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Convex Analysisの二,三の進展について

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Offering a unifying theoretical perspective not readily available in any other text, this innovative guide to econometrics uses simple geometrical arguments to develop students' intuitive understanding of basic and advanced topics, emphasizing throughout the practical applications of modern theory and nonlinear techniques of estimation. One theme of the text is the use of artificial regressions for estimation, reference, and specification testing of nonlinear models, including diagnostic tests for parameter constancy, serial correlation, heteroscedasticity, and other types of mis-specification. Explaining how estimates can be obtained and tests can be carried out, the authors go beyond a mere algebraic description to one that can be easily translated into the commands of a standard econometric software package. Covering an unprecedented range of problems with a consistent emphasis on those that arise in applied work, this accessible and coherent guide to the most vital topics in econometrics today is indispensable for advanced students of econometrics and students of statistics interested in regression and related topics. It will also suit practising econometricians who want to update their skills. Flexibly designed to accommodate a variety of course levels, it offers both complete coverage of the basic material and separate chapters on areas of specialized interest.

Estimation and Inference in Econometrics

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

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

This paper looks at adaptive applications that can switch between a small number of different levels of service, with switching decisions made solely by the originating end-system. Typical of such applications are real time streaming protocols which can use different coding rates. The end-systems probe the network with sample traffic to determine congestion, and decide at what rate to enter according to the fate of their “probe” packets. During the lifetime of a connection, the procedure is repeated to reassess and possibly readjust the rate. We derive analytic models, based on diffusion limits under a natural scaling, to quantify the benefits of in-call probing. We then use simulation to compare the results in a number of scenarios, and show that this simple theory is remarkably accurate in predicting large-scale behaviour. The results also show that a small amount of in-call probing produces significant benefits to the system.

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Modelling the Performance of In-Call Probing for Multi-Level Adaptive Applications

In this paper, we propose new dynamic decentralized multi-channel multiple access (MAC) protocols and study their performance. Our protocols build on slotted Aloha, but extend it in several ways to improve flow completion time and throughput, as follows: (i) channels are assigned to flows rather than packets to eliminate per packet collisions, thus the total number of collisions is reduced, and (ii) each flow owns or drops channels dynamically considering successful transmissions, thus the number of owned channels adapts to varying traffic. We present an analysis of the stability region and of flow completion times, for our algorithms, and show that one of them can achieve close to 100% throughput if flow sizes are large. We demonstrate by extensive simulations that, compared to current multi-channel MAC protocols, our algorithms improve flow completion time and throughput in wireless local area and mesh networks.

https://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/ITA08MU.pdf

Dynamic decentralized multi-channel MAC protocols

In a recent ruling, FCC has enabled an unlicensed use of the vacated analogue broadcast TV spectrum, called “White Spaces”, provided that the unlicensed devices detect and avoid primary users (remaining TV channels and wireless MICs). The offered frequency space is very wide and has excellent propagation characteristics, hence the ruling offers a huge potential to develop high-quality, cheap and ubiquitous wireless access, and it also poses several novel design challenges (detecting primaries, network asymmetries, channel selections, etc). In this work we focus on the problem of channel and rate selections for white-spaces. We start by presenting measurements using an indoor testbed operating in the 500 to 600MHz band, and demonstrate that the key challenge is handling variable channel quality. The measurements show that there is significant potential benefit from selecting the best channel and adapting the transmission rate while also revealing non-stationary channel characteristics. The proposed learning algorithms are well suited to non-stationary environments and attempt to minimize “regret” with respect to a best or oracle-like algorithm. The novel aspects of the algorithms includes (i) accounting for the cost incurred when switching channels and synchronization costs, (ii) exploiting inherent correlations between the effective throughputs achieved when selecting the same channel but different transmission rates. We further extend the algorithms to account for fairness issues in network scenarios where several users are served by an access point.

/pdf/exploiting-channel-diversity-in-white-spaces-3zhjdwtsm2.pdf

Exploiting Channel Diversity in White Spaces

In this paper we explore some of the connections between cooperative game theory and the utility maximization framework for routing and flow control in networks. Central to both approaches are the allocation of scarce resources between the various users of a network and the importance of discovering distributed mechanisms that work well. The specific setting of our study is ad-hoc networks where a game-theoretic approach is particularly appealing. We discuss the underlying motivation for the primal and dual algorithms that assign routes and flows within the network and coordinate resource usage between the users. Important features of this study are the stochastic nature of the traffic pattern offered to the network and the use of a dynamic scheme to vary a user's ability to send traffic. We briefly review coalition games defined by a characteristic function and the crucial notion of the Shapley value to allocate resources between players. We present a series of experiments with several test networks that illustrate how a distributed scheme of flow control and routing can in practice be aligned with the Shapley values which capture the influence or market power of individual users within the network.

/pdf/coalition-games-and-resource-allocation-in-ad-hoc-networks-4zejsqpnir.pdf

Coalition Games and Resource Allocation in Ad-Hoc Networks

Various actively probing techniques on a network allow an estimation of the sustainable capacity available in the network. One implementation probes the network using two probe sequences having different loads. Probe estimates associated with each sequence are used to estimate the sustainable capacity in the network. Another implementation probes the network using two probe sequences, in which either the probe packet size or the inter-probe gap are different between the sequences. Calculation of a delay trend for each sequence leads to estimate of the maximum network capacity and the background load on the network. Yet another implementation uses a windowing technique to probe the sustainable capacity in the network. The window size is increased until the network is saturated. The number of bytes transferred within the window during a given time period is an estimate of the sustainable capacity of the network.

Peter Key

Papers

Modelling the Performance of In-Call Probing for Multi-Level Adaptive Applications

Dynamic decentralized multi-channel MAC protocols

Exploiting Channel Diversity in White Spaces

Coalition Games and Resource Allocation in Ad-Hoc Networks

Sustainable capacity estimation