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Introduction to game dynamics Pierre Auger IRD UR Geodes, Centre d’île de France et Institut Systèmes Complexes, ENS Lyon
Summary z Hawk-dove game z Generalized replicator equations z Rock-cissor-paper game z Hawk-dove-retaliator and hawk-dove-bully z Bi-matrix games
Fighting for resources Dominique Allainé, Lyon 1
Hawk-Dove game z Payoff matrix z Gain z Cost
Playing against a population z Hawk reward z Dove reward z Average reward
Replicator equations With
Replicator equations Because Leading to then
Hawk-dove phase portraits
Replicator equations z GC, pure hawk equilibrium Butterflies
Replicator equations : n tactics (n>2) z Payoff matrix z a ij reward when playing i against j
Replicator equations With z Average reward z Reward player i
Equilibrium With z Unique interior equilibrium (linear) z Corner
Rock-Scissor-Paper game z Payoff matrix
Four equilibrium points z Unique interior equilibrium
Local stability analysis z Unique interior equilibrium saddle center
zLinear 2D systems (hyperbolic)
R-C-P phase portrait z First integral
Hawk-Dove-Retaliator game z Payoff matrix
H-D-R phase portrait
Hawk-Dove-Bully game z Payoff matrix
H-D-B phase portrait
Bimatrix games (two populations) z Pop 1 against pop 2 z Pop 2 against pop 1
Bimatrix games (2 tactics) z Average reward z Reward player i
Adding any column of constant terms z Pop 1 against pop 2 z Pop 2 against pop 1
Five equilibrium points z Unique interior equilibrium (possibility)
Jacobian matrix at (x*,y*)
Local stability analysis z Unique interior equilibrium (trJ=0 ; center, saddle) z Corners (Stable or unstable nodes, saddle)
zLinear 2D systems (hyperbolic)
Battle of the sexes z Females : Fast (Fa) or coy (Co) z Males : Faithful (F) or Unfaithful (UF)
Battle of the sexes z Males against females
Battle of the sexes z Females against males
Adding C/2-G in second column
Five equilibrium points z Unique interior equilibrium : C
Local stability analysis (center) z Existence of a first integral H(x,y) :
Phase portrait (existence of periodic solutions)
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