Game Equilibrium Models I: Evolution and Game Dynamics

338 Pages·1991·9.72 MB·English

by  Reinhard Selten (auth.), Professor Dr. Reinhard Selten (eds.)

Reinhard Selten (Ed.)
Game Equilibrium
Models I
Evolution and Game Dynamics
With Contributions by
I. Eshel, J. W. Friedman, R. Gardner, P. Hammerstein
P. F. Hoekstra, Y. lwasa, D. Messick, M. Morris
H. J. Poethke, R. Selten, A. Shmida, F. J. Weissing
With 41 Figures
Springer-Verlag Berlin Heidelberg GmbH
Professor Dr. Reinhard Selten
Institut ftir Gesellschaft-und
Wirtschaftswissenschaften
der Universitat Bonn
Wirtschaftstheoretische Abteilung I
Adenauerallee 24-42
0-5300 Bonn 1; FRG
ISBN 978-3-642-08108-8 ISBN 978-3-662-02674-8 (eBook)
DOI 10.1007/978-3-662-02674-8
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©Springer-Verlag Berlin Heidelberg 1991
Originally published by Springer-Verlag Berlin Heidelberg New York Tokyo in 1991
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214217130-543210
Preface to the Series "Game Equilibrium Models"
The four volumes of the series "Game Equilibrium Models" are the result of a research
year at the Center for Interdisciplinary Research of the University of Bielefeld,
Germany. The German name of this center is Zentrum fur interdisziplinare Forschung,
but everybody who is familiar with this unique institution refers to it by the
official acronym ZiF.
In the time from October 1, 1987, to September 30, 1988, the ZiF was the home of
the interdisciplinary research group which produced the papers in the four volumes of
this series. Participants coming from many parts of the world lived in the guest
apartments of the Zi F for the whole time or part of it and worked on a common
project. The name of tt.e project was "Game Theory in the Behavioral Sciences". It
occurred to me only later that "Game Equilibrium Models" - the title of the series -
even more appropriately indicates the unifying theme of the research group.
Among the participants were economists, biologists, mathematicians, political
scientists, psychologists and a philosopher. A lively interaction resulted from the
mix of disciplines. The common methodological basis of non-cooperative theory was the
shared culture which facilitated communication across disciplines. The intense
exchange of ideas coming from different fields had a profound influence on the
thinking of many among the participants.
It was not easy to find a coherent way to group the papers into the four volumes
and to find appropriate titles for the books. These and other difficult decisions
have been made by an editorial committee consisting of Wulf Albers, Rudolf Avenhaus,
Eric van Damme, Werner Guth, Peter Hammerstein, Ronald Harstad, Franz Weissing, and
myself.
In the behalf of the whole research group I want to thank all those who helped
to make the research year possible. We owe special thanks to the staff of the ZiF and
in particular to Mrs. lilo Jegerlehner for her technical help in the preparation of
the four volumes.
Finally, I want to express my gratitude to all those who assisted me in the
organizational and editorial work, especially to Franz Weissing whose efforts were
indispensable.
Bielefeld/Bonn, January 1991 Reinhard Selten
Contents
Preface . . . . • • . . . . . . . . . . . . . . . . • . . . . • . . . . . . . • . • . • . . • • • . • . . . . . • . • . . . . . . . . . . . . • . . . V
Contributors . • . . . . . . • . . . • . • . . . . • • • . . . . • . . . . . . . . . . . . . . . . . . . • . • . . . . . . . . . . . . . . . • . IX
Introduction to the Series "Game Equilibrium Models" .•.•.........•.•...........
I nt roduct ion to Vo 1u me I : "Evo 1u ti on and Game Dynamics • • . . . . . . . . . . . . . . . . . . . . . • . 3
I1an Eshel
GAME THEORY AND POPULATION DYNAMICS IN COMPLEX GENETICAL SYSTEMS:
THE ROLE OF SEX IN SHORT TERM AND IN LONG TERM EVOLUTION....................... 6
I. Game Theory and the Dynamics of Evolutionary Changes
in Complex Sexual Systems- The Problem.................................. 6
2. Selective Gene Substitution and Long Term Convergence to an ESS in a
Viability Population Game • . •• . • . . • . •• .. . . . . • . . . • . . . . . . .. . . . . .. . . . . • . • . . . 9
3. Other Relevant Payment Functions and Evolutionary Genetic Stability (EGS) I2
4. Is It Possible to Infer a (Long Term) Game Structure
From the (Short Term) Population Dynamics? .•......•••.....•.•••.......... I4
5. The Role of Sex in Short and in Long Term Evolution...................... I8
6. Summary.................................................................. 22
Discussion . • . . . . • . . . . . . . . . • . • . . . . . . • . . . . • • . • . • . . • • • . . . . . . • . . . . . . • . . . . . . . . . . . 22
Franz J. Weissing
EVOLUTIONARY STABILITY AND DYNAMIC STABILITY IN A CLASS OF EVOLUTIONARY
NORMAL FORM GAMES . . • • • . . . . . . . . . . • . • . . . . . . . • . • . • • . . . • • . . . . . • . • . . . . • . • . • . . . . . . . . . 29
I. Introduction............................................................. 30
2. Notation and Basic Definitions........................................... 34
3. Nash Equilibrium Strategies of RSP-Games .........•.•...•........•........ 36
4. Evolutionary Stability in RSP-Games .•.•••..•••••...••....•...•........... 42
5. Stability with Respect to the Continuous Replicator Dynamics ...•.•....... 47
6. Stability with Respect to the Discrete Replicator Dynamics .•...•......... 60
7. Complex Attractors of the Discrete Replicator Dynamics ...............•... 76
8. Are RSP-Games Played in Biological Populations?.......................... 93
Reinhard Selten
ANTICIPATORY LEARNING IN TWO-PERSON GAMES ..•..•...•.....•..••.•.•.•............ 98
I. Introduction............................................................. 99
2. Regular Equilibrium Points ••......•..•..••....•...•..••.•.•.•....••....•• IOI
3. The Preliminary Model •.......••..•••....•••....••••....•........•••...•.• I05
4. The Anticipatory Learning Process •••.•.••.•••.....•.................•.... II8
5. Special Stability Conditions ...••.•.••••.•...•••.................•..•.... I40
6. Anticipated Anticipations .•..•.••...••••...•.••...•••....•.•.•.•......... I48
7. Concluding Remark .•.....•.•..•......•••.....•.....•••....•.•...•......... I 53
Rolf F. Hoekstra, Yoh Iwasa and Franz J. Weissing
THE ORIGIN OF ISOGAMOUS SEXUAL DIFFERENTIATION ..........•.••...•••.•.....•..... I55
I. Introduction ...•.•.•.•....••••••....••..•••.•......•.••••...•.•.......•.• I55
2. A Genera 1 Mode 1 for the Evo 1u ti on of Mating Types . . • . • . • . . . • . . • . . . . . . . . . . I 57
3. Some Specific Models .•......••......•••.•...•......•.•.•.•.••.•.......... I7!'1
4. Conclusions ••.•.•.....•.•.•....•...•......••.•.•.•••.....•..•...•.•.•.•.. I80
VIII
Roy Gardner and Holly Norris
THE EVOLUTIONARY STABILITY OF BLUFFING IN A CLASS OF EXTENSIVE FORM GAMES 182
I. Introduction ......................•.................................... 182
II. Evolutionary Dynamics of an Extensive Form Game ...............•......•• 183
III. Evolutionarily Stable Strategies ....................................•.. 185
IV. B~ologi~al Interpretations ............................................. 189
V. D1scuss1on ...................•......•.................................• 192
Reinhard Se1ten and Avi Shmida
POLLINATOR FORAGING AND FLOWER COMPETITION IN A GAME EQUILIBRIUM MODEL ......... 195
I. Introduction .....•..................................................... 195
2. The Decision Problem of a Pollinator ................................... 202
3. Heuristic Discussion of Optimal Pollinator Foraging .................... 209
4. Derivation of Results on Optimal Foraging .............................. 213
5. The Critical level of the Microhabitat Quality .....•................... 216
6. Pollen Flow ............................................................ 220
7. The Investment Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
8. Necessary Conditions for Strong Evolutionary Stability in the
Investment Game ........................................................ 231
9. Sufficient Conditions for Strong Evolutionary Stability in the
Investment Game ..............................•......................... 234
10. Parameter Influences ................................................... 242
11. Discussion ............................................................. 246
James W. Friedman and Peter Hammerstein
TO TRADE, OR NOT TO TRADE; THAT IS THE QUESTION ................................ 257
1. Introduction .............................•............................. 257
2. A Summary of the Field Observation of the Hamlet ....................... 258
3. A Simplified Model of Mating Behavior .................................. 260
4. Optimal Behavior and Equilibrium ....................................... 265
5. The Scope for Male Specialization ....................................... 268
6. Concluding Comments .................................................... 274
Hans J. Poethke and Franz J. Weissing
COMPETITION AVOIDANCE IN A DRAGONFLY MATING SYSTEM ............................. 276
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
2. The Information Situation at the Mating Place .......................... 278
3. The low Density Case . . . . . . • . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . 281
4. A General Normal Form Game ............................................. 283
5. Evolutionary Stability in Games with Strategy-Payoff Feedback .......... 293
6. Numerical Results .....•................................................ 297
7. A Game Model with Switching Costs ...................................... 301
David Messick
ON THE EVOLUTION OF GROUP-BASED AlTRUISM ....................................... 304
1. Introduction ........................................................... 304
2. Mechanisms for the Evolution of Altruism ............................... 305
3. The Model .............................................................. 308
4. Psychological Implications ............................................. 320
Contributors
I7an Eshe7, Faculty of Mathematical Sciences, Tel Aviv University Raymond and
Beverly Sackler, Ramat Aviv, 69978 Tel Aviv, Israel
James W. Friedman, Department of Economics, University of North Carol ina, Gardner
Hall 01719, Chapel Hill, NC 27514, USA
Roy Gardner, Department of Economics, Indiana University, Ballantine Hall, Blooming
ton, IN 47405, USA
Peter Hammerstein, Max-Planck-lnstitut fUr Verhaltensphysiologie, Abteilung Wickler,
D-8131 Seewiesen, FRG
Rolf Hoekstra, Universitat of Wageningen, Department of Genetics, Dreijenlaan 2,
6703-HA Wageningen, The Netherlands
Yoh Iwasa, Kyushu University, Department of Biology, Faculty of Science 33, Fukkoka
812, Japan
David Messick, Department of Psychology, University of California Santa Barbara,
Santa Barbara, CA 93106, USA
Mo77y Morris, Department of Biology, Indiana University, Bloomington, IN 47405, USA
Hans-Joachim Poethke, Universitat Mainz, lnstitut fUr Zoologie, AG Populations
biologie, Postfach 3980, D-6500 Main, FRG
Reinhard Se7ten, Institut fUr Gesellschafts- und Wirtschaftswissenschaften der Uni
versitat Bonn, Wirtschaftstheoretische Abteilung I, Adenauerallee 24-42, D-5300
Bonn I, FRG
Avi Shmida, Department of Botany, The Hebrew University, Givat Ram, Jerusalem 91904,
Israel
Franz J.Weissing, Rijksuniversiteit Groningen, Biologisch Centrum, Kerklaan 30,
9751-NN Haren, The Netherlands
Introduction to the Series •Game Equilibrium Model's"
Game equilibrium models are descriptions of interactive decision situations by gamE
in extensive or normal form. The analysis of such models is based on the equilibrit
point concept, often refined by additional requirements like subgame perfectness. T~
series consists of four volumes:
I: Evolution and Game Dynamics
II: Methods, Morals and Markets
III: Strategic Bargaining
IV: Social and Political Interaction.
The game equilibrium models presented in these books deal with a wide variety o
topics. Just one example from each of the volumes may provide an illustration: Eg
trading in hermaphrodite fish (Friedman and Hammerstein in Volume I), the social or
ganization of irrigation systems (Weissing and Ostrom in Volume II), wage bargainin'
(Ha77er in Volume III), and beheading games in mediaeval 1 iterature (O'Neill i1
Volume IV).
Non-cooperative game theory is a useful research tool not only in economics anc
the social sciences, but also in biology. Game theory has been created as a theory 01
conflict and cooperation among rational individuals. For a long time strong rational·
ity assumptions seemed to be indispensable at the foundations of game theory. In thi!
respect, biological applications have changed our perspectives. Game equilibrium ma)
be reached as the result of processes of natural selection without any rational de·
liberation. Therefore, game equilibrium models can contribute to the explanation of
behavioral and structural features of animals and plants.
The interpretation of game equilibrium as the result of evo 1 uti on raises the
question of dynamic stability with respect to evolutionary processes. Similar prob
lems also arise in theories of game learning. The first volume contains three papers
on game dynamics. Two of them are concerned with the dynamic foundations of evol
utionary game theory and the third one explores stability in a model of anticipatory
learning. The remaining papers in the first volume present evolutionary game equilib
rium models ranging from abstract investigations of phenomena like bluffing or group
based altruism to the examination of concrete systems observed in nature like
"competition avoidance in a dragonfly mating system". Not only theoretical clarifica
tions of the foundations of evolutionary game theory and related research can be
found in Evolution and Game Dynamics, but also exciting new biological applications.
The title of the second volume, Methods, Morals, and Markets, points to several
areas of research which attract the interest mainly of economists, but also of pol
itical scientists, mathematicians and philosophers. The first paper is a sophisti
cated mathematical contribution which applies new tools to basic questions of non-co
operative game theory. The word "method" mainly refers to this paper, but to some ex-
2
tent also to the next three contributions, which discuss basic conceptual problems in
the interpretation of game equilibrium. Two papers relate to the philosophical notion
of the social contract and its exploration with the help of game theoretical models.
This work in concerned with "morals", a theme which is also touched by a paper on ir
rigation institutions. The remaining four papers of the second volume explore game
equilibrium models of markets; two of these contributions are experimental and com
pare theoretical solutions with experimental data.
The third volume on Strategic Bargaining collects ten papers on game equilibrium
models of bargaining. All these papers look at bargaining situations as non-coopera
tive games. Unlike in cooperative game theory, cooperation is not taken for granted,
but explained as an outcome of equilibrium analysis. General models of two-person and
n-person bargaining are explored, sometimes enriched by institutional detail like the
availability of long-term contracts. Other papers explore bargaining in special con
texts like wage negotiations. Two contributions concern spatial games; one of these
contributions is experimental.
The exploration of strategic models of bargaining is an active field of research
which attracts the attention of many game theorists and economists. The ten papers in
the third volume contribute to the progress in this field.
The fourth volume on Social and Political Interaction mainly presents game equi
librium models in the area of political science. Three of the papers concern topics
in other fields: the distribution of foreign language skills, altruism as a social
dilemma (an experimental paper) and beheading games in mediaeval 1 iterature. Five
contributions to the area of i nternat ion a1 re 1a t ions de a1 with game theoret i ca 1
models of the balance of power, of alliance formation, and of an issue in armament
policy. An investigation of inspection problems like those arising in connection with
the non-proliferation treaty also touches the area of international relations. Other
papers on problems of political science deal with the game theoretical resolution of
the Condorcet paradox by equilibrium selection, the modelling of political pressure
exerted by firms on the government and the draft resistance problem.
The main emphasis is on biology in Volume I, on economics in Volumes II and Ill,
and on political science in Volume IV. This is the result of an attempt to group the
great variety of papers resulting from a year long interdisciplinary research project
in a reasonably coherent way. However, not only biologists, but also economists and a
psychologist have contributed to Volume I. Similarly, not only economists and mathe
maticians, but also political scientists as well as a biologist and a psychologist
are among the authors of Volumes II and III. All four volumes are the result of the
cooperation of researchers from many disciplines united by a common interest in game
equilibrium models within and beyond the borders of their fields.
Bielefeld/Bonn, January 1991 Reinhard Selten