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

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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

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Modelling aggressiveness

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Fighting for resources Dominique Allainé, Lyon 1

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Hawk-Dove game z Payoff matrix z Gain z Cost

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Playing against a population z Hawk reward z Dove reward z Average reward

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Replicator equations With

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Replicator equations Because Leading to then

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Hawk-dove phase portraits

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Replicator equations z G

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Replicator equations : n tactics (n>2) z Payoff matrix z a ij reward when playing i against j

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Replicator equations With z Average reward z Reward player i

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Equilibrium With z Unique interior equilibrium (linear) z Corner

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Rock-Scissor-Paper game z Payoff matrix

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Replicator equations

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Four equilibrium points z Unique interior equilibrium

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Replicator equations

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Local stability analysis z Unique interior equilibrium saddle center

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zLinear 2D systems (hyperbolic)

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R-C-P phase portrait z First integral

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Hawk-Dove-Retaliator game z Payoff matrix

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H-D-R phase portrait

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Hawk-Dove-Bully game z Payoff matrix

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H-D-B phase portrait

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Bimatrix games (two populations) z Pop 1 against pop 2 z Pop 2 against pop 1

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Bimatrix games (2 tactics) z Average reward z Reward player i

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Adding any column of constant terms z Pop 1 against pop 2 z Pop 2 against pop 1

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Replicator equations

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Five equilibrium points z Unique interior equilibrium (possibility)

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Jacobian matrix at (x*,y*)

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Local stability analysis z Unique interior equilibrium (trJ=0 ; center, saddle) z Corners (Stable or unstable nodes, saddle)

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zLinear 2D systems (hyperbolic)

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Battle of the sexes z Females : Fast (Fa) or coy (Co) z Males : Faithful (F) or Unfaithful (UF)

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Battle of the sexes z Males against females

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Battle of the sexes z Females against males

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Adding C/2-G in second column

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Replicator equations

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Five equilibrium points z Unique interior equilibrium : C

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

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Phase portrait (existence of periodic solutions)

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