We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

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

In the Hamadryas baboon, males are substantially larger than females. A troop of baboons is subdivided into a number of ‘one-male groups’, consisting of one adult male and one or more females with their young. The male prevents any of ‘his’ females from moving too far from him. Kummer (1971) performed the following experiment. Two males, A and B, previously unknown to each other, were placed in a large enclosure. Male A was free to move about the enclosure, but male B was shut in a small cage, from which he could observe A but not interfere. A female, unknown to both males, was then placed in the enclosure. Within 20 minutes male A had persuaded the female to accept his ownership. Male B was then released into the open enclosure. Instead of challenging male A , B avoided any contact, accepting A’s ownership.

Evolution and the Theory of Games

Sphere Packings, Lattices and Groups

Hybridization may influence evolution in a variety of ways. If hybrids are less fit, the geographical range of ecologically divergent populations may be limited, and prezygotic reproductive isolation may be reinforced. If some hybrid genotypes are fitter than one or both parents, at least in some environments, then hybridization could make a positive contribution. Single alleles that are at an advantage in the alternative environment and genetic background will introgress readily, although such introgression may be hard to detect. ‘Hybrid speciation’, in which fit combinations of alleles are established, is more problematic; its likelihood depends on how divergent populations meet, and on the structure of epistasis. These issues are illustrated using Fisher’s model of stabilizing selection on multiple traits, under which reproductive isolation evolves as a side-effect of adaptation in allopatry. This confirms a priori arguments that while recombinant hybrids are less fit on average, some gene combinations may be fitter than the parents, even in the parental environment. Fisher’s model does predict heterosis in diploid F1s, asymmetric incompatibility in reciprocal backcrosses, and (when dominance is included) Haldane’s Rule. However, heterosis arises only when traits are additive, whereas the latter two patterns require dominance. Moreover, because adaptation is via substitutions of small effect, Fisher’s model does not generate the strong effects of single chromosome regions often observed in species crosses.

The role of hybridization in evolution

We introduce a class of one-dimensional discrete space-discrete time stochastic growth models described by a height function ht(x) with corner initialization. We prove, with one exception, that the limiting distribution function of ht(x) (suitably centered and normalized) equals a Fredholm determinant previously encountered in random matrix theory. In particular, in the universal regime of large x and large t the limiting distribution is the Fredholm determinant with Airy kernel. In the exceptional case, called the critical regime, the limiting distribution seems not to have previously occurred. The proofs use the dual RSK algorithm, Gessel's theorem, the Borodin–Okounkov identity and a novel, rigorous saddle point analysis. In the fixed x, large t regime, we find a Brownian motion representation. This model is equilvalent to the Seppalainen–Johansson model. Hence some of our results are not new, but the proofs are.

/pdf/limit-theorems-for-height-fluctuations-in-a-class-of-ulcsgomnop.pdf

Limit Theorems for Height Fluctuations in a Class of Discrete Space and Time Growth Models

Percolation on the fitness hypercube and the evolution of reproductive isolation.

Discontinuities in effective subsurface transport properties commonly arise (l) at abrupt contacts between geologic materials (i.e., in composite porous media) and (2) in discrete velocity fields of numerical groundwater-flow solutions. However, standard random-walk methods for simulating transport and the theory on which they are based (diffusion theory and the theory of stochastic differential equations (SDEs)) only apply when effective transport properties are sufficiently smooth. Limitations of standard theory have precluded development of random-walk methods (diffusion processes) that obey advection dispersion equations in composite porous media. In this paper we (1) generalize SDEs to the case of discontinuous coefficients (i.e., step functions) and (2) develop random-walk methods to numerically integrate these equations. The new random-walk methods obey advection-dispersion equations, even in composite media. The techniques retain many of the computational advantages of standard random-walk methods, including the ability to efficiently simulate solute-mass distributions and arrival times while suppressing errors such as numerical dispersion. Examples relevant to the simulation of subsurface transport demonstrate the new theory and methods. The results apply to problems found in many scientific disciplines and offer a unique contribution to diffusion theory and the theory of SDEs.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/1999WR900224

Diffusion processes in composite porous media and their numerical integration by random walks: Generalized stochastic differential equations with discontinuous coefficients

Two-dimensional bootstrap percolation is a cellular automaton in which sites become ‘infected’ by contact with two or more already infected nearest neighbours. We consider these dynamics, which can be interpreted as a monotone version of the Ising model, on an n × n square, with sites initially infected independently with probability p. The critical probability p
 c
 is the smallest p for which the probability that the entire square is eventually infected exceeds 1/2. Holroyd determined the sharp first-order approximation: p
 c
 ~ π
 2/(18 log n) as n → ∞. Here we sharpen this result, proving that the second term in the expansion is −(log n)−3/2+o(1), and moreover determining it up to a poly(log log n)-factor. The exponent −3/2 corrects numerical predictions from the physics literature.

/pdf/a-sharper-threshold-for-bootstrap-percolation-in-two-4otdx5txos.pdf

A sharper threshold for bootstrap percolation in two dimensions

[1] During the last decade, isotopic fractionation has gained acceptance as an indicator of microbiological and chemical transformations of contaminants in groundwater. These transformation processes typically favor isotopically light, compared to isotopically heavy, contaminants, resulting in enrichment of the latter in the residual aqueous phase. In these isotope applications, it has been generally presumed that physical transport processes in groundwater have a negligible effect on isotopic enrichment. It is well known, however, that aqueous phase diffusion generally proceeds faster for isotopically light, compared to isotopically heavy, solute molecules, often resulting in isotopic fractionation in groundwater. This paper considers the potential for isotopic fractionation during transport in groundwater resulting from minute isotopic effects on aqueous diffusion coefficients. Analyses of transport in heterogeneous systems delimit the viable range of isotopic fractionation by diffusion in groundwater. Results show that diffusion can result in similar degrees of depletion and enrichment of isotopically heavy solutes during transport in heterogeneous systems with significant diffusion rate–limited mass transfer between fast- and slow-flow zones. Additional analyses and examples explore conditions that attenuate the development of significant fractionation. Examples are presented for 13C methyl tertiary butyl ether and deuterated and nondeuterated isopropanol and tertiary butyl alcohol using aqueous diffusion coefficients measured by the Taylor dispersion method with refractive index profiling as a part of this study. Examples elucidate the potential for diffusive fractionation as a confounder in isotope applications and emphasize the importance of hydrogeologic analysis for assessing the role of diffusive fractionation in isotope applications at contaminant field sites.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2006WR005264

Janko Gravner

Papers

Limit Theorems for Height Fluctuations in a Class of Discrete Space and Time Growth Models

Percolation on the fitness hypercube and the evolution of reproductive isolation.

Diffusion processes in composite porous media and their numerical integration by random walks: Generalized stochastic differential equations with discontinuous coefficients

A sharper threshold for bootstrap percolation in two dimensions

Isotopic fractionation by diffusion in groundwater