Post on 14-Jan-2016
transcript
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)