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Game theory models: Solutions and dynamics. Steven Hamblin

MSc Thesis

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The talk for my MSc thesis defence in 2007.

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Page 1: MSc Thesis

Game theory models:Solutions and dynamics.

Steven Hamblin

Page 2: MSc Thesis

Dynamics

Game

Complexity

Technique

and results

Page 3: MSc Thesis

Using game theory...

Page 4: MSc Thesis

Frequency dependence:

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Frequency dependence: Optimization...

Game theory....

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Frequency dependence: Optimization...

Game theory....

... fitness.

Page 7: MSc Thesis

Evolutionarily Stable Strategy (ESS)

n John Maynard Smith (1973).

n “No regrets” - a strategy which, if the entire population adopts it, no individual can beat by switching to another.

Page 8: MSc Thesis

Animal communication models

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1

22

1 1 1 1

2 2 2 2 2 2 2 2

Page 13: MSc Thesis

I. Game complexity.

Dynamics

Game

Complexity

Technique

and results

Page 14: MSc Thesis

Problems:

n Size of the strategy space.

n Pervasiveness.

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Strategy space...

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1

22

1 1 1 1

2 2 2 2 2 2 2 2

Page 17: MSc Thesis

1

22

1 1 1 1

2 2 2 22 2 2 2

Supported path

Unreached branches

Page 18: MSc Thesis

II. Game dynamics.

Dynamics

Game

Complexity

Technique

and results

Page 19: MSc Thesis

Population equilibria..

n Will a population go to an ESS from a non-equilibrium starting point?

n What happens if there are multiple solutions to the model?

Page 20: MSc Thesis

Replicator dynamics...

Page 21: MSc Thesis

Models too

simple

Model dynamics

are important.

Page 22: MSc Thesis

Models too simple

Model dynamics

are important.

Make realistic models

Solve them analytically.

Page 23: MSc Thesis

Models too simple

Model dynamics

are important.

Make realistic models

Solve them analytically.

ESS conditions

Strategy space

RD requires simple models.

Page 24: MSc Thesis

Models too simple

Model dynamics

are important.

Make realistic models

Solve them analytically.

ESS conditions

Strategy space

RD requires simple models.

Page 25: MSc Thesis

Models too simple

Model dynamics

are important.

Make realistic models

Solve them analytically.

ESS conditions

Strategy space

RD requires simple models.

Genetic algorithms!

Page 26: MSc Thesis

Genetic Algorithmsn Algorithms that

simulate evolution to solve optimization problems.

n GAs are a search method, not a faithful replication of natural selection.

Page 27: MSc Thesis

III. Results.

Dynamics

Game

Complexity

Technique

and results

Page 28: MSc Thesis

Two models...

n E85 - a model of conventional signalling.

n SPS - a simpler model of handicapped signalling.

Page 29: MSc Thesis

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)

Page 30: MSc Thesis

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 Attack Pause-Attack Flee

(4,-6) (5,-6) (10,0)

A,P,FA,P,FA,P,FA, P, F"S"/

"W"

"S"/

"W"

Opp

Signals

"W"

Opp

Signals

"W"

Opp

Signals

"S"

Opp

Signals

"S"

If Weak,

signal

If Strong,

signal:

WeakStrongWeakStrongEgo State:

Strategy chromosome:

ESS is "S"/"W"/A/F/P/A

Page 31: MSc Thesis

Graph shows strategy evolution over time.

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Previously unknown ES Set solution

Page 37: MSc Thesis

Previously unknown ES Set solution

ESS disappears very rapidly.

Page 38: MSc Thesis

IV. Conclusions.

Page 39: MSc Thesis

Models too simple

Model dynamics

are important.

Make realistic models

Solve them analytically.

ESS conditions

Strategy space

RD requires simple models.

Genetic algorithms!

Results

Page 40: MSc Thesis

Fin

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

Page 41: MSc Thesis

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

Page 42: MSc Thesis

{1,0}{1,0}{1,0}{1,0}{1,0}{1,0}

Donor strategy:

Another borderline example is the Johnstone & Grafen(1993) version of the Sir Philip Sidney game (Fig. 8). Thisgame models two different types of signaller (differing inrelatedness to the receiver), which are then assigned totwo levels of thirst. This arrangement decomposes toa four-state (type ! thirst level) action–response game.The receiver receives no information other than the sig-nal, so each of the types then collapses into an average ex-pected type according to the signal chosen (MaynardSmith 1994). This game then simplifies into a single-stateaction–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

The averaging out of opponent types through signalpooling will occur whenever players cannot modifybehavioural choices with later moves. The single-state

and dual-state action–response games each allow only onemove per player. The stateless signalling and conventionalsignalling games comprise a signalling phase and a re-sponse phase. The response phase allows players tomodify their initial choice, the signal choice, given sub-sequently gained information. Players may choose behav-iours that will either maximize benefit in best cases, orminimize costs in worse cases. This fine-tuning providesa mechanism for types that were pooled earlier in thegame to be subsequently separated. This level of complex-ity allows for conventional signalling between playerswith conflicting interests (Hurd & Enquist 1998).

Stateless Mutual Signalling Gamesand Handicaps

Stateless signalling games (Fig. 7) begin with an initialmixed ESS signalling move. With no underlying state,the signalling move must be a choice made between

1

1 1

2 2

m0 m1

m0 m1 m0 m1

H D H D H D

H D H D H D H D H D H D H D

2 2

1 1

2 2

H D

H D

w1 P T N Co P T N Co P–H T–H N–H Co–H P–H T–H N–H Co–H

w2 P N T Co P–H N–H T–H Co– H P N T Co P–H N–H T–H Co–H

Figure 7. Kim’s (1995) stateless signalling game. Payoffs for the Hawk–Dove (H–D) subgames have been replaced by the generic variables T, Co,N, P (temptation, coordination, neutral and punishment); the results are identical to the traditional Hawk–Dove payoffs (with V and C ). A signal-ling ESS exists when T O Co R N O P. Generic payoffs for Hawk–Dove players are equivalent to: T Z V, Co Z V/C, N Z 0, P Z (V ! C )/2(where the generic payoffs are on the left, and classical Hawk–Dove payoffs are on the right). H is the handicap cost of using signalm1. The signalsin this game are effectively simultaneous. The second player chooses the signal to be used before learning of the first player’s choice of signal.

N

N N

S S

RR

type 1 type 2(q) (1 − q)

T T T T

(P) (1 − P) (P) (1 − P)

s sns ns s sns ns

g gng ng g gng ng g gng ng g gng 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, eitherclosely 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!), andsignal (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 noway of ever learning whether the signaller is of type 1 or type 2, so the distinction between them can be ignored and all signallers treated asone weighted average type. This reduces the game to a basic action–response game.

HURD & ENQUIST: TAXONOMY OF BIOLOGICAL COMMUNICATION 1165

Beneficiary strategy:

Page 43: MSc Thesis

Class of Beneficiary:

1 2

Thirsty

Not

1 2 1 2

1 2

Thirsty

Not

1 2 1 2

SignallingStrategies

DonationStrategies

Donor: Give when signal.

B1: Signal when thirsty.

B2: Always signal.

Donor: Give when no signal.

B1: Signal when not thirsty.

B2: Never signal.

Donor: Always give.

B1: Never signal.

B2: Never signal.

Beneficiary(no signal)

Beneficiary(signal)

Donor Donor gives resource.

Page 44: MSc Thesis

Model features...

n Sequential / multiple signals.

n Strength states.

n Asymmetries:

n Role asymmetries.

n Uncorrelated asymmetries.

Page 45: MSc Thesis

Sir Philip Sydney

n Maynard Smith (1991)

n Johnstone & Grafen (1993)

Page 46: MSc Thesis

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

Page 47: MSc Thesis

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).

Page 48: MSc Thesis

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

Page 49: MSc Thesis

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

Page 50: MSc Thesis

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

Page 51: MSc Thesis

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.

Page 52: MSc Thesis

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

Page 53: MSc Thesis

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

Page 54: MSc Thesis

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

Page 55: MSc Thesis

V = resource value.C = cost of fighting.

Page 56: MSc Thesis

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

V = resource value.C = cost of fighting.

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