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Analyzing Game Dynamics: Hawk-Dove and Beyond in Evolutionary Strategies

This study explores key concepts in game dynamics, focusing on the Hawk-Dove game, generalized replicator equations, and the rock-paper-scissors game. It examines interactions in bi-matrix games, including the Hawk-Dove-Retaliator and Hawk-Dove-Bully models. The analysis includes payoff matrices, equilibrium points, and local stability in evolutionary strategies, highlighting how tactics evolve in populations competing for resources. The findings contribute to understanding aggressiveness and behavioral strategies in ecological contexts.

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Analyzing Game Dynamics: Hawk-Dove and Beyond in Evolutionary Strategies

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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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