestations of special interaction systems. The treatment of the theory is purely mathematical and does not require any special knowledge of physics. It is shown how standard notions in cooperative game theory arise naturally in this context. In particular, states of general interaction systems are seen to arise as linear superpositions of pure quantum states and Fourier transformation to become meaningful. Moreover, quantum games fall into this framework. Finally, a theory of Markov evolution of interaction states is presented that generalizes classical homogeneous Markov chains to the present context. Keywords: cooperative game; decision system; evolution; Fourier transform; interaction system; measurement; quantum game 1. Introduction In an interaction system, economic (or general physical) agents interact pairwise, but do not necessarily cooperate towards a common goal. However, this model arises as a natural generalization of the model of cooperative TU games, for which already Owen [1] introduced the co-value as an assessment of the pairwise interaction of two cooperating players1. -
Stable Matching: Theory, Evidence, and Practical Design
THE PRIZE IN ECONOMIC SCIENCES 2012 INFORMATION FOR THE PUBLIC Stable matching: Theory, evidence, and practical design This year’s Prize to Lloyd Shapley and Alvin Roth extends from abstract theory developed in the 1960s, over empirical work in the 1980s, to ongoing efforts to fnd practical solutions to real-world prob- lems. Examples include the assignment of new doctors to hospitals, students to schools, and human organs for transplant to recipients. Lloyd Shapley made the early theoretical contributions, which were unexpectedly adopted two decades later when Alvin Roth investigated the market for U.S. doctors. His fndings generated further analytical developments, as well as practical design of market institutions. Traditional economic analysis studies markets where prices adjust so that supply equals demand. Both theory and practice show that markets function well in many cases. But in some situations, the standard market mechanism encounters problems, and there are cases where prices cannot be used at all to allocate resources. For example, many schools and universities are prevented from charging tuition fees and, in the case of human organs for transplants, monetary payments are ruled out on ethical grounds. Yet, in these – and many other – cases, an allocation has to be made. How do such processes actually work, and when is the outcome efcient? Matching theory The Gale-Shapley algorithm Analysis of allocation mechanisms relies on a rather abstract idea. If rational people – who know their best interests and behave accordingly – simply engage in unrestricted mutual trade, then the outcome should be efcient. If it is not, some individuals would devise new trades that made them better of. -
Nudging Cooperation in Public Goods Provision
A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Barron, Kai; Nurminen, Tuomas Article — Accepted Manuscript (Postprint) Nudging cooperation in public goods provision Journal of Behavioral and Experimental Economics Provided in Cooperation with: WZB Berlin Social Science Center Suggested Citation: Barron, Kai; Nurminen, Tuomas (2020) : Nudging cooperation in public goods provision, Journal of Behavioral and Experimental Economics, ISSN 2214-8043, Elsevier, Amsterdam, Vol. 88, Iss. (Article No.:) 101542, http://dx.doi.org/10.1016/j.socec.2020.101542 This Version is available at: http://hdl.handle.net/10419/216878 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 -
Competition and Cooperation on Predation: Bifurcation Theory of Mutualism Author: Srijana Ghimire Xiang-Sheng Wang University of Louisiana at Lafayette
Competition and Cooperation on Predation: Bifurcation Theory Of Mutualism Author: Srijana Ghimire Xiang-Sheng Wang University of Louisiana at Lafayette Introduction Existence and Stability of E1,E+ and E− Existence and property of Hopf bifurcation 3. R > 1 and R > 3R − 2R2. In this case, Q = Q , 1 2 1 1 c 1 points We investigate two predator-prey models which take into con- xc = 1/R1, E1 always exists, E1 is locally asymptotically H E + H E E sideration the cooperation between two different predators and stable if and only if Q < Q1, E− does not exist, and E+ exits + + R H within one predator species, respectively. Local and global dy- if and only if Q > Q1. 2.0 E E E - namics are studied for the model systems. By a detailed bi- - - Q Q Q Q Q Q furcation analysis, we investigate the dependence of predation + + h + h1 h2 1.5 no Hopf bifurcation Existence conditions of positive equilibria. (a) case 1(a) (b) case 1(b) (c) case 1(c) dynamics on mutualism (cooperative predation). H H E R + 2 E E two supercritical + + 1.0 R2 > 1 Q H E+ E - E Q1 - E y y - First Predator-Prey Model with Competition 1 E 1 E 1 1 y 1 E 1 0.5 one supercritical NA Q Q Q Q Q Q Q Q Q and Co-operation 1 + 1 + 1 h + h1 1 h2 R2 < one Q R1 NA if Q < Q1 (d) case 2(a) (e) case 2(b) (f) case 2(c) E± subcritical E+ if Q ≥ Q1 E+ d 0 H 0 2 4 6 8 E x = 1 − x − p xy − p xz − 2qxyz, (1) + 1 2 Q+ E Q1 + 0 Figure: Existence and property of Hopf bifurcation points in the (d, R) y = p xy + qxyz − d y, (2) 2 1 1 R2 = 3R1 − 2R NA 1 H parameter space. -
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. -
Stability and Median Rationalizability for Aggregate Matchings
games Article Stability and Median Rationalizability for Aggregate Matchings Federico Echenique 1, SangMok Lee 2, Matthew Shum 1 and M. Bumin Yenmez 3,* 1 Division of the Humanities and Social Sciences, California Institute of Technology, Pasadena, CA 91125, USA; [email protected] (F.E.); [email protected] (M.S.) 2 Department of Economics, Washington University in St. Louis, St. Louis, MO 63130, USA; [email protected] 3 Department of Economics, Boston College, Chestnut Hill, MA 02467, USA * Correspondence: [email protected] Abstract: We develop the theory of stability for aggregate matchings used in empirical studies and establish fundamental properties of stable matchings including the result that the set of stable matchings is a non-empty, complete, and distributive lattice. Aggregate matchings are relevant as matching data in revealed preference theory. We present a result on rationalizing a matching data as the median stable matching. Keywords: aggregate matching; median stable matching; rationalizability; lattice 1. Introduction Following the seminal work of [1], an extensive literature has developed regarding matching markets with non-transferable utility. This literature assumes that there are agent-specific preferences, and studies the existence of stable matchings in which each Citation: Echenique, F.; Lee, S.; agent prefers her assigned partner to the outside option of being unmatched, and there Shum, M.; Yenmez, M.B. Stability and are no pairs of agents that would like to match with each other rather than keeping their Median Rationalizability for assigned partners. Aggregate Matchings. Games 2021, 12, In this paper, we develop the theory of stability for aggregate matchings, which we 33. -
PIER Working Paper 11-035
Penn Institute for Economic Research Department of Economics University of Pennsylvania 3718 Locust Walk Philadelphia, PA 19104-6297 [email protected] http://economics.sas.upenn.edu/pier PIER Working Paper 11-035 “Identification and Estimation of Stochastic Bargaining Models Fourth Version” by Antonio Merlo and Xun Tang http://ssrn.com/abstract=1946367 Identification and Estimation of Stochastic Bargaining Models∗ Antonio Merlo† Xun Tang‡ Revised, October 2011 Abstract Stochastic sequential bargaining models (Merlo and Wilson (1995, 1998)) have found wide applications in different fields including political economy and macroeconomics due to their flexibility in explaining delays in reaching an agreement. This paper presents new results in nonparametric identification and estimation of such models under different data scenarios. Key words: Nonparametric identification and estimation, non-cooperative bargaining, stochastic sequential bargaining, rationalizable counterfactual outcomes. JEL codes: C14, C35, C73, C78. ∗We thank co-editor Jean-Marc Robin and three anonymous referees for their helpful suggestions. Steven Berry, Herman Bierens, Hanming Fang, Philip Haile, Isabelle Perrigne, Katja Seim, Elie Tamer, Petra Todd, Quang Vuong, Ken Wolpin and seminar and conference participants at several institutions provided useful comments. We thank Hulya Eraslan for sharing her data with us and Chamna Yoon for providing excellent research assistance. The usual disclaimer applies. †Department of Economics, University of Pennsylvania, CEPR, CESifo and NBER, [email protected]. ‡Department of Economics, University of Pennsylvania, [email protected]. 1 1 Introduction Starting with the seminal contributions of Stahl (1972) and Rubinstein (1982), noncooper- ative (or strategic) bargaining theory has flourished in the past thirty years. The original model of bilateral bargaining with alternating offers and complete information has been extended in a number of directions allowing for more general extensive forms, information structure and more than two players (e.g. -
Rationalizable Implementation of Correspondences∗
Rationalizable Implementation of Correspondences∗ Takashi Kunimotoyand Roberto Serranoz This Version: May 2016 Abstract We come close to characterizing the class of social choice correspon- dences that are implementable in rationalizable strategies. We identify a new condition, which we call set-monotonicity, and show that it is necessary and almost sufficient for rationalizable implementation. Set-monotonicity is much weaker than Maskin monotonicity, which is the key condition for Nash implementation and which also had been shown to be necessary for rationalizable implementation of social choice functions. Set-monotonicity reduces to Maskin monotonicity in the case of functions. We conclude that the conditions for rationalizable implementation are not only starkly differ- ent from, but also much weaker than those for Nash implementation, when we consider social choice correspondences. JEL Classification: C72, D78, D82. Keywords: Complete information, implementation, Maskin monotonicity, rationalizability, set-monotonicity, social choice correspondence. 1 Introduction The design of institutions to be used by rational agents has been an important research agenda in economic theory. As captured by the notion of Nash equilib- rium, rationality is encapsulated in two aspects: these are (i) the best responses of agents to their beliefs, and (ii) that those beliefs are correct, the so-called rational ∗Financial support from the Japan Society for the Promotion of Science (24330078, 25780128) (Kunimoto) is gratefully acknowledged. All remaining errors are our own. ySchool of Economics, Singapore Management University, 90 Stamford Road, Singapore 178903; [email protected] zDepartment of Economics, Brown University, Providence, RI 02912, U.S.A.; roberto [email protected] 1 expectations assumption. One can drop the latter and retain the former, moving then into the realm of rationalizability.