A talk that I gave to the Cog Sem seminar series at the University of Alberta Psychology department in 2007 on my master's research.

1. Solving Game Theory Models
(and other sordid affairs).
Steven Hamblin and Peter L. Hurd.

2. What just happened?
(Part I)

3. Oskar Morgenstern
(1902 - 1977)

4. John von Neumann
(1903-1957)

5. Theory of Games and
Economic Behavior
(1944)

12. John Nash (1928-)
Nash Equilibrium (1950)

13. Not John Nash

14. John Nash (1928-)
Nash Equilibrium (1950)

15. Left
Right
Left
10,10
-100,-100
Right
-100,-100
10,10

16. Left
Right
Left
10,10
-100,-100
Right
-100,-100
10,10

17. Left
Right
Left
10,10
-100,-100
Right
-100,-100
10,10

18. W. D. Hamilton (1936-2000)
“Unbeatable Strategy”
(1967)

19. John
Maynard Smith
(1920-2004)

20. Evolution and the
Theory of Games
(1982)
Evolutionarily
Stable
Strategy(ESS)

21. Why are animal conflicts
“Limited” so often?

22. Why are animal conflicts
“Limited” so often?

23. Why are animal conflicts
“Limited” so often?

24. E(I, I)
E(J, I)

25. Nash equilibrium condition
E(I, I)
E(J, I)

26. Nash equilibrium condition
E(I, I) > E(J, I)
or
E(I, I) = E(J, I) and
E(I, J) > E(J, J)

27. Nash equilibrium condition
E(I, I) > E(J, I)
or
E(I, I) = E(J, I) and
E(I, J) > E(J, J)
Stability condition

28. Hawk
Dove
Hawk
1/2(V-C)
V
Dove
0
V/2
E(I, I) > E(J, I)
or
E(I, I) = E(J, I) and
E(I, J) > E(J, J)

29. V = 20
C = 10
Hawk
Dove
Hawk
5
20
Dove
0
10
E(I, I) > E(J, I)
or
E(I, I) = E(J, I) and
E(I, J) > E(J, J)

30. V = 20
C = 10
Hawk
Dove
Hawk
5
20
Dove
0
10
E(I, I) > E(J, I)
or
E(I, I) = E(J, I) and
E(I, J) > E(J, J)

31. V = 20
C = 10
Hawk
Dove
Hawk
5
20
Dove
0
10
E(I, I) > E(J, I)
or
E(I, I) = E(J, I) and
E(I, J) > E(J, J)

32. V = 20
C = 10
Hawk
Dove
Hawk
5
20
Dove
0
10
E(I, I) > E(J, I)
E(Hawk,Hawk) = 5
E(Dove,Hawk) = 0
or
E(I, I) = E(J, I) and
E(I, J) > E(J, J)

33. V = 20
C = 10
Hawk
Dove
Hawk
5
20
Dove
0
10
E(I, I) > E(J, I)
E(Hawk,Hawk) = 5
E(Dove,Hawk) = 0
or
E(I, I) = E(J, I) and
E(I, J) > E(J, J)

34. V = 20
C = 10
Hawk
Dove
Hawk
5
20
Dove
0
10
E(I, I) > E(J, I)
or
E(I, I) = E(J, I) and
E(I, J) > E(J, J)

35. V = 20
C = 40
Hawk
Dove
Hawk
5
20
Dove
0
10
E(I, I) > E(J, I)
or
E(I, I) = E(J, I) and
E(I, J) > E(J, J)

36. V = 20
C = 40
Hawk
Dove
Hawk
-10
20
Dove
0
10
E(I, I) > E(J, I)
or
E(I, I) = E(J, I) and
E(I, J) > E(J, J)

37. V = 20
C = 40
Hawk
Dove
Hawk
-10
20
Dove
0
10
E(I, I) > E(J, I)
Mixed ESS:
50% Hawk / 50% Dove
or
E(I, I) = E(J, I) and
E(I, J) > E(J, J)

38. V = 20
C = 40
Hawk
E(I, I) > E(J, I)
Dove
Hawk
-10
20
Dove
0
10
Mixed ESS:
50% Hawk / 50% Dove
or
E(I, I) = E(J, I) and
E(I, J) > E(J, J)

39. Questions:

40. Strategy A
Strategy B
Strategy C
Strategy A
10,-6
-6,2
2,6
Strategy B
5,2
4,4
3,3
Strategy C
-4,1
1,1
0,6
Strategy D
10,12
-5,-10
4,4
1,7
2,1
Questions:
Strategy E
1. Complexity?
4,-2
S

41. Strategy A
Strategy B
Strategy C
Strategy A
10,-6
-6,2
2,6
Strategy B
5,2
4,4
3,3
Strategy C
-4,1
1,1
0,6
Strategy D
10,12
-5,-10
4,4
1,7
2,1
Questions:
Strategy E
1. Complexity?
4,-2
S

42. Questions:
1. Complexity?
2. Population not at equilibrium?

43. Questions:
1. Complexity?
2. Population not at equilibrium?

44. That was then.
This is now.
(Part II)

46. 1
Hawk
Dove
2
2
Hawk
Dove
Hawk
Dove
(V-C) / 2
V
0
V/2
Player 1 payoffs
(V-C) / 2
0
V
V/2
Player 2 payoffs

47. 1
2
2
1
2
1
2
2
1
2
2
1
2
2
2

48. 1
2
2
1
2
1
2
2
1
2
2
Supported path
Unreached branches
1
2
2
2

49. E(I, I) > E(J, I)
or
E(I, I) = E(J, I) and
E(I, J) > E(J, J)

50. E(I, I) > E(J, I)
or
E(I, I) = E(J, I) and
E(I, J) = E(J, J)
(for some I = J)

51. 1 = ESS
Strong
Strong
Strong
Weak
1
"S"
1
"W"
"S" "W"
2
2
2
1
1
"S""W"
2
Signal
Strong
Weak
1
Signal
Weak
Signal
Strong
2
2
Weak
2
2
Signal
Weak
1
Full Attack
2
Pause-Attack
Flee
2
Full Attack
Pause-Attack
2
Flee
(Enquist, 1985)

52. Genetic Algorithms
• Algorithms that
simulate evolution
to solve
optimization
problems.

53. 0
20
40
60
80
Strategy when strong
Graph shows strategy evolution over time.
0
20
40
60
80
Strategy when weak
Tracked Generations

55. 0
20
40
60
80
100
Strategy when strong
0
20
40
60
80
100
Strategy when weak
Tracked Generations

56. 0
20
40
60
80
100
Strategy when strong
Pink / Red: Previously
unknown ES Set solution
0
20
40
60
80
100
Strategy when weak
Tracked Generations

57. 0
20
40
60
80
100
Strategy when strong
ESS / Red: Previously
Pinkdisappears very
rapidly.
unknown ES Set solution
0
20
40
60
80
100
Strategy when weak
Tracked Generations

58. So far...

59. So far...
1 = ESS
• e85 is too complex the ESS formalism
has broken down.
Strong
Strong
Weak
1
"S"
1
Signal
Weak
Signal
Strong
"W"
2
2
2
2
Signal
Strong
1
Signal
Weak
1
Full Attack
2
Pause-Attack
Flee
2
Full Attack
Pause-Attack
2
Flee

60. So far...
• e85 is too complex the ESS formalism
has broken down.
• Populations not
already at the ESS
evolve more easily
to the ES Set.

61. Sir Philip Sydney
Maynard Smith (1991)
Johnstone & Grafen (1993)

62. Thirsty
Give
B
1,0
1,SB
SD , S B
1
B
Signal
No Signal
Signal
No Signal
D
D
D
D
Give
SD,1
0
Not
Thirsty
Thirsty
SD,1
Don't
B
Not
Thirsty
Give
Don't
Don't
Give
Give
Don't
Don't

63. Donor and beneficiary are related, and
signalling is costly (reduces payoff).
Give
B
SD,1
1,0
1,SB
SD , S B
1
B
Signal
No Signal
Signal
No Signal
D
D
D
D
Give
SD,1
0
Not
Thirsty
Thirsty
Not
Thirsty
Don't
B
Thirsty
Give
Don't
Don't
Give
Give
Don't
Don't

64. 1 = ESS
Closely related
Thirsty
Thirsty
Not Thirsty
2
Signal
2
No Signal
Signal
No Signal
Don't
Signal
No Signal
Signal
Don't
Give
Don't
1
1
Give
Don't
No Signal
1
1
Give
2
1
1
Not Thirsty
2
1
Give
Distantly related
Give
Don't
1
Give
Don't
Give
Johnstone and Grafen (1993)
Don't
Give
Don't

65. Beneficiary
1 = ESS
Closely related
Thirsty
Thirsty
Not Thirsty
2
Signal
2
No Signal
Signal
No Signal
Don't
Signal
No Signal
Signal
Don't
Give
Don't
1
1
Give
Don't
No Signal
1
1
Give
2
1
1
Not Thirsty
2
1
Give
Distantly related
Give
Don't
1
Give
Don't
Give
Johnstone and Grafen (1993)
Don't
Give
Don't

66. Donor
1 = ESS
Closely related
Thirsty
Thirsty
Not Thirsty
2
Signal
2
No Signal
Signal
No Signal
Don't
Signal
No Signal
Signal
Don't
Give
Don't
1
1
Give
Don't
No Signal
1
1
Give
2
1
1
Not Thirsty
2
1
Give
Distantly related
Give
Don't
1
Give
Don't
Give
Johnstone and Grafen (1993)
Don't
Give
Don't

67. ESS:
Donors give if a signal is received.
Closely related beneficiaries signal if thirsty.
Distantly related beneficiaries always signal.
1 = ESS
Closely related
Thirsty
Thirsty
Not Thirsty
2
Signal
2
No Signal
Signal
No Signal
Don't
Signal
No Signal
Signal
Don't
Give
Don't
1
1
Give
Don't
No Signal
1
1
Give
2
1
1
Not Thirsty
2
1
Give
Distantly related
Give
Don't
1
Give
Don't
Give
Johnstone and Grafen (1993)
Don't
Give
Don't

68. 1.0
0.8
0.6
0.2
0.4
Always give
Give when signal
Give when no signal
Never give
0.0
Proportion of total strategies
Donor strategies over time
0
100
200
300
Generation
400
500

69. 0.6
0.8
1.0
Class 1 Beneficiary strategies
0.0
0.2
0.4
Always signal
Signal when thirsty
Signal when not thirsty
Never signal
0
100
200
300
Generation
400
500

70. 0.6
0.8
1.0
Class 2 Beneficiary strategies
0.0
0.2
0.4
Always signal
Signal when thirsty
Signal when not thirsty
Never signal
0
100
200
300
Generation
400
500

71. Parameters
• Solutions to the
game are fragile;
changing the
parameters of the
model generates
multiple different
solutions.

72. So far...

73. So far...
• Sir Philip Sydney is
simpler than e85 but still breaks the
ESS formalism.
1 = ESS
Class 1
Thirsty
Give
Don't
2
Signal
No Signal
1
1
Give
Thirsty
Not Thirsty
2
Signal
Class 2
Don't
2
No Signal
1
Give
Don't
Signal
1
Give
Don't
No Signal
1
Give
Don't
1
Give
Don't

74. So far...
• Sir Philip Sydney is
simpler than e85 but still breaks the
ESS formalism.
• Again, populations
not already at the
ESS evolve more
easily to the ES Set.

75. When all is said and done...
• ESS and related theory was a paradigm shift in
theoretical biology.
• ESS is useful intuitively, but limited practically.
• Most games with temporal sequence / underlying
state / etc., won’t have an ESS.
• Even more useful solution tools (e.g. ES Sets) are too
complicated to calculate for larger, more realistic
games.
• Genetic algorithms are a sensible choice to solve
complex game theory models.

76. Thanks to Pete
and the Hurd Lab!

77. Questions?

78. Genetic algorithm outcomes
0.001
ES
O E
0.002
ES
O E
0.003
ES
O E
0.004
ES
45
50
55
60
65
70
75
80
85
90
100 95
Seed
40
35
30
25
20
15
10
5
0
E
MutationRate
O E
0.005
ES
O E
0.006
ES
O
E ES
0.007
O