Evolutionary Games and Population Dynamics

1

• Evolutionary Games and Population Dynamics

2
(No Transcript)
3
(No Transcript)
4

5

6

7

8

9

10

•  

11
Oskar Morgenstern (1902-1977)John von Neumann
(1903-1957)John Nash (b. 1930)
12
Nash-Equilibrium

• Arbitrarily many players
• each has arbitrarily many strategies
• there always exists an equilibrium solution
• no player can improve payoff by deviating
• each strategy best reply to the others

13
Nash equilibria can be inefficient
14
John Maynard Smith (1920-2004)
15
Evolutionary Game Theory

• Population of players
• (not necessarily rational)
• Subgroups meet and interact
• Strategies Types of behaviour
• Successful strategies spread in population

16
Population setting
17
Population Dynamics
18
Example Moran Process
19
Discrete time
20
Continuous time
21
Replicator Dynamics
22
Replicator dynamics and Nash equilibria
23
Replicator equation

24
Replicator equation for n2
25
Replicator equation for n2

• Dominance
• Bistability
• stable coexistence

26
Example dominance
27
Vampire Bat (Desmodus rotundus)
28
Vampire Bat (Desmodus rotundus)
29
Vampire Bats

• Blood donation as a Prisoners Dilemma?
• Wilkinson, Nature 1990
• The trait should vanish
• Repeated Interactions? (or kin selection?)

30
Example bistability
31
Example bistability
32
Example coexistence
33
Example coexistence
34
Innerspecific conflicts

• Ritual fighting
• Konrad Lorenz
• arterhaltende Funktion

35
Maynard Smith and Price, 1974

36
Example neutrality
37
If n3 strategies

• Example Rock-Paper-Scissors

38
(No Transcript)
39
(No Transcript)
40
(No Transcript)
41
(No Transcript)
42
Rock-Paper-Scissors
43
Rock-Paper-Scissors

44
Generalized Rock-Paper-Scissors
45
Generalized Rock-Paper-Scissors
46
Bacterial Game Dynamics

• Escherichia coli
• Type A wild type

47
Bacterial Game Dynamics

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

48
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

49
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)

50
Mating behavior

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

51
(No Transcript)
52
Mating behavior

• males 3 morphs (inheritable)

53
(No Transcript)
54
Rock-Paper-Scissors in Nature

• males 3 morphs (inheritable)
• A monogamous, guards female

55
Rock-Paper-Scissors in Nature

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

56
Rock Paper Scissors in human interactions

• Example three players divide some goods
• Any pair forms a majority
• Shifting coalitions

57
Phase portraits of Replicator equations

58
(No Transcript)