An introduction to game dynamics

1
An introduction to game dynamics

2
(No Transcript)
3
Nash equilibria
4
Nash equilibria
5
Nash equilibria
6

• The greatest conceptual revolution in
  biology...the replacement of typological
  thinking by population thinking.
• Ernst Mayr

7
John Maynard Smith
8
Symmetric Games
9
Chicken Game
10
Population setting
11
Population setting
12
Replicator equation
13
Replicator dynamics and Nash equilibria
14
Folk theorem of evolutionary game theory

• Nash equilibria are rest points
• strict Nash equilibria are attractors
• stable rest points are Nash equilibria
• limits of interior orbits are Nash equilibria

15
Existence of Nash equilibria
16
Existence of Nash equilibria
17
Existence of Nash equilibria
18
Existence of Nash equilibria
19
Existence of Nash equilibria
20
Replicator equation for n2
21
Replicator equation for n2

• Dominance
• Bistability
• stable coexistence

22
Example dominance
23
Example bistability
24
Example coexistence
25
Replicator equations for n3

• Possibility for
• rock-scissors-paper
• heteroclinic cycles

26
Rock -Paper- Scissors
for the adult player

27
(No Transcript)
28
(No Transcript)
29
(No Transcript)
30
Rock-Paper-Scissors
31
Rock-Paper-Scissors

32
(No Transcript)
33
Generalized Rock-Paper-Scissors
34
Generalized Rock-Paper-Scissors
35
Generalized Rock-Paper-Scissors
36
Generalized Rock-Paper-Scissors
37
Generalized Rock-Paper-Scissors
38
Proof (Zeeman, Hofbauer,)
39

40

41
In the Stone Age
42
Bacterial Game Dynamics

• Escherichia coli
• Type A wild type
• Type B mutant producing colicin (toxic) and an
  immunity protein
• Type C produces only the immunity protein

43
Bacterial Game Dynamics

• Escherichia coli
• Rock-Paper-Scissors cycle
• Not permanent!
• Serial transfer (from flask to flask)
• only one type can survive!
• (Kerr et al, Nature 2002)

44
Mating behavior

• Uta stansburiana (lizards)
• (Sinervo and Lively, Nature, 1998)

45
(No Transcript)
46
Mating behavior

• males 3 morphs (inheritable)

47
(No Transcript)
48
Rock-Paper-Scissors in Nature

• males 3 morphs (inheritable)
• A monogamous, guards female
• B polygamous, guards harem (less efficiently)
• C loose males, sneaky matings

49
Other examples of RPS-cycles
In models of ecological competition
(May,) Non-permanent community can be
stabilised by fourth population (competitor or
predator) (Hofbauer, Schreiber,)
50
Other examples of RPS-cycles

• In the evolution of virulence
• Three strains of a pathogen
• Different virulence (extra mortality)
• Super-infection of host organism
  (most virulent takes over)
• Different infectivity
• (Nowak, May,)

51
Other examples of RPS-cycles

• In genetics
• Segregation distorters (outlaw genes)
• Harm their opposite chromosome before gametes are
  formed
• (Crow, Grafen, Stadler)

52
Iterated Prisoners Dilemma

• infinitely repeated, with errors
• All D
• Tit For Tat
• Firm But Fair

53
Iterated Prisoners Dilemma
Heteroclinic network A Tit For Tat B Firm But
Fair C Bully D AllD
54
Other game dynamics

• Transmission of strategies through
  inheritance, imitation, learning

55
Imitation dynamics
56
Imitation dynamics
57
Imitate the better
58
Best Reply Dynamics

59
Best Reply
60
Best reply dynamics
61
Best Reply
62
Best Reply
63
Are there game dynamics which always converge
to Nash equilibria?

64
Myopic adjustment (Hofbauer)
65
Myopic adjustment (Hofbauer)
66
Phase portraits of Replicator equations
67
Public Goods

• all do better if there are more cooperators in
  the group
• within each group, defectors do better than
  cooperators
• temptation to free-ride and exploit co-players
• no public good!

68
Public Goods Experiments

• 6 players
• 1 dollar each
• contribute to common pool?
• Experimenter triples amount in pool
• divides it equally among all players

69
Public Goods Experiments

• 6 players
• 1 dollar each
• contribute to common pool?
• Experimenter triples amount in pool
• divides it equally among all players
• return for each dollar only 50 cents!
• not to contribute is dominant

70
Optional Public Goods

• (Hauert et al, Science 2002)
• large population
• N players offered to participate (sample)
• S accept (group)
• N-S decline (loners, risk-averse)
• loners have fallback-solution

71
Optional Public Goods

• three strategies
• loner
• cooperator
• defector
• (if only one participates loner)

72
Game Dynamics
73
Payoff for optional game
74
Optional Public Goods
75
Replicator Dynamics
76
Rock-Paper-Scissors cycle

• If most cooperate, best to defect
• if most defect, best to abstain
• if most abstain, best to cooperate
• (for small groups, Simpsons paradox)

77
Simpsons paradox
78
Change in variables
79
Hamiltonian system
80
Replicator dynamics
81
Time averages
82
Red Queen Dynamics
83
Optional Public Goods

• Well-mixed population (random encounters)

84
Spatial Games

• Interaction with nearest neighbors only
  
• best takes over

85
Optional Public Goods

• Local interactions

86
looking back to bacterial games

• Rock-Paper-Scissors cycle
• Not permanent!
• Serial transfer (from flask to flask)
• only one type can survive!

87
looking back to bacterial games

• Rock-Paper-Scissors cycle
• Not permanent!
• Serial transfer (from flask to flask)
• only one type can survive!
• Serial transfer from plate to plate
• all types survive!
• (Kerr et al, Nature 2002)

88
Asymmetric Games
89
Asymmetric Games

• Owner vs. Intruder
• Female vs. Male
• Young vs. Old
• Buyer vs. Seller
• Host vs. Parasite

90
Male vs. Female Battle of the Sexes

• Males invest little
• Females invest much
• Females must be choosy
• long engagement periods?
• Females coy or fast
• Males faithful or philandering

91
Battle of the Sexes

• Females coy
• Males faithful
• Females fast
• Males philanderers
• Females coy
• ...

92
Battle of the Sexes
93
Battle of the Sexes
94
Battle of the Sexes
95
Example Owner-Intruder
96
Example Owner-Intruder
97
Example Owner-Intruder
98
Example Owner-Intruder

• (1,0) Bourgeois strategy
• (0,1) paradox strategy (Prudhon)
• asymmetric Nash equilibria

99
Conditional Strategies

• If owner, escalate if intruder, display
• If male, philanderer if female, coy

100
Conditional Strategies
101
Conditional Strategies
102
Conditional Strategies
103
Conditional Strategies
104
Punishment

• after public goods game, players can fine
  co-players
• punishment is costly
• punishment is efficient
• works even if players never meet again!
• (Fehr and Gächter, 2002)

105
(No Transcript)
106
(No Transcript)
107
Punishment

• introduces a
• second order social dilemma!

108
Evolutionary Game Dynamics
109
Strategies for Public Goods with Punishment

110
Payoff for Public Goods with Punishment

111

112

113
Public Goods with Punishment

114
Public Goods with Punishment

115
Reputation and Temptation

116
additional payoff terms

117
Reputation effect

118
Bifurcation
119
Morals?

• More freedom yields more cooperation
• Individuals that are less social
• make better societies

120

• Christoph Hauert
• Josef Hofbauer
• Martin Nowak
• Silvia Demonte
• Hannelore Brandt