The Stability Concept of Evolutionary Game Theory: A Dynamic Approach

ISBN-13: 9783540554196 / Angielski / Miękka / 1992 / 129 str.

Ross Cressman

Opis

These Notes grew from my research in evolutionary biology, specifically on the theory of evolutionarily stable strategies (ESS theory), over the past ten years. Personally, evolutionary game theory has given me the opportunity to transfer my enthusiasm for abstract mathematics to more practical pursuits. I was fortunate to have entered this field in its infancy when many biologists recognized its potential but were not prepared to grant it general acceptance. This is no longer the case. ESS theory is now a rapidly expanding (in both applied and theoretical directions) force that no evolutionary biologist can afford to ignore. Perhaps, to continue the life-cycle metaphor, ESS theory is now in its late adolescence and displays much of the optimism and exuberance of this exciting age. There are dangers in writing a text about a theory at this stage of development. A comprehensive treatment would involve too many loose ends for the reader to appreciate the central message. On the other hand, the current central message may soon become obsolete as the theory matures. Although the restricted topics I have chosen for this text reflect my own research bias, I am confident they will remain the theoretical basis of ESS theory. Indeed, I feel the adult maturity of ESS theory is close at hand and I hope the text will play an important role in this achievement.

Spis treści

1. Introduction.- 2. Frequency-Dependent Evolution in a Single Haploid Species.- 1. Pure and Mixed Strategies.- 2. Monomorphic ESS’s and Stability.- 3. The Hawk-Dove Game.- 4. The Static Characterization of an ESS.- (A) Biological Considerations.- (B) Mathematical Considerations.- 5. Stability for the Continuous Dynamic.- 6. The Strong Stability Concept and the Dynamic Characterization of an ESS.- 7. Stability for the Discrete Dynamic.- 8. Alternative Proof of Theorem 2.5.2.- (A) The Centre Manifold for the Pure-Strategy Dynamic.- (B) The Centre Manifold for the Mixed-Strategy Dynamic.- 9. Nonlinear Fitness Functions.- (A) The Sex-Ratio Game.- (B) ESS’s and Stability.- 10. Appendix on Centre Manifold Theory.- 3. Frequency-Dependent Evolution in a Two-Species Haploid System.- 1. Frequency-Dependent Fitness.- 2. Monomorphic ESS’s and Stability.- 3. Examples: Battle-of-the-Sexes and Edgeworth Market Games.- (A) Battle-of-the Sexes.- (B) Edgeworth Market Games.- 4. The Static Characterization of a Two-Species ESS.- 5. Strong Stability for the Continuous Dynamic.- 6. Multi-Species Frequency-Dependent Evolution.- 4. Frequency-Dependent Evolution in a Randomly-Mating Diploid Species.- 1. Natural Selection as an Evolutionary Game.- (A) The Frequency-Dependent Haploid Perspective.- (B) The Frequency-Independent Diploid Perspective.- 2. Single-Locus Models.- 3. Two-Phenotype, Frequency-Dependent Evolution at a Single Locus.- 4. Multi-Phenotype, Frequency-Dependent Evolution at a Single Locus.- (A) Examples.- (B) Linearization.- 5. A Two-Locus, Two-Allele, Two-Phenotype Example.- 5. Frequency- and Density-Dependent Evolution in a Haploid Species.- 1. Frequency- and Density-Dependent Fitness (and the Haploid Dynamic).- 2. Monomorphic DDESS’s and Stability.- 3. The Density-Dependent Hawk-Dove Game.- (A) Non-Aggressive Behavior Enhanced at High Density.- (B) Aggressive Behavior Enhanced at High Density.- 4. The DDESS Conditions and Strong Stability.- 5. Density-Dependent Natural Selection as a Haploid Evolutionary Game.- 6. Evolutionary Stability in Multi-Species Population-Dynamic Models.- An Intermission.- 6. Evolutionary Stable Sets and Contestant Information.- 1. A Mixed-Strategy Hawk-Dove Game.- 2. The Static Characterization of an ES Set.- 3. The Dynamic Characterization of an ES Set.- 4. The Hawk-Dove Game with Varying Resource.- 5. ES Sets for Games in Extensive Form.- 6. The Owner-Intruder Game.- (A) A Single-Species Extensive-Game Perspective.- (B) The Two-Species Perspective.- 7. Multi-Stage Games.- (A) A Two-Trial Game.- (B) The Iterated Prisoner’s Dilemma Game.- 7. References.- 8. Index.