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Introduction to game dynamics

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

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

  2. Summary • Hawk-dove game • Generalized replicator equations • Rock-cissor-paper game • Hawk-dove-retaliator and hawk-dove-bully • Bi-matrix games

  3. Modelling aggressiveness

  4. Fighting for resources Dominique Allainé, Lyon 1

  5. Hawk-Dove game • Payoff matrix • Gain • Cost

  6. Playing against a population • Hawk reward • Dove reward • Average reward

  7. Replicator equations With

  8. Replicator equations then Because Leading to

  9. Hawk-dove phase portraits

  10. Replicator equations • G<C, dimorphic equilibrium • G>C, pure hawk equilibrium J. Hofbauer & K. Sigmund, 1988 Butterflies

  11. Replicator equations : n tactics (n>2) • Payoff matrix • aij reward when playing i against j

  12. Replicator equations With • Reward player i • Average reward

  13. Equilibrium With • Corner • Unique interior equilibrium (linear)

  14. Rock-Scissor-Paper game • Payoff matrix

  15. Replicator equations

  16. Four equilibrium points • Unique interior equilibrium

  17. Replicator equations

  18. Local stability analysis saddle • Unique interior equilibrium center

  19. Linear 2D systems (hyperbolic)

  20. R-C-P phase portrait • First integral

  21. Hawk-Dove-Retaliator game • Payoff matrix

  22. H-D-R phase portrait

  23. Hawk-Dove-Bully game • Payoff matrix

  24. H-D-B phase portrait

  25. Bimatrix games (two populations) • Pop 1 against pop 2 • Pop 2 against pop 1

  26. Bimatrix games (2 tactics) • Reward player i • Average reward

  27. Adding any column of constant terms • Pop 1 against pop 2 • Pop 2 against pop 1

  28. Replicator equations

  29. Five equilibrium points • Unique interior equilibrium (possibility)

  30. Jacobian matrix at (x*,y*)

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

  32. Linear 2D systems (hyperbolic)

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

  34. Battle of the sexes • Males against females

  35. Battle of the sexes • Females against males

  36. Adding C/2-G in second column

  37. Replicator equations

  38. Five equilibrium points • Unique interior equilibrium : C<G<T+C/2

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

  40. Phase portrait (existence of periodic solutions)

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