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Solving Game Theory Models (and other sordid affairs).Steven Hamblin and Peter L. Hurd.
What just happened?(Part I)
Oskar Morgenstern (1902 - 1977)
John von Neumann (1903-1957)
Theory of Games and Economic Behavior
(1944)
John Nash (1928-) Nash Equilibrium (1950)
Not John Nash
John Nash (1928-) Nash Equilibrium (1950)
Right
10,10
-100,-100
-100,-100Right
10,10Left
Left
Right
10,10
-100,-100
-100,-100Right
10,10Left
Left
Right
10,10
-100,-100
-100,-100Right
10,10Left
Left
W. D. Hamilton (1936-2000) “Unbeatable Strategy”
(1967)
John Maynard Smith
(1920-2004)
Evolution and the Theory of Games
(1982)
Evolutionarily Stable
Strategy(ESS)
© 1973 Nature Publishing Group
Why are animal conflicts “Limited” so often?
© 1973 Nature Publishing Group
Why are animal conflicts “Limited” so often?
Why are animal conflicts “Limited” so often?
E(I, I) � E(J, I)
Nash equilibrium condition
E(I, I) � E(J, I)
Nash equilibrium condition
E(I, I) > E(J, I)
or
E(I, I) = E(J, I) andE(I, J) > E(J, J)
Stability condition
Nash equilibrium condition
E(I, I) > E(J, I)
or
E(I, I) = E(J, I) andE(I, J) > E(J, J)
Dove V/20
V1/2(V-C)Hawk
DoveHawkE(I, I) > E(J, I)
or
E(I, I) = E(J, I) andE(I, J) > E(J, J)
Dove 100
205Hawk
DoveHawk
E(I, I) > E(J, I)
or
E(I, I) = E(J, I) andE(I, J) > E(J, J)
V = 20
C = 10
Dove 100
205Hawk
DoveHawk
E(I, I) > E(J, I)
or
E(I, I) = E(J, I) andE(I, J) > E(J, J)
V = 20
C = 10
Dove 100
205Hawk
DoveHawk
E(I, I) > E(J, I)
or
E(I, I) = E(J, I) andE(I, J) > E(J, J)
V = 20
C = 10
Dove 100
205Hawk
DoveHawkE(I, I) > E(J, I)
or
E(I, I) = E(J, I) andE(I, J) > E(J, J)
V = 20
C = 10
E(Hawk,Hawk) = 5
E(Dove,Hawk) = 0
Dove 100
205Hawk
DoveHawkE(I, I) > E(J, I)
or
E(I, I) = E(J, I) andE(I, J) > E(J, J)
V = 20
C = 10
E(Hawk,Hawk) = 5
E(Dove,Hawk) = 0
Dove 100
205Hawk
DoveHawk
E(I, I) > E(J, I)
or
E(I, I) = E(J, I) andE(I, J) > E(J, J)
V = 20
C = 10
Dove 100
205Hawk
DoveHawk
E(I, I) > E(J, I)
or
E(I, I) = E(J, I) andE(I, J) > E(J, J)
V = 20
C = 40
Dove 100
20-10Hawk
DoveHawk
E(I, I) > E(J, I)
or
E(I, I) = E(J, I) andE(I, J) > E(J, J)
V = 20
C = 40
Dove 100
20-10Hawk
DoveHawk
E(I, I) > E(J, I)
or
E(I, I) = E(J, I) andE(I, J) > E(J, J)
V = 20
C = 40
Mixed ESS:
50% Hawk / 50% Dove
Dove 100
20-10Hawk
DoveHawk
E(I, I) > E(J, I)
or
E(I, I) = E(J, I) andE(I, J) > E(J, J)
V = 20
C = 40
Mixed ESS:
50% Hawk / 50% Dove
Questions:
-5,-2
6,0
2,3
12,11
Strategy D
6,61,7 2,14,-2Strategy E
4,410,12Strategy D -5,-10
1,1Strategy C 0,6-4,1
5,2 4,4Strategy B 3,3
Strategy A 2,610,-6 -6,2
Strategy CStrategy BStrategy A
Questions: 1. Complexity?
-5,-2
6,0
2,3
12,11
Strategy D
6,61,7 2,14,-2Strategy E
4,410,12Strategy D -5,-10
1,1Strategy C 0,6-4,1
5,2 4,4Strategy B 3,3
Strategy A 2,610,-6 -6,2
Strategy CStrategy BStrategy A
Questions: 1. Complexity?
Questions: 1. Complexity?
2. Population not at equilibrium?
Questions: 1. Complexity?
2. Population not at equilibrium?
That was then.This is now.
(Part II)
1
2 2
(V-C) / 2
(V-C) / 2
V
0
0
V
V/2
V/2
Hawk
Hawk Hawk
Dove
DoveDove
Player 1 payoffs
Player 2 payoffs
1
22
1 1 1 1
2 2 2 2 2 2 2 2
1
22
1 1 1 1
2 2 2 22 2 2 2
Supported path
Unreached branches
E(I, I) > E(J, I)
or
E(I, I) = E(J, I) andE(I, J) > E(J, J)
E(I, I) > E(J, I)
or
E(I, I) = E(J, I) andE(I, J) = E(J, J)(for some I �= J)
1 1 1 1
2 2
2 2
2 2
2 2
Strong
Strong Weak
"S" "W"
Signal
Strong
Signal
Weak
Weak
Strong Weak
"S""W""S" "W"
1 1
2 2 2
Signal
Strong
Signal
Weak
Full Attack Pause-Attack Flee
Full AttackPause-Attack
Flee
= ESS 1
(Enquist, 1985)
Genetic Algorithms
• Algorithms that simulate evolution to solve optimization problems.
Strategy when strong0
20
40
60
80
Strategy when weak
Tracked Generations
020
40
60
80
Graph shows strategy evolution over time.
Strategy when strong
02
040
60
80
100
Strategy when weak
Tracked Generations
020
40
60
80
100
Strategy when strong
02
040
60
80
100
Strategy when weak
Tracked Generations
020
40
60
80
100
Pink / Red: Previously unknown ES Set solution
Strategy when strong
02
040
60
80
100
Strategy when weak
Tracked Generations
020
40
60
80
100
Pink / Red: Previously unknown ES Set solutionESS disappears very rapidly.
So far...
1 1 1 1
2 2
2 2
2 2
2 2
Strong
Strong Weak
"S" "W"
Signal
Strong
Signal
Weak
Weak
Strong Weak
"S""W""S" "W"
1 1
2 2 2
Signal
Strong
Signal
Weak
Full Attack Pause-Attack Flee
Full AttackPause-Attack
Flee
= ESS 1
• e85 is too complex - the ESS formalism has broken down.
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.
So far...
Sir Philip Sydney
Maynard Smith (1991)Johnstone & Grafen (1993)
B
D
Thirsty
Give
Not Thirsty
Don't
B B
D
Give Don'tD
Give Don't
D
Give Don't
Signal No Signal Signal No Signal
Don't 1,SB1,0
SD,1SD,1Give
Not Thirsty
Thirsty
0 � SD, SB � 1
B
D
Thirsty
Give
Not Thirsty
Don't
B B
D
Give Don'tD
Give Don't
D
Give Don't
Signal No Signal Signal No Signal
Don't 1,SB1,0
SD,1SD,1Give
Not Thirsty
Thirsty
0 � SD, SB � 1
Donor and beneficiary are related, and signalling is costly (reduces payoff).
Johnstone and Grafen (1993)
2 2 2 2
1
1
1
1
1
1
1
1
Closely related
Thirsty Not Thirsty
SignalNo Signal
SignalNo Signal
Distantly related
Thirsty Not Thirsty
SignalNo Signal
SignalNo Signal
Give Don't
= ESS 1
Give Don't Give Don'tGive Don't
Give Don'tGive Don't
Give Don'tGive Don't
Johnstone and Grafen (1993)
Beneficiary
2 2 2 2
1
1
1
1
1
1
1
1
Closely related
Thirsty Not Thirsty
SignalNo Signal
SignalNo Signal
Distantly related
Thirsty Not Thirsty
SignalNo Signal
SignalNo Signal
Give Don't
= ESS 1
Give Don't Give Don'tGive Don't
Give Don'tGive Don't
Give Don'tGive Don't
Johnstone and Grafen (1993)
Donor
2 2 2 2
1
1
1
1
1
1
1
1
Closely related
Thirsty Not Thirsty
SignalNo Signal
SignalNo Signal
Distantly related
Thirsty Not Thirsty
SignalNo Signal
SignalNo Signal
Give Don't
= ESS 1
Give Don't Give Don'tGive Don't
Give Don'tGive Don't
Give Don'tGive Don't
Johnstone and Grafen (1993)
2 2 2 2
1
1
1
1
1
1
1
1
Closely related
Thirsty Not Thirsty
SignalNo Signal
SignalNo Signal
Distantly related
Thirsty Not Thirsty
SignalNo Signal
SignalNo Signal
Give Don't
= ESS 1
Give Don't Give Don'tGive Don't
Give Don'tGive Don't
Give Don'tGive Don't
ESS: Donors give if a signal is received.Closely related beneficiaries signal if thirsty.Distantly related beneficiaries always signal.
0 100 200 300 400 500
0.0
0.2
0.4
0.6
0.8
1.0
Donor strategies over time
Generation
Prop
ortio
n of
tota
l stra
tegi
es
Always giveGive when signalGive when no signalNever give
0 100 200 300 400 500
0.0
0.2
0.4
0.6
0.8
1.0
Class 1 Beneficiary strategies
Generation
Prop
ortio
n of
tota
l stra
tegi
es
Always signalSignal when thirstySignal when not thirstyNever signal
0 100 200 300 400 500
0.0
0.2
0.4
0.6
0.8
1.0
Class 2 Beneficiary strategies
Generation
Prop
ortio
n of
tota
l stra
tegi
es
Always signalSignal when thirstySignal when not thirstyNever signal
Parameters
• Solutions to the game are fragile; changing the parameters of the model generates multiple different solutions.
So far...
2 2 2 2
1 1 1 1 1 1 1 1
Class 1
Thirsty Not Thirsty
Signal No Signal Signal No Signal
Class 2
Thirsty Not Thirsty
Signal No Signal Signal No Signal
Give Don't
= ESS 1
Give Don't Give Don't Give Don't Give Don't Give Don't Give Don'tGive Don't
• Sir Philip Sydney is simpler than e85 - but still breaks the ESS formalism.
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.
So far...
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.
Thanks to Pete and the Hurd Lab!
Questions?
Genetic algorithm outcomes
MutationRate
Seed
0.001 0.002 0.003 0.004 0.005 0.006 0.007
05
1015
2025
3035
4045
5055
6065
7075
8085
9095
100
E ES O E ES O E ES O E ES O E ES O E ES O E ES O