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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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Game Theory Assignment For all of these games, P1 chooses between the columns, and P2 chooses between the rows.

Game Theory Assignment For all of these games, P1 chooses between the columns, and P2 chooses between the rows.

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