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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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Presentation on theme: "Introduction to game dynamics Pierre Auger IRD UR Geodes, Centre d’île de France et Institut Systèmes Complexes, ENS Lyon."— Presentation transcript:

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

2 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

3 Modelling aggressiveness

4 Fighting for resources Dominique Allainé, Lyon 1

5 Hawk-Dove game z Payoff matrix z Gain z Cost

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

7 Replicator equations With

8 Replicator equations Because Leading to then

9 Hawk-dove phase portraits

10 Replicator equations z GC, pure hawk equilibrium Butterflies

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

12 Replicator equations With z Average reward z Reward player i

13 Equilibrium With z Unique interior equilibrium (linear) z Corner

14 Rock-Scissor-Paper game z Payoff matrix

15 Replicator equations

16 Four equilibrium points z Unique interior equilibrium

17 Replicator equations

18 Local stability analysis z Unique interior equilibrium saddle center

19 zLinear 2D systems (hyperbolic)

20 R-C-P phase portrait z First integral

21 Hawk-Dove-Retaliator game z Payoff matrix

22 H-D-R phase portrait

23 Hawk-Dove-Bully game z Payoff matrix

24 H-D-B phase portrait

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

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

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

28 Replicator equations

29 Five equilibrium points z Unique interior equilibrium (possibility)

30 Jacobian matrix at (x*,y*)

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

32 zLinear 2D systems (hyperbolic)

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

34 Battle of the sexes z Males against females

35 Battle of the sexes z Females against males

36 Adding C/2-G in second column

37 Replicator equations

38 Five equilibrium points z Unique interior equilibrium : C

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

40 Phase portrait (existence of periodic solutions)


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