Introduction to game dynamics

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Introduction to game dynamics

• Pierre Auger
• IRD UR Geodes, Centre dîle de France et
• Institut Systèmes Complexes, ENS Lyon

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Summary

• Hawk-dove game
• Generalized replicator equations
• Rock-cissor-paper game
• Hawk-dove-retaliator and hawk-dove-bully
• 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

• Payoff matrix

• Gain

• Cost

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Playing against a population

• Hawk reward

• Dove reward

• Average reward

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Replicator equations
With
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Replicator equations
then
Because
Leading to
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Hawk-dove phase portraits
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Replicator equations

• GltC, dimorphic equilibrium

• GgtC, pure hawk equilibrium

J. Hofbauer K. Sigmund, 1988
Butterflies
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Replicator equations n tactics (ngt2)

• Payoff matrix

• aij reward when playing i against j

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Replicator equations
With

• Reward player i

• Average reward

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Equilibrium
With

• Corner

• Unique interior equilibrium (linear)

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Rock-Scissor-Paper game

• Payoff matrix

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Replicator equations
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Four equilibrium points

• Unique interior equilibrium

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Replicator equations
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Local stability analysis
saddle

• Unique interior equilibrium

center
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• Linear 2D systems (hyperbolic)

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R-C-P phase portrait

• First integral

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Hawk-Dove-Retaliator game

• Payoff matrix

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H-D-R phase portrait
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Hawk-Dove-Bully game

• Payoff matrix

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H-D-B phase portrait
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Bimatrix games (two populations)

• Pop 1 against pop 2

• Pop 2 against pop 1

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Bimatrix games (2 tactics)

• Reward player i

• Average reward

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Adding any column of constant terms

• Pop 1 against pop 2

• Pop 2 against pop 1

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Replicator equations
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Five equilibrium points

• Unique interior equilibrium (possibility)

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Jacobian matrix at (x,y)
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Local stability analysis

• Corners (Stable or unstable nodes, saddle)

• Unique interior equilibrium (trJ0 center,
  saddle)

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• Linear 2D systems (hyperbolic)

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Battle of the sexes

• Females Fast (Fa) or coy (Co)

• Males Faithful (F) or Unfaithful (UF)

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Battle of the sexes

• Males against females

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Battle of the sexes

• 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

• Unique interior equilibrium

CltGltTC/2
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Local stability analysis (center)

• Existence of a first integral H(x,y)

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Phase portrait (existence of periodic solutions)