The talk for my MSc thesis defence in 2007.

1. Game theory models:
Solutions and dynamics.
Steven Hamblin

2. Game
Complexity
Dynamics
Technique
and results

3. Using game theory...

4. Frequency dependence:

5. Frequency dependence:
Optimization...
Game theory....

6. Frequency dependence:
Optimization...
Game theory....
... fitness.

7. Evolutionarily Stable Strategy (ESS)
n
n
John Maynard Smith (1973).
“No regrets” - a strategy which, if the entire
population adopts it, no individual can beat by
switching to another.

8. Animal communication models

12. 1
2
2
1
2
1
2
2
1
2
2
1
2
2
2

13. I. Game complexity.
Game
Complexity
Dynamics
Technique
and results

14. Problems:
n
Size of the strategy space.
n
Pervasiveness.

15. Strategy space...

16. 1
2
2
1
2
1
2
2
1
2
2
1
2
2
2

17. 1
2
2
1
2
1
2
2
1
2
2
Supported path
Unreached branches
1
2
2
2

18. II. Game dynamics.
Game
Complexity
Dynamics
Technique
and results

19. Population equilibria..
n
n
Will a population go to an ESS from a nonequilibrium starting point?
What happens if there are multiple solutions to the
model?

20. Replicator
dynamics...

21. Models too
simple
Model dynamics
are important.

22. Models too
simple
Make realistic
models
Model dynamics
are important.
Solve them
analytically.

23. ESS conditions
Models too
simple
Make realistic
models
Strategy space
Model dynamics
are important.
Solve them
analytically.
RD requires
simple models.

24. ESS conditions
Models too
simple
Make realistic
models
Strategy space
Model dynamics
are important.
Solve them
analytically.
RD requires
simple models.

25. ESS conditions
Models too
simple
Make realistic
models
Strategy space
Model dynamics
are important.
Solve them
analytically.
RD requires
simple models.
Genetic
algorithms!

26. Genetic Algorithms
n
n
Algorithms that
simulate evolution to
solve optimization
problems.
GAs are a search
method, not a faithful
replication of natural
selection.

27. III. Results.
Game
Complexity
Dynamics
Technique
and results

28. Two models...
n
E85 - a model of conventional signalling.
n
SPS - a simpler model of handicapped signalling.

29. 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)

30. If Strong,
signal:
Opp
Signals
"S"
Opp
Signals
"W"
Opp
Signals
"W"
"S"
/
"W"
A, P, F
A,P,F
A,P,F
A,P,F
Ego State: Strong
ESS is "S"/"W"/A/F/P/A
Opp
Signals
"S"
"S"
/
"W"
Strategy chromosome:
If Weak,
signal
Weak
Strong
Weak
Strong
Strong
Strong
Weak
1
"S"
Weak
1
"W"
1
Signal
Weak
Signal
Strong
2
2
"S" "W"
2
2
2
1
2
Pause-Attack
Flee
2
Full Attack
(4,-6)
Signal
Weak
1
Full Attack
1
"S""W"
2
Signal
Strong
Pause-Attack
(5,-6)
Weak
2
Flee
(10,0)
2
2

31. Graph shows strategy evolution over time.

36. Previously unknown
ES Set solution

37. ESS disappears very
Previously unknown
rapidly. Set solution
ES

38. IV. Conclusions.

39. ESS conditions
Models too
simple
Make realistic
models
Strategy space
Model dynamics
are important.
Solve them
analytically.
Genetic
algorithms!
RD requires
simple models.
Results

40. Fin
Thanks to the Hurd Lab (and especially Pete)
for the support over these past two years.

41. 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

42. sponse phase. The response phase allows players to
modify their initial choice, the signal choice, given subsequently gained information. Players may choose behaviours that will either maximize benefit in best cases, or
Beneficiary strategy:
minimize costs in worse cases. This fine-tuning provides
a mechanism for types that were pooled earlier in the
game to be subsequently separated. This level of complex{1,0}
{1,0}
{1,0}
ity allows for conventional signalling between players
with conflicting interests (Hurd & Enquist 1998).
relatedness to the receiver), which are then assigned to
two levels of thirst. This arrangement decomposes to
a four-state (type ! thirst level) action–response game.
The receiver receives no information other than the sigDonor strategy:
nal, so each of the types then collapses into an average expected type according to the signal chosen (Maynard
Smith 1994). This game then simplifies into a single-state
{1,0}
{1,0}
{1,0}
action–response game. In other words, these two models,
the Sir Philip Sidney game and the action–response game,
are really the same thing.
Mutual Signalling Games and Signal Type
Stateless Mutual Signalling Games
and Handicaps
The averaging out of opponent types through signal
pooling will occur whenever players cannot modify
behavioural choices with later moves. The single-state
Stateless signalling games (Fig. 7) begin with an initial
mixed ESS signalling move. With no underlying state,
the signalling move must be a choice made between
N
type 1
N
T
(P)
S
s
type 2
(1 − q)
(q)
T
N
T
(1 − P)
ns
T
(P)
s
ns
s
(1 − P)
ns
S
s
ns
R
g
R
ng
g
ng g
ng
g
ng g
ng
g
ng g
ng
g
ng
Figure 8. The ‘Sir Philip Sidney’ game as modelled by Johnstone & Grafen (1993). A signaller (S, the beneficiary) is of one of two types, either
closely related (type 1), or less closely related (type 2), chosen in a move by nature (N). These signallers are either thirsty (T ) or not (T ), and
signal (s) or do not (ns). The receiver (R, the donor) then either gives ( g) the resource to the beneficiary, or does not (ng). The receiver has no

43. Class of Beneficiary:
1
2
1
2
1
2
1
2
1
2
1
2
Thirsty
Signalling
Strategies
Not
Thirsty
Donation
Strategies
Not
Donor: Give when signal.
B1: Signal when thirsty.
B2: Always signal.
Beneficiary
(no signal)
Donor: Give when no signal.
B1: Signal when not thirsty.
B2: Never signal.
Beneficiary
(signal)
Donor
Donor: Always give.
B1: Never signal.
B2: Never signal.
Donor gives resource.

44. Model features...
n
Sequential / multiple signals.
n
Strength states.
n
Asymmetries:
n
Role asymmetries.
n
Uncorrelated asymmetries.

45. Sir Philip Sydney
n
Maynard Smith (1991)
n
Johnstone Grafen (1993)

46. 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

47. 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

48. 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

49. 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

50. 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

51. 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

52. 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

53. 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

54. 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

55. V = resource value.
C = cost of fighting.

56. 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
V = resource value.
C = cost of fighting.