Introduction to game dynamics Pierre Auger IRD UR Geodes, Centre d’île de France et Institut Systèmes Complexes, ENS Lyon.

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Presentation transcript:

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

Modelling aggressiveness

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 G<C, dimorphic equilibrium J. Hofbauer & K. Sigmund, 1988 z G>C, 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

Replicator equations

Four equilibrium points z Unique interior equilibrium

Replicator equations

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

Replicator equations

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

Replicator equations

Five equilibrium points z Unique interior equilibrium : C<G<T+C/2

Local stability analysis (center) z Existence of a first integral H(x,y) :

Phase portrait (existence of periodic solutions)