Game Dynamics in Extensive Form Dietmar Berwanger LSV, ENS Cachan - - PowerPoint PPT Presentation

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Game Dynamics in Extensive Form Dietmar Berwanger LSV, ENS Cachan - - PowerPoint PPT Presentation

Sep 18, 2022 165 likes •635 views

Game Dynamics in Extensive Form Dietmar Berwanger LSV, ENS Cachan & CNRS ICLA - Logic & Social Interaction, 2009 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 1 / 13 Evolution in repeated games 0,1 2,1 1,2

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Game Dynamics in Extensive Form

Dietmar Berwanger

LSV, ENS Cachan & CNRS

ICLA - Logic & Social Interaction, 2009

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 1 / 13

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SLIDE 2

Evolution in repeated games

0,1 2,1 1,2 0,3 1,1 2,0

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 2 / 13

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SLIDE 3

Evolution in repeated games

0,1 2,1 1,2 0,3 1,1 2,0

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 2 / 13

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SLIDE 4

Evolution in repeated games

0,1 2,1 1,2 0,3 1,1 2,0

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 2 / 13

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SLIDE 5

Evolution in repeated games

0,1 2,1 1,2 0,3 1,1 2,0

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 2 / 13

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SLIDE 6

Evolution in repeated games

0,1 2,1 1,2 0,3 1,1 2,0

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 2 / 13

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SLIDE 7

Evolution in repeated games

0,1 2,1 1,2 0,3 1,1 2,0

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 2 / 13

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SLIDE 8

Evolution in repeated games

0,1 2,1 1,2 0,3 1,1 2,0

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 2 / 13

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SLIDE 9

Evolution in repeated games

0,1 2,1 1,2 0,3 1,1 2,0

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 2 / 13

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SLIDE 10

Evolution in repeated games

0,1 2,1 1,2 0,3 1,1 2,0

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 2 / 13

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SLIDE 11

Evolution in repeated games

0,1 2,1 1,2 0,3 1,1 2,0

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 2 / 13

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SLIDE 12

Evolution in repeated games

0,1 2,1 1,2 0,3 1,1 2,0

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 2 / 13

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SLIDE 13

Evolution in repeated games

0,1 2,1 1,2 0,3 1,1 2,0

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 2 / 13

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SLIDE 14

Evolution in repeated games

0,1 2,1 1,2 0,3 1,1 2,0

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 2 / 13

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SLIDE 15

Evolution in repeated games

0,1 2,1 1,2 0,3 1,1 2,0 Justification of Nash Equilibrium: definite state selective pressure ◮ sequential dynamics Project: Reveal extensive structure from the normal-form representation of a game.

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 2 / 13

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SLIDE 16

Game dynamics graph

0,1 2,1 1,2 0,3 1,1 2,0 0,1 2,1 1,2 0,3 1,1 2,0 0,1 2,1 1,2 0,3 1,1 2,0 0,1 2,1 1,2 0,3 1,1 2,0 0,1 2,1 1,2 0,3 1,1 2,0 0,1 2,1 1,2 0,3 1,1 2,0

Game Γ in normal form: n players strategy sets Si -- finite utility ui : S → N ◮ Game dynamics graph G(Γ): nodes: profiles in S edges: switches s → s′ by any subset of players We consider pure strategies.

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 3 / 13

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SLIDE 17

Best-response dynamics

0,1 2,1 1,2 0,3 1,1 2,0 0,1 2,1 1,2 0,3 1,1 2,0 0,1 2,1 1,2 0,3 1,1 2,0 0,1 2,1 1,2 0,3 1,1 2,0 0,1 2,1 1,2 0,3 1,1 2,0 0,1 2,1 1,2 0,3 1,1 2,0

Greedy walks converge to Nash Equilibrium if well-founded e.g., in potential games.

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 4 / 13

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Best-response dynamics

0,1 2,1 1,2 0,3 1,1 2,0 0,1 2,1 1,2 0,3 1,1 2,0 0,1 2,1 1,2 0,3 1,1 2,0 0,1 2,1 1,2 0,3 1,1 2,0 0,1 2,1 1,2 0,3 1,1 2,0 0,1 2,1 1,2 0,3 1,1 2,0

Greedy walks converge to Nash Equilibrium if well-founded e.g., in potential games. What if not?

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 4 / 13

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Sink equilibria [Vetta & al, 2005]

[Goemans, Mirokni, Vetta 2005] Sink: terminal connected component of best-response graph. ◮ Price of sinking -- social cost of lack of coordination vs price of anarchy. Σui(♦♣t) Σui(✇♦rst ❙✐♥❦❊q) Σui(♦♣t) Σui(✇♦rst ◆❛s❤❊q)

  • Theorem. The price of anarchy can underestimate

the price of sinking by a factor of n. Consequences for convergence speed of random best-response walks.

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 5 / 13

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Salience of the definite state

Example: Tragedy of the commons - variant each player has one responsible and n irresponsible strategies

◮ responsible strategies guarantee 1 Rp ◮ irresponsible strategies pay off 2 Rs for one player ◮ all other irresponsibles -- 0 Rs

who wins depends on all chosen strategies. Pareto optimum: n+1 · Nash Eq: n+ǫ · Sink Eq: 2 ◮ Effect hides when mixing strategies relies on perfect information about the current state

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 6 / 13

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Strategising equilibrium selection

1,1 0,0 0,0 2,1 1,1 0,0 0,0 2,1 1,1 0,0 0,0 2,1 1,1 0,0 0,0 2,1 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 7 / 13

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Strategising equilibrium selection

1,1 0,0 0,0 2,1 1,1 0,0 0,0 2,1 1,1 0,0 0,0 2,1 1,1 0,0 0,0 2,1 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 7 / 13

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Efficient orbit supported by a non-greedy switch

1,1,1 0,0,0 1,1,0 2,0,1 0,0,0 2,2,2 1,2,1 1,3,3

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 8 / 13

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Efficient orbit supported by a non-greedy switch

1,1,1 0,0,0 1,1,0 2,0,1 0,0,0 2,2,2 1,2,1 1,3,3

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 8 / 13

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Efficient orbit supported by a non-greedy switch

1,1,1 0,0,0 1,1,0 2,0,1 0,0,0 2,2,2 1,2,1 1,3,3

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 8 / 13

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Efficient orbit supported by a non-greedy switch

1,1,1 0,0,0 1,1,0 2,0,1 0,0,0 2,2,2 1,2,1 1,3,3

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 8 / 13

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SLIDE 27

Efficient orbit supported by a non-greedy switch

1,1,1 0,0,0 1,1,0 2,0,1 0,0,0 2,2,2 1,2,1 1,3,3

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 8 / 13

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SLIDE 28

Efficient orbit supported by a non-greedy switch

1,1,1 0,0,0 1,1,0 2,0,1 0,0,0 2,2,2 1,2,1 1,3,3

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 8 / 13

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SLIDE 29

Efficient orbit supported by a non-greedy switch

1,1,1 0,0,0 1,1,0 2,0,1 0,0,0 2,2,2 1,2,1 1,3,3

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 8 / 13

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Efficient orbit supported by a non-greedy switch

1,1,1 0,0,0 1,1,0 2,0,1 0,0,0 2,2,2 1,2,1 1,3,3

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 8 / 13

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SLIDE 31

Efficient orbit supported by a non-greedy switch

1,1,1 0,0,0 1,1,0 2,0,1 0,0,0 2,2,2 1,2,1 1,3,3

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 8 / 13

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SLIDE 32

Efficient orbit supported by a non-greedy switch

1,1,1 0,0,0 1,1,0 2,0,1 0,0,0 2,2,2 1,2,1 1,3,3

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 8 / 13

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SLIDE 33

Efficient orbit supported by a non-greedy switch

1,1,1 0,0,0 1,1,0 2,0,1 0,0,0 2,2,2 1,2,1 1,3,3

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 8 / 13

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Efficient orbit supported by a non-greedy switch

1,1,1 0,0,0 1,1,0 2,0,1 0,0,0 2,2,2 1,2,1 1,3,3

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 8 / 13

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Efficient orbit supported by a non-greedy switch

1,1,1 0,0,0 1,1,0 2,0,1 0,0,0 2,2,2 1,2,1 1,3,3

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 8 / 13

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Efficient orbit supported by a non-greedy switch

1,1,1 0,0,0 1,1,0 2,0,1 0,0,0 2,2,2 1,2,1 1,3,3

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 8 / 13

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Dynamics metagame

1,0 0,0 0,0 0,1 0,0 0,0

Game graph: G(Γ); utility: cumulative

◮ mean payoff,

discounted

perfect information

◮ no procedural rules ◮ fair tie breaking Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 9 / 13

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SLIDE 38

Dynamics metagame

1,0 0,0 0,0 0,1 0,0 0,0

Game graph: G(Γ); utility: cumulative

◮ mean payoff,

discounted

perfect information

◮ no procedural rules ◮ fair tie breaking Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 9 / 13

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Basic case: two players

1,0 0,0 0,0 0,1 0,0 0,0 1,0 0,0 0,0 0,1 0,0 0,0 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 10 / 13

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Basic case: two players

1,0 0,0 0,0 0,1 0,0 0,0 1,0 0,0 0,0 0,1 0,0 0,0

  • Theorem. [Ehrenfeucht, Mycielski]

Mean-payoff zero-sum games are determined with memoryless strategies. ◮ feasible outcomes ◭

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 10 / 13

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Basic case: two players

1,0 0,0 0,0 0,1 0,0 0,0 1,0 0,0 0,0 0,1 0,0 0,0

  • Theorem. [Ehrenfeucht, Mycielski]

Mean-payoff zero-sum games are determined with memoryless strategies. ◮ feasible outcomes ◭

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 10 / 13

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Basic case: two players

1,0 0,0 0,0 0,1 0,0 0,0 1,0 0,0 0,0 0,1 0,0 0,0

  • Theorem. [Ehrenfeucht, Mycielski]

Mean-payoff zero-sum games are determined with memoryless strategies. ◮ feasible outcomes ◭

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 10 / 13

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Basic case: two players

1,0 0,0 0,0 0,1 0,0 0,0 1,0 0,0 0,0 0,1 0,0 0,0

  • Theorem. [Ehrenfeucht, Mycielski]

Mean-payoff zero-sum games are determined with memoryless strategies. ◮ feasible outcomes ◭

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 10 / 13

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Challenges

◮ Games of infinite duration: non-zero sum more than two players

  • rder of moves

◮ Folk Theorem

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 11 / 13

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SLIDE 45

Applications

◮ Extensive games of perfect information: best response to path signalling backwards-inductive outcome subgame-perfect equilibria ◮ Symmetric games: regret minimisation, full signalling iterated elimination of weakly dominated strategies

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 12 / 13

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Conclusion

◮ Lifting strategic games to extensive meta-games infinte games with numeric payoffs (can be perturbed) plausible when playing with automata captures some standard concepts ◮ Outlook: Special structure of metagames Value iteration Incomplete information, dynamic objects

Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 13 / 13