Introduction to game dynamics

Pierre Auger

IRD UR Geodes, Centre d’île de France etInstitut Systèmes Complexes, ENS Lyon

Summary

Hawk-dove game Generalized replicator equations Rock-cissor-paper game Hawk-dove-retaliator and hawk-dove-

bully Bi-matrix games

Modelling aggressiveness

Fighting for resources

Dominique Allainé, Lyon 1

Hawk-Dove game

Payoff matrix

20

2G

GCG

A

G

C

Gain

Cost

H D

H

D

Playing against a population

Hawk reward

x

xAH 1

0,1

x

xAD 1

1,0 Dove reward

x

xAxx

11, Average reward

Replicator equations

Hxdtdx

Dydtdy

With 1 yx

Replicator equations

DHxxdtdx 1 DH xx 1Because

Leading to CxGxxdtdx 1

21

then

Hawk-dove phase portraits

Replicator equations

G<C, dimorphic equilibrium CG

x *

1* x

J. Hofbauer & K. Sigmund, 1988

G>C, pure hawk equilibrium

CxGxxdtdx 1

21

Butterflies

Replicator equations : n tactics (n>2)

Payoff matrix ijaA

aij reward when playing i against j

Replicator equations

iii xdtdx

With 1i

ix

Ni xxxxu ,...,,...,, 21

TuAu Average reward

0,...,0,1,0,...,0,0iu

T

iAuu Reward player i

Equilibrium

0,...,0,1,0,...,0,0* iM

iii xdtdx

With 1i

ix

***

2

*

1

* ,...,,...,, Ni xxxxM

Unique interior equilibrium (linear)

Corner

ii;

Rock-Scissor-Paper game

Payoff matrix

R

011

101

110

A

C P

R

C

P

Replicator equations

yxzdtdz

xzydtdy

zyxdtdx

Four equilibrium points

0,1,0 1,0,0 0,0,1

Unique interior equilibrium

31

,31

,31

Replicator equations

xyyydtdy

xyxxdtdx

2

2

2

2

Local stability analysis

0,1,0 1,0,0 0,0,1

Unique interior equilibrium

31

,31

,31

saddle

center

Linear 2D systems (hyperbolic)

R-C-P phase portrait

First integral xyzzyxH ),,(

Hawk-Dove-Retaliator game

Payoff matrix

H

222

220

22

GGCG

GG

CGG

CG

A

D R

H

D

R

H-D-R phase portrait

Hawk-Dove-Bully game

Payoff matrix

H

20

02

0

2

GG

G

GGCG

A

D B

H

D

B

H-D-B phase portrait

Bimatrix games (two populations)

Pop 1 against pop 2

2221

1211

aa

aaA

Pop 2 against pop 1

2221

1211

bb

bbB

Bimatrix games (2 tactics)

1ydtdy

TxxByy )1,()1,(

1xdtdx

TyyAxx )1,()1,(

Average reward

TyyA )1,()0,1(1

Reward player i

TxxB )1,()0,1(1

Adding any column of constant terms

Pop 1 against pop 2

0

0

21

12

A

Pop 2 against pop 1

0

0

21

12

B

Replicator equations

xyydtdy

yxxdtdx

211212

211212

1

1

Five equilibrium points

Unique interior equilibrium (possibility)

0,1 1,0 0,0 1,1

2112

12

2112

12 ,

Jacobian matrix at (x*,y*)

*))(*)(21()*)(1(*

)*)(1(**))(*)(21(*

2112122112

2112211212

xyyy

xxyxJ

Local stability analysis

Unique interior equilibrium (trJ=0 ; center, saddle)

0,1 1,0 0,0 1,1

2112

12

2112

12 ,

Corners (Stable or unstable nodes, saddle)

Linear 2D systems (hyperbolic)

Battle of the sexes

Females : Fast (Fa) or coy (Co)

Males : Faithful (F) or Unfaithful (UF)

Battle of the sexes

Males against females

22

0C

GTC

G

GA

F

FaCo

UF

Battle of the sexes

Females against males

2

20

CGCG

TC

GB

F

Fa

Co

UF

Adding C/2-G in second column

0

0

CG

TB

02

20

TC

G

C

A

Replicator equations

xGTCTyydtdy

yTGC

xxdtdx

1

21

Five equilibrium points

Unique interior equilibrium :

0,1 1,0 0,0 1,1

TGC

TGCT

2,

C<G<T+C/2

Local stability analysis (center)

Existence of a first integral H(x,y) :

)1ln()ln()1ln()ln(),( 21122112 xxyyyxH

Phase portrait (existence of periodic solutions)