introduction to game dynamics
DESCRIPTION
Introduction to game dynamics. Pierre Auger IRD UR Geodes, Centre d’île de France et Institut 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. - PowerPoint PPT PresentationTRANSCRIPT
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
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
G<C, dimorphic equilibrium CG
x *
1* x
J. Hofbauer & K. Sigmund, 1988
G>C, pure hawk equilibrium
CxGxxdtdx 1
21
Butterflies
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;
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
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)
Local stability analysis (center)
Existence of a first integral H(x,y) :
)1ln()ln()1ln()ln(),( 21122112 xxyyyxH