/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

We introduce a new algorithm named WGAN, an alternative to traditional GAN training. In this new model, we show that we can improve the stability of learning, get rid of problems like mode collapse, and provide meaningful learning curves useful for debugging and hyperparameter searches. Furthermore, we show that the corresponding optimization problem is sound, and provide extensive theoretical work highlighting the deep connections to different distances between distributions.

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Wasserstein Generative Adversarial Networks

Preface 1. Decision-Theoretic Foundations 1.1 Game Theory, Rationality, and Intelligence 1.2 Basic Concepts of Decision Theory 1.3 Axioms 1.4 The Expected-Utility Maximization Theorem 1.5 Equivalent Representations 1.6 Bayesian Conditional-Probability Systems 1.7 Limitations of the Bayesian Model 1.8 Domination 1.9 Proofs of the Domination Theorems Exercises 2. Basic Models 2.1 Games in Extensive Form 2.2 Strategic Form and the Normal Representation 2.3 Equivalence of Strategic-Form Games 2.4 Reduced Normal Representations 2.5 Elimination of Dominated Strategies 2.6 Multiagent Representations 2.7 Common Knowledge 2.8 Bayesian Games 2.9 Modeling Games with Incomplete Information Exercises 3. Equilibria of Strategic-Form Games 3.1 Domination and Ratonalizability 3.2 Nash Equilibrium 3.3 Computing Nash Equilibria 3.4 Significance of Nash Equilibria 3.5 The Focal-Point Effect 3.6 The Decision-Analytic Approach to Games 3.7 Evolution. Resistance. and Risk Dominance 3.8 Two-Person Zero-Sum Games 3.9 Bayesian Equilibria 3.10 Purification of Randomized Strategies in Equilibria 3.11 Auctions 3.12 Proof of Existence of Equilibrium 3.13 Infinite Strategy Sets Exercises 4. Sequential Equilibria of Extensive-Form Games 4.1 Mixed Strategies and Behavioral Strategies 4.2 Equilibria in Behavioral Strategies 4.3 Sequential Rationality at Information States with Positive Probability 4.4 Consistent Beliefs and Sequential Rationality at All Information States 4.5 Computing Sequential Equilibria 4.6 Subgame-Perfect Equilibria 4.7 Games with Perfect Information 4.8 Adding Chance Events with Small Probability 4.9 Forward Induction 4.10 Voting and Binary Agendas 4.11 Technical Proofs Exercises 5. Refinements of Equilibrium in Strategic Form 5.1 Introduction 5.2 Perfect Equilibria 5.3 Existence of Perfect and Sequential Equilibria 5.4 Proper Equilibria 5.5 Persistent Equilibria 5.6 Stable Sets 01 Equilibria 5.7 Generic Properties 5.8 Conclusions Exercises 6. Games with Communication 6.1 Contracts and Correlated Strategies 6.2 Correlated Equilibria 6.3 Bayesian Games with Communication 6.4 Bayesian Collective-Choice Problems and Bayesian Bargaining Problems 6.5 Trading Problems with Linear Utility 6.6 General Participation Constraints for Bayesian Games with Contracts 6.7 Sender-Receiver Games 6.8 Acceptable and Predominant Correlated Equilibria 6.9 Communication in Extensive-Form and Multistage Games Exercises Bibliographic Note 7. Repeated Games 7.1 The Repeated Prisoners Dilemma 7.2 A General Model of Repeated Garnet 7.3 Stationary Equilibria of Repeated Games with Complete State Information and Discounting 7.4 Repeated Games with Standard Information: Examples 7.5 General Feasibility Theorems for Standard Repeated Games 7.6 Finitely Repeated Games and the Role of Initial Doubt 7.7 Imperfect Observability of Moves 7.8 Repeated Wines in Large Decentralized Groups 7.9 Repeated Games with Incomplete Information 7.10 Continuous Time 7.11 Evolutionary Simulation of Repeated Games Exercises 8. Bargaining and Cooperation in Two-Person Games 8.1 Noncooperative Foundations of Cooperative Game Theory 8.2 Two-Person Bargaining Problems and the Nash Bargaining Solution 8.3 Interpersonal Comparisons of Weighted Utility 8.4 Transferable Utility 8.5 Rational Threats 8.6 Other Bargaining Solutions 8.7 An Alternating-Offer Bargaining Game 8.8 An Alternating-Offer Game with Incomplete Information 8.9 A Discrete Alternating-Offer Game 8.10 Renegotiation Exercises 9. Coalitions in Cooperative Games 9.1 Introduction to Coalitional Analysis 9.2 Characteristic Functions with Transferable Utility 9.3 The Core 9.4 The Shapkey Value 9.5 Values with Cooperation Structures 9.6 Other Solution Concepts 9.7 Colational Games with Nontransferable Utility 9.8 Cores without Transferable Utility 9.9 Values without Transferable Utility Exercises Bibliographic Note 10. Cooperation under Uncertainty 10.1 Introduction 10.2 Concepts of Efficiency 10.3 An Example 10.4 Ex Post Inefficiency and Subsequent Oilers 10.5 Computing Incentive-Efficient Mechanisms 10.6 Inscrutability and Durability 10.7 Mechanism Selection by an Informed Principal 10.8 Neutral Bargaining Solutions 10.9 Dynamic Matching Processes with Incomplete Information Exercises Bibliography Index

博弈论 : 矛盾冲突分析 = Game theory : analysis of conflict

Information Theory and Reliable Communication

A new era of theoretical computer science addresses fundamental problems about auctions, networks, and human behavior.

http://www.maths.lse.ac.uk/Personal/stengel/TEXTE/agt-stengel.pdf

Algorithmic Game Theory

The standard envelope theorems apply to choice sets with convex and topological structure, providing sufficient conditions for the value function to be differentiable in a parameter and characterizing its derivative. This paper studies optimization with arbitrary choice sets and shows that the traditional envelope formula holds at any differentiability point of the value function. We also provide conditions for the value function to be, variously, absolutely continuous, left- and right-differentiable, or fully differentiable. These results are applied to mechanism design, convex programming, continuous optimization problems, saddle-point problems, problems with parameterized constraints, and optimal stopping problems.

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Envelope Theorems for Arbitrary Choice Sets

The paper considers a hold-up model where only one of n future trading opportunities will prove to be efficient, and where ex post renegotiation of the ex ante contract cannot be prevented. As the environment becomes more complex (n→∞), the outcome under any message-contingent long-term contract converges to that of the "incomplete contracting" model where trade is contractible ex post, but not ex ante. When trades are costly to describe, both ex ante and ex post, the incomplete contracting result is extended to a broader class of environments.

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Complexity and Renegotiation: A Foundation for Incomplete Contracts

The ability of an incumbent firm to deter entry by writing exclusionary contracts with customers has been a subject of contention in the antitrust literature. The courts’ concern with such exclusionary contracts has been challenged by those who argue that an incumbent, faced with buyers whose interest is to promote entry and competition, would have to pay buyers more for the inclusion of exclusionary provisions than it could possibly gain from exclusion. In a provocative article, Eric B. Rasmusen et al. (1991) (henceforth, RRW) have argued that an incumbent may in fact be able to exclude rivals profitably using such contractual provisions because it can exploit buyers’ lack of coordination. In essence, if buyers expect other buyers to sign such provisions, then they may see little reason not to do so themselves. The RRW argument is potentially an important one because most alleged instances of entry deterrence through exclusionary contracts with customers occur in situations with multiple buyers. Unfortunately, however, RRW’s two main results contain errors. In this Note, we reconsider the RRW model, providing correct characterizations of the likelihood and cost of exclusion for an incumbent firm. Our results indicate that while the intuition suggested by RRW is valid, the equilibrium likelihood and cost of exclusion differ from what RRW derive. Moreover, our analysis illuminates some further aspects of exclusionary contracting with multiple buyers. Among these issues, we focus on how an incumbent can use discriminatory offers to exploit the externalities that exist among buyers in the provision of competition. In Section I, we review the basic assumptions of the RRW model. In Sections II and III we then consider, in turn, the cases of simultaneous and sequential offers studied by RRW. For the simultaneous model, we distinguish between settings in which the incumbent can and cannot discriminate in its offers to different buyers. (This distinction is the source of the problem in RRW’s analysis: their simultaneous-offer model assumes no discrimination, but the proof of their Proposition 2 assumes—at times—that discrimination is feasible.) We show that absent the ability to discriminate, the incumbent can exclude profitably only when buyers fail to coordinate on their most preferred continuation equilibrium. In contrast, we show that when discrimination is possible, the incumbent need not rely on a lack of buyer coordination to * Segal: Department of Economics, Landau Building, Stanford University, Stanford, CA 94305 (e-mail: ilya.segal@stanford.edu); Whinston: Department of Economics, Northwestern University, 2003 Sheridan Road, Evanston, IL 60208 (e-mail: mwhinston@nwu.edu). We thank Eric Rasmusen, John Wiley, and participants in the Berkeley-Stanford IOfest, the University of Copenhagen Vertical Restraints Conference, and a seminar at the Antitrust Division of the U.S. Department of Justice for helpful comments. We thank Federico Echenique for excellent research assistance. 1 For example, a recent Department of Justice investigation concerned the use of exclusive contracts by the leading provider of computerized ticketing services, Ticketmaster. Most major cities have several large concert/sports venues. In many cities, Ticketmaster has exclusive contracts with a very large share of these venues. 2 The other leading response to the argument that contracts with buyers cannot be a profitable method of exclusion is given in Philippe Aghion and Patrick Bolton (1987 Section I). One factor limiting the applicability of the Aghion and Bolton theory, however, is that it involves partial exclusion: the contracts signed are not fully exclusionary (they involve finite stipulated damages) and the profitability of the strategy arises from extracting rents from an entrant when entry occurs. Indeed, in the Aghion and Bolton model, contracts that fully exclude offer no benefits over writing no contract at all. In Section III of their paper, Aghion and Bolton also discuss a two-buyer version of their model in which externalities arise across buyers. In contrast to the analysis here and in RRW, they allow the incumbent to make an offer to a buyer i that is conditional on the acceptance decision of the other buyer. 3 R. Innes and R. J. Sexton (1994) also discuss the use of discriminatory offers in the context of a model in which buyers can form coalitions with the entrant.

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Naked Exclusion: Comment

The paper studies inefficiencies arising in contracting between one principal and N agents when the utility of each agent depends on all agents' trades with the principal. When the principal commits to a set of publicly observable bilateral contract offers, the arising inefficiency is due entirely to the externalities imposed on non-signers. In contrast, when the principal's offers are privately observed, the distortion is due to the externalities given agents' equilibrium trades. Comparison of the two externalities determines the relative efficiency of the two contracting regimes. In both cases, we show that when N is large, each agent can be treated as non-pivotal, provided that appropriate continuity assumptions are satisfied. We also study the case in which the principal can condition each agent's trade on other agents' messages. We characterize the set of such mechanisms in which each agent's participation is voluntary. When the principal can commit to any such mechanism, she implements the first-best outcome, while threatening each deviator with the harshest possible punishment. However, in the presence of noise that goes to zero slower than N goes to infinity, in the limit we obtain a (generally inefficient) outcome in which each agent feels non-pivotal.

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Contracting with Externalities

We study mechanism design in dynamic quasilinear environments where private information arrives over time and decisions are made over multiple periods. We make three contributions. First, we provide a necessary condition for incentive compatibility that takes the form of an envelope formula for the derivative of an agent's equilibrium expected payoff with respect to his current type. It combines the familiar marginal effect of types on payoffs with novel marginal effects of the current type on future ones that are captured by �impulse response functions.� The formula yields an expression for dynamic virtual surplus that is instrumental to the design of optimal mechanisms and to the study of distortions under such mechanisms. Second, we characterize the transfers that satisfy the envelope formula and establish a sense in which they are pinned down by the allocation rule (�revenue equivalence�). Third, we characterize perfect Bayesian equilibrium-implementable allocation rules in Markov environments, which yields tractable sufficient conditions that facilitate novel applications. We illustrate the results by applying them to the design of optimal mechanisms for the sale of experience goods (�bandit auctions�).

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Ilya Segal

Papers

Envelope Theorems for Arbitrary Choice Sets

Complexity and Renegotiation: A Foundation for Incomplete Contracts

Naked Exclusion: Comment

Contracting with Externalities

Dynamic Mechanism Design: A Myersonian Approach