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Epistemic Game Theory Lecture 3 ESSLLI12, Opole Eric Pacuit - - PowerPoint PPT Presentation

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Epistemic Game Theory Lecture 3 ESSLLI12, Opole Eric Pacuit Olivier Roy TiLPS, Tilburg University MCMP, LMU Munich ai.stanford.edu/~epacuit http://olivier.amonbofis.net August 8, 2012 Eric Pacuit and Olivier Roy 1 Plan for the week 1.

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SLIDE 1

Epistemic Game Theory

Lecture 3

ESSLLI’12, Opole

Eric Pacuit Olivier Roy TiLPS, Tilburg University MCMP, LMU Munich ai.stanford.edu/~epacuit http://olivier.amonbofis.net August 8, 2012

Eric Pacuit and Olivier Roy 1

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SLIDE 2

Plan for the week

  • 1. Monday Basic Concepts.
  • 2. Tuesday Epistemics.
  • 3. Wednesday Fundamentals of Epistemic Game Theory.
  • Models of all-out attitudes (cnt’d).
  • Common knowledge of Rationality and iterated strict

dominance in the matrix.

  • (If time, o/w tomorrow.) Common knowledge of Rationality

and backward induction (strict dominance in the tree).

  • 4. Thursday Puzzles and Paradoxes.
  • 5. Friday Extensions and New Directions.

Eric Pacuit and Olivier Roy 2

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SLIDE 3

A family of attitudes

Eric Pacuit and Olivier Roy 3

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SLIDE 4

A family of attitudes

◮ Conditional Beliefs: M, w |

= Bϕ

i ψ iff M, w′ |

= ψ for all w′ ∈ maxi(πi(w) ∩ ||ϕ||).

Eric Pacuit and Olivier Roy 3

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SLIDE 5

A family of attitudes

◮ Conditional Beliefs: M, w |

= Bϕ

i ψ iff M, w′ |

= ψ for all w′ ∈ maxi(πi(w) ∩ ||ϕ||).

◮ Safe Belief: M, w |

= []iϕ iff M, w′ | = ϕ for all w′ i w.

Eric Pacuit and Olivier Roy 3

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SLIDE 6

A family of attitudes

◮ Conditional Beliefs: M, w |

= Bϕ

i ψ iff M, w′ |

= ψ for all w′ ∈ maxi(πi(w) ∩ ||ϕ||).

◮ Safe Belief: M, w |

= []iϕ iff M, w′ | = ϕ for all w′ i w.

◮ Knowledge: M, w |

= Kiϕ iff M, w′ | = ϕ for all w′ such that w′ ∼i w.

Eric Pacuit and Olivier Roy 3

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SLIDE 7

A family of attitudes

◮ Conditional Beliefs: M, w |

= Bϕ

i ψ iff M, w′ |

= ψ for all w′ ∈ maxi(πi(w) ∩ ||ϕ||).

◮ Safe Belief: M, w |

= []iϕ iff M, w′ | = ϕ for all w′ i w.

◮ Knowledge: M, w |

= Kiϕ iff M, w′ | = ϕ for all w′ such that w′ ∼i w. Plain beliefs defined: Biψ ⇔df B⊤

i ϕ

Eric Pacuit and Olivier Roy 3

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SLIDE 8

A family of attitudes

◮ Conditional Beliefs: M, w |

= Bϕ

i ψ iff M, w′ |

= ψ for all w′ ∈ maxi(πi(w) ∩ ||ϕ||).

◮ Safe Belief: M, w |

= []iϕ iff M, w′ | = ϕ for all w′ i w.

◮ Knowledge: M, w |

= Kiϕ iff M, w′ | = ϕ for all w′ such that w′ ∼i w. Plain beliefs defined: Biψ ⇔df B⊤

i ϕ

Conditional beliefs defined: Bϕ

i ψ ⇔df Kiϕ → Ki(ϕ ∧ []i(ϕ → ψ))

Eric Pacuit and Olivier Roy 3

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SLIDE 9

Soft attitudes

w v1 v0 v2 Suppose that w is the current state. Knowledge (KP) Belief (BP) Safe Belief (✷P) Strong Belief (BsP) Graded Beliefs (Brϕ)

Eric Pacuit and Olivier Roy 4

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SLIDE 10

Soft attitudes

w v1 v0 v2 Suppose that w is the current state. Knowledge (KP) Belief (Bip) Safe Belief ([]iP) Strong Belief (BsP) Graded Beliefs (Brϕ)

Eric Pacuit and Olivier Roy 4

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SLIDE 11

Soft attitudes

¬P w ¬P v1 P v0 P v2 Suppose that w is the current state.

◮ Belief (Bip)

Safe Belief (✷P) Strong Belief (BsP) Knowledge (KP) Graded Beliefs (Brϕ)

Eric Pacuit and Olivier Roy 4

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SLIDE 12

Soft attitudes

P w ¬P v1 P v0 P v2 Suppose that w is the current state.

◮ Belief (Bip) ◮ Safe Belief ([]ip)

Strong Belief (BsP) Knowledge (KP) Graded Beliefs (Brϕ)

Eric Pacuit and Olivier Roy 4

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SLIDE 13

Soft attitudes

P w P v1 P v0 P v2 Suppose that w is the current state.

◮ Belief (Bip) ◮ Safe Belief ([]ip) ◮ Knowledge (Kip)

Graded Beliefs (Brϕ)

Eric Pacuit and Olivier Roy 4

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SLIDE 14

Properties of Soft Attitudes

Beliefs and conditional beliefs can be mistaken. ¬P w ¬P v1 P v0 P v2 | = Biϕ → ϕ

Eric Pacuit and Olivier Roy 5

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SLIDE 15

Properties of Soft Attitudes

Beliefs and conditional beliefs are fully introspective. ¬P w ¬P v1 P v0 P v2 | = Biϕ → BiBiϕ | = ¬Biϕ → Bi¬Biϕ

Eric Pacuit and Olivier Roy 6

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SLIDE 16

Eric Pacuit and Olivier Roy 7

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SLIDE 17

Properties of Soft Attitudes

Safe Belief is truthful and positively introspective. P w ¬P v1 P v0 P v2 | = []iϕ → ϕ | = []iϕ → []i[]iϕ

Eric Pacuit and Olivier Roy 8

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SLIDE 18

Properties of Soft Attitudes

Safe Belief is not negatively introspective. ¬P, Q v1 P, Q w P, Q v0 P, ¬Q v2 | = ¬[]iϕ → []i¬[]iϕ but... | = Biϕ ↔ Bi[]iϕ

Eric Pacuit and Olivier Roy 9

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SLIDE 19

Higher-order attitudes and common knowledge.

Eric Pacuit and Olivier Roy 10

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SLIDE 20

“Common Knowledge” is informally described as what any fool would know, given a certain situation: It encompasses what is relevant, agreed upon, established by precedent, assumed, being attended to, salient, or in the conversational record. It is not Common Knowledge who “defined” Common Knowledge!

Eric Pacuit and Olivier Roy 11

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SLIDE 21

“Common Knowledge” is informally described as what any fool would know, given a certain situation: It encompasses what is relevant, agreed upon, established by precedent, assumed, being attended to, salient, or in the conversational record. It is not Common Knowledge who “defined” Common Knowledge!

Eric Pacuit and Olivier Roy 11

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SLIDE 22

The first formal definition of common knowledge?

  • M. Friedell. On the Structure of Shared Awareness. Behavioral Science (1969).
  • R. Aumann. Agreeing to Disagree. Annals of Statistics (1976).

The first rigorous analysis of common knowledge

  • D. Lewis. Convention, A Philosophical Study. 1969.

Fixed-point definition: γ := i and j know that (ϕ and γ)

  • G. Harman. Review of Linguistic Behavior. Language (1977).
  • J. Barwise. Three views of Common Knowledge. TARK (1987).

Shared situation: There is a shared situation s such that (1) s entails ϕ, (2) s entails everyone knows ϕ, plus other conditions

  • H. Clark and C. Marshall. Definite Reference and Mutual Knowledge. 1981.
  • M. Gilbert. On Social Facts. Princeton University Press (1989).

Eric Pacuit and Olivier Roy 11

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SLIDE 23

The first formal definition of common knowledge?

  • M. Friedell. On the Structure of Shared Awareness. Behavioral Science (1969).
  • R. Aumann. Agreeing to Disagree. Annals of Statistics (1976).

The first rigorous analysis of common knowledge

  • D. Lewis. Convention, A Philosophical Study. 1969.

Fixed-point definition: γ := i and j know that (ϕ and γ)

  • G. Harman. Review of Linguistic Behavior. Language (1977).
  • J. Barwise. Three views of Common Knowledge. TARK (1987).

Shared situation: There is a shared situation s such that (1) s entails ϕ, (2) s entails everyone knows ϕ, plus other conditions

  • H. Clark and C. Marshall. Definite Reference and Mutual Knowledge. 1981.
  • M. Gilbert. On Social Facts. Princeton University Press (1989).

Eric Pacuit and Olivier Roy 11

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SLIDE 24

The first formal definition of common knowledge?

  • M. Friedell. On the Structure of Shared Awareness. Behavioral Science (1969).
  • R. Aumann. Agreeing to Disagree. Annals of Statistics (1976).

The first rigorous analysis of common knowledge

  • D. Lewis. Convention, A Philosophical Study. 1969.

Fixed-point definition: γ := i and j know that (ϕ and γ)

  • G. Harman. Review of Linguistic Behavior. Language (1977).
  • J. Barwise. Three views of Common Knowledge. TARK (1987).

Shared situation: There is a shared situation s such that (1) s entails ϕ, (2) s entails everyone knows ϕ, plus other conditions

  • H. Clark and C. Marshall. Definite Reference and Mutual Knowledge. 1981.
  • M. Gilbert. On Social Facts. Princeton University Press (1989).

Eric Pacuit and Olivier Roy 11

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SLIDE 25

The first formal definition of common knowledge?

  • M. Friedell. On the Structure of Shared Awareness. Behavioral Science (1969).
  • R. Aumann. Agreeing to Disagree. Annals of Statistics (1976).

The first rigorous analysis of common knowledge

  • D. Lewis. Convention, A Philosophical Study. 1969.

Fixed-point definition: γ := i and j know that (ϕ and γ)

  • G. Harman. Review of Linguistic Behavior. Language (1977).
  • J. Barwise. Three views of Common Knowledge. TARK (1987).

Shared situation: There is a shared situation s such that (1) s entails ϕ, (2) s entails everyone knows ϕ, plus other conditions

  • H. Clark and C. Marshall. Definite Reference and Mutual Knowledge. 1981.
  • M. Gilbert. On Social Facts. Princeton University Press (1989).

Eric Pacuit and Olivier Roy 11

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SLIDE 26
  • P. Vanderschraaf and G. Sillari. “Common Knowledge”, The Stanford Encyclo-

pedia of Philosophy (2009). http://plato.stanford.edu/entries/common-knowledge/.

Eric Pacuit and Olivier Roy 12

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SLIDE 27

The “Standard” Account

E W

  • R. Aumann. Agreeing to Disagree. Annals of Statistics (1976).
  • R. Fagin, J. Halpern, Y. Moses and M. Vardi. Reasoning about
  • Knowledge. MIT Press, 1995.

Eric Pacuit and Olivier Roy 13

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SLIDE 28

The “Standard” Account

E W An event/proposition is any (definable) subset E ⊆ W

Eric Pacuit and Olivier Roy 13

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SLIDE 29

The “Standard” Account

E W blabla blabla

Eric Pacuit and Olivier Roy 13

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SLIDE 30

The “Standard” Account

E W

w

w | = KA(E) and w | = KB(E)

Eric Pacuit and Olivier Roy 13

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SLIDE 31

The “Standard” Account

E W

w

The model also describes the agents’ higher-order knowledge/beliefs

Eric Pacuit and Olivier Roy 13

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SLIDE 32

The “Standard” Account

E W

w

Everyone Knows: K(E) =

i∈A Ki(E), K 0(E) = E,

K m(E) = K(K m−1(E))

Eric Pacuit and Olivier Roy 13

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SLIDE 33

The “Standard” Account

E W

w

Common Knowledge: C(E) =

  • m≥0

K m(E)

Eric Pacuit and Olivier Roy 13

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SLIDE 34

The “Standard” Account

E W

w

w | = K(E) w | = C(E)

Eric Pacuit and Olivier Roy 13

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SLIDE 35

The “Standard” Account

E W

w

w | = C(E)

Eric Pacuit and Olivier Roy 13

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SLIDE 36
  • Fact. For all i ∈ A and E ⊆ W , KiC(E) = C(E).

Eric Pacuit and Olivier Roy 14

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SLIDE 37
  • Fact. For all i ∈ A and E ⊆ W , KiC(E) = C(E).

Suppose you are told “Ann and Bob are going together,”’ and respond “sure, that’s common knowledge.” What you mean is not only that everyone knows this, but also that the announcement is pointless, occasions no surprise, reveals nothing new; pause in effect, that the situation after the announcement does not differ from that before. ... the event “Ann and Bob are going together” — call it E — is common knowledge if and only if some event — call it F — happened that entails E and also entails all players’ knowing F (like all players met Ann and Bob at an intimate party). (Aumann, 1999 pg. 271, footnote 8)

Eric Pacuit and Olivier Roy 14

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SLIDE 38
  • Fact. For all i ∈ A and E ⊆ W , KiC(E) = C(E).

An event F is self-evident if Ki(F) = F for all i ∈ A.

  • Fact. An event E is commonly known iff some self-evident event

that entails E obtains.

Eric Pacuit and Olivier Roy 14

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SLIDE 39
  • Fact. For all i ∈ A and E ⊆ W , KiC(E) = C(E).

An event F is self-evident if Ki(F) = F for all i ∈ A.

  • Fact. An event E is commonly known iff some self-evident event

that entails E obtains.

  • Fact. w ∈ C(E) if every finite path starting at w ends in a state

in E The following axiomatize common knowledge:

◮ C(ϕ → ψ) → (Cϕ → Cψ) ◮ Cϕ → (ϕ ∧ ECϕ)

(Fixed-Point)

◮ C(ϕ → Eϕ) → (ϕ → Cϕ)

(Induction) With Eϕ :=

i∈Ag Kiϕ.

Eric Pacuit and Olivier Roy 14

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SLIDE 40

Some General Remarks

Eric Pacuit and Olivier Roy 15

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SLIDE 41

Some General Remarks

◮ Two broad families of models of higher-order information:

  • Type spaces. (probabilistic)
  • Plausibility models. (all-out)

◮ There’s also a natural notion of qualitative type spaces, just

like a natural probabilistic version of plausibility models. No strict separation between the two ways of thinking about information in interaction.

Eric Pacuit and Olivier Roy 15

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Some General Remarks

◮ Two broad families of models of higher-order information:

  • Type spaces. (probabilistic)
  • Plausibility models. (all-out)

◮ There’s also a natural notion of qualitative type spaces, just

like a natural probabilistic version of plausibility models. No strict separation between the two ways of thinking about information in interaction.

◮ In both the notion of a state is crucial. A state encodes:

  • 1. The “non-epistemic facts”. Here, mostly: what the agents are

playing.

  • 2. What the agents know and/or believe about 1.
  • 3. What the agents know and/or believe about 2.
  • 4. ...

Eric Pacuit and Olivier Roy 15

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SLIDE 43

Now let’s do epistemics in games...

Eric Pacuit and Olivier Roy 16

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SLIDE 44

Basics again

The Epistemic or Bayesian View on Games

◮ Traditional game theory:

Actions, outcomes, preferences, solution concepts.

◮ Epistemic game theory:

Actions, outcomes, preferences, beliefs, choice rules.

Eric Pacuit and Olivier Roy 17

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SLIDE 45

Basics again

The Epistemic or Bayesian View on Games

◮ Traditional game theory:

Actions, outcomes, preferences, solution concepts.

◮ Epistemic game theory:

Actions, outcomes, preferences, beliefs, choice rules. := (interactive) decision problem: choice rule and higher-order information.

Eric Pacuit and Olivier Roy 17

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SLIDE 46

Basics again

The Epistemic or Bayesian View on Games

◮ Traditional game theory:

Actions, outcomes, preferences, solution concepts.

◮ Decision theory:

Actions, outcomes, preferences, beliefs, choice rules.

◮ Epistemic game theory:

Actions, outcomes, preferences, beliefs, choice rules. := (interactive) decision problem: choice rule and higher-order information.

Eric Pacuit and Olivier Roy 17

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SLIDE 47

Basics again

Beliefs, Choice Rules, Rationality

What do we mean when we say that a player chooses rationally? That she follows some given choice rules.

◮ Maximization of expected utility, (Strict) dominance

reasoning, Admissibility, etc.

Eric Pacuit and Olivier Roy 18

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SLIDE 48

Basics again

Beliefs, Choice Rules, Rationality

What do we mean when we say that a player chooses rationally? That she follows some given choice rules.

◮ Maximization of expected utility, (Strict) dominance

reasoning, Admissibility, etc. In game models:

◮ The model describes the choices and (higher-order)

beliefs/attitudes at each state.

◮ It is the choice rules that determine whether the choice made

at each state is ”rational” or not.

  • An agent can be rational at a state given one choice rule, but

irrational given the other.

  • Rationality in this sense is not built in the models.

Eric Pacuit and Olivier Roy 18

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SLIDE 49

Basics again

Rationality

Let G = N, {Si}i∈N, {ui}i∈N be a strategic game and T = {Ti}i∈N, {λi}i∈N, S a type space for G. For each ti ∈ Ti, we can define a probability measure pti ∈ ∆(S−i): pti(s−i) =

  • t−i∈T−i

λi(ti)(s−i, t−i) The set of states (pairs of strategy profiles and type profiles) where player i chooses rationally is: Rati := {(si, ti) | si is a best response to pti} The event that all players are rational is Rat = {(s, t) | for all i, (si, ti) ∈ Rati}.

Eric Pacuit and Olivier Roy 19

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SLIDE 50

Basics again

Rationality

Let G = N, {Si}i∈N, {ui}i∈N be a strategic game and T = {Ti}i∈N, {λi}i∈N, S a type space for G. For each ti ∈ Ti, we can define a probability measure pti ∈ ∆(S−i): pti(s−i) =

  • t−i∈T−i

λi(ti)(s−i, t−i) The set of states (pairs of strategy profiles and type profiles) where player i chooses rationally is: Rati := {(si, ti) | si is a best response to pti} The event that all players are rational is Rat = {(s, t) | for all i, (si, ti) ∈ Rati}.

◮ Types, as opposed to players, are rational or not at a

given state.

Eric Pacuit and Olivier Roy 19

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SLIDE 51

Rationality and common belief of rationality (RCBR) in the matrix

Eric Pacuit and Olivier Roy 20

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SLIDE 52

RCBR in the Matrix

IESDS

2 1 l c r t 3, 3 1, 1 0, 0 m 1,1 3, 3 1, 0 m 0, 4 0, 0 4, 0

Eric Pacuit and Olivier Roy 21

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SLIDE 53

RCBR in the Matrix

IESDS

2 1 l c r t 3, 3 1, 1 0, 0 m 1,1 3, 3 1, 0 m 0, 4 0, 0 4, 0

֌

2 1 l c t 3, 3 1, 1 m 1,1 3, 3 b 0, 4 0, 0

Eric Pacuit and Olivier Roy 21

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SLIDE 54

RCBR in the Matrix

IESDS

2 1 l c r t 3, 3 1, 1 0, 0 m 1,1 3, 3 1, 0 m 0, 4 0, 0 4, 0

֌

2 1 l c t 3, 3 1, 1 m 1,1 3, 3 b 0, 4 0, 0

֌

2 1 l c t 3, 3 1, 1 m 1,1 3, 3

Eric Pacuit and Olivier Roy 21

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SLIDE 55

RCBR in the Matrix

1’s types

λ1(t1) l c r s1 0.5 0.5 s2 s3 λ1(t2) l c r s1 0.5 s2 0.5 s3

Eric Pacuit and Olivier Roy 22

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SLIDE 56

RCBR in the Matrix

2’s types

λ2(s1) t m b t1 0.5 0.5 t2 λ2(s2) t m b t1 0.25 0.25 t2 0.25 0.25 λ2(s3) t m b t1 0.5 t2 0.5

Eric Pacuit and Olivier Roy 23

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SLIDE 57

RCBR in the Matrix

2 1 l c r t 3, 3 1, 1 0, 0 m 1,1 3, 3 1, 0 b 0, 4 0, 0 4, 0 λ2(s1) t m b t1 0.5 0.5 t2 λ2(s2) t m b t1 0.25 0.25 t2 0.25 0.25 λ2(s3) t m b t1 0.5 t2 0.5

Eric Pacuit and Olivier Roy 24

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SLIDE 58

RCBR in the Matrix

2 1 l c r t 3, 3 1, 1 0, 0 m 1,1 3, 3 1, 0 b 0, 4 0, 0 4, 0 λ2(s1) t m b t1 0.5 0.5 t2 λ2(s2) t m b t1 0.25 0.25 t2 0.25 0.25 λ2(s3) t m b t1 0.5 t2 0.5

◮ l and c are rational for both s1 and s2.

Eric Pacuit and Olivier Roy 24

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SLIDE 59

RCBR in the Matrix

2 1 l c r t 3, 3 1, 1 0, 0 m 1,1 3, 3 1, 0 b 0, 4 0, 0 4, 0 λ2(s1) t m b t1 0.5 0.5 t2 λ2(s2) t m b t1 0.25 0.25 t2 0.25 0.25 λ2(s3) t m b t1 0.5 t2 0.5

◮ l and c are rational for both s1 and s2.

Eric Pacuit and Olivier Roy 24

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SLIDE 60

RCBR in the Matrix

2 1 l c r t 3, 3 1, 1 0, 0 m 1,1 3, 3 1, 0 b 0, 4 0, 0 4, 0 λ2(s1) t m b t1 0.5 0.5 t2 λ2(s2) t m b t1 0.25 0.25 t2 0.25 0.25 λ2(s3) t m b t1 0.5 t2 0.5

◮ l and c are rational for both s1 and s2. ◮ l is the only rational action for s3.

Eric Pacuit and Olivier Roy 24

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SLIDE 61

RCBR in the Matrix

2 1 l c r t 3, 3 1, 1 0, 0 m 1,1 3, 3 1, 0 b 0, 4 0, 0 4, 0 λ2(s1) t m b t1 0.5 0.5 t2 λ2(s2) t m b t1 0.25 0.25 t2 0.25 0.25 λ2(s3) t m b t1 0.5 t2 0.5

◮ l and c are rational for both s1 and s2. ◮ l is the only rational action for s3. ◮ Whatever her type, it is never rational to play r for 2.

Eric Pacuit and Olivier Roy 24

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SLIDE 62

RCBR in the Matrix

2 1 l c r t 3, 3 1, 1 0, 0 m 1,1 3, 3 1, 0 b 0, 4 0, 0 4, 0 λ1(t1) l c r s1 0.5 0.5 s2 s3 λ1(t2) l c r s1 0.5 s2 0.5 s3

Eric Pacuit and Olivier Roy 25

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SLIDE 63

RCBR in the Matrix

2 1 l c r t 3, 3 1, 1 0, 0 m 1,1 3, 3 1, 0 b 0, 4 0, 0 4, 0 λ1(t1) l c r s1 0.5 0.5 s2 s3 λ1(t2) l c r s1 0.5 s2 0.5 s3

◮ t and m are rational for t1.

Eric Pacuit and Olivier Roy 25

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SLIDE 64

RCBR in the Matrix

2 1 l c r t 3, 3 1, 1 0, 0 m 1,1 3, 3 1, 0 b 0, 4 0, 0 4, 0 λ1(t1) l c r s1 0.5 0.5 s2 s3 λ1(t2) l c r s1 0.5 s2 0.5 s3

◮ t and m are rational for t1.

Eric Pacuit and Olivier Roy 25

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SLIDE 65

RCBR in the Matrix

2 1 l c r t 3, 3 1, 1 0, 0 m 1,1 3, 3 1, 0 b 0, 4 0, 0 4, 0 λ1(t1) l c r s1 0.5 0.5 s2 s3 λ1(t2) l c r s1 0.5 s2 0.5 s3

◮ t and m are rational for t1. ◮ m and b are rational for t2.

Eric Pacuit and Olivier Roy 25

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SLIDE 66

RCBR in the Matrix

λ2(s1) t m b t1 0.5 0.5 t2 λ2(s2) t m b t1 0.25 0.25 t2 0.25 0.25 λ2(s3) t m b t1 0.5 t2 0.5

Eric Pacuit and Olivier Roy 26

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SLIDE 67

RCBR in the Matrix

λ2(s1) t m b t1 0.5 0.5 t2 λ2(s2) t m b t1 0.25 0.25 t2 0.25 0.25 λ2(s3) t m b t1 0.5 t2 0.5

◮ All of 2’s types believe that 1 is rational.

Eric Pacuit and Olivier Roy 26

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SLIDE 68

RCBR in the Matrix

λ1(t1) l c r s1 0.5 0.5 s2 s3 λ1(t2) l c r s1 0.5 s2 0.5 s3

Eric Pacuit and Olivier Roy 27

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SLIDE 69

RCBR in the Matrix

λ1(t1) l c r s1 0.5 0.5 s2 s3 λ1(t2) l c r s1 0.5 s2 0.5 s3

◮ Type t1 of 1 believes that 2 is rational.

Eric Pacuit and Olivier Roy 27

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SLIDE 70

RCBR in the Matrix

λ1(t1) l c r s1 0.5 0.5 s2 s3 λ1(t2) l c r s1 0.5 s2 0.5 s3

◮ Type t1 of 1 believes that 2 is rational. ◮ But type t2 doesn’t! (1/2 probability that 2 is playing r.)

Eric Pacuit and Olivier Roy 27

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SLIDE 71

RCBR in the Matrix

λ2(s1) t m b t1 0.5 0.5 t2 λ2(s2) t m b t1 0.25 0.25 t2 0.25 0.25 λ2(s3) t m b t1 0.5 t2 0.5

Eric Pacuit and Olivier Roy 28

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SLIDE 72

RCBR in the Matrix

λ2(s1) t m b t1 0.5 0.5 t2 λ2(s2) t m b t1 0.25 0.25 t2 0.25 0.25 λ2(s3) t m b t1 0.5 t2 0.5

◮ Only type s1 of 2 believes that 1 is rational and that 1

believes that 2 is also rational.

Eric Pacuit and Olivier Roy 28

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SLIDE 73

RCBR in the Matrix

λ1(t1) l c r s1 0.5 0.5 s2 s3 λ1(t2) l c r s1 0.5 s2 0.5 s3

Eric Pacuit and Olivier Roy 29

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SLIDE 74

RCBR in the Matrix

λ1(t1) l c r s1 0.5 0.5 s2 s3 λ1(t2) l c r s1 0.5 s2 0.5 s3

◮ Type t1 of 1 believes that 2 is rational and that 2 believes

that 1 believes that 2 is rational.

Eric Pacuit and Olivier Roy 29

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SLIDE 75

RCBR in the Matrix

2 1 l c r t 3, 3 1, 1 0, 0 m 1,1 3, 3 1, 0 b 0, 4 0, 0 4, 0 λ1(t1) l c r s1 0.5 0.5 s2 s3 λ2(s1) t m b t1 0.5 0.5 t2

Eric Pacuit and Olivier Roy 30

slide-76
SLIDE 76

RCBR in the Matrix

2 1 l c r t 3, 3 1, 1 0, 0 m 1,1 3, 3 1, 0 b 0, 4 0, 0 4, 0 λ1(t1) l c r s1 0.5 0.5 s2 s3 λ2(s1) t m b t1 0.5 0.5 t2

◮ No further iteration of mutual belief in rationality eliminate

some types or strategies.

Eric Pacuit and Olivier Roy 30

slide-77
SLIDE 77

RCBR in the Matrix

2 1 l c r t 3, 3 1, 1 0, 0 m 1,1 3, 3 1, 0 b 0, 4 0, 0 4, 0 λ1(t1) l c r s1 0.5 0.5 s2 s3 λ2(s1) t m b t1 0.5 0.5 t2

◮ No further iteration of mutual belief in rationality eliminate

some types or strategies.

◮ So at all the states in {(t1, s1)} × {t, m} × {l, c} we have

rationality and common belief in rationality.

Eric Pacuit and Olivier Roy 30

slide-78
SLIDE 78

RCBR in the Matrix

2 1 l c r t 3, 3 1, 1 0, 0 m 1,1 3, 3 1, 0 b 0, 4 0, 0 4, 0 λ1(t1) l c r s1 0.5 0.5 s2 s3 λ2(s1) t m b t1 0.5 0.5 t2

◮ No further iteration of mutual belief in rationality eliminate

some types or strategies.

◮ So at all the states in {(t1, s1)} × {t, m} × {l, c} we have

rationality and common belief in rationality.

◮ But observe that {t, m} × {l, c} is precisely the set of profiles

that survive IESDS.

Eric Pacuit and Olivier Roy 30

slide-79
SLIDE 79

RCBR in the Matrix

The general result: RCBR ⇒ IESDS

Suppose that G is a strategic game and T is any type space for G. If (s, t) is a state in T in which all the players are rational and there is common belief of rationality, then s is a strategy profile that survives iteratively removal of strictly dominated strategies.

  • D. Bernheim. Rationalizable strategic behavior. Econometrica, 52:1007-1028,

1984.

  • D. Pearce.

Rationalizable strategic behavior and the problem of perfection. Econometrica, 52:1029-1050, 1984.

  • A. Brandenburger and E. Dekel.

Rationalizability and correlated equilibria. Econometrica, 55:1391-1402, 1987.

Eric Pacuit and Olivier Roy 31

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SLIDE 80

RCBR in the Matrix

Proof: RCBR ⇒ IESDS

◮ We show by induction on n that the if the players have n-level of

mutual belief in rationality then they do not play strategies that would be eliminated at the n + 1th round of IESDS.

slide-81
SLIDE 81

RCBR in the Matrix

Proof: RCBR ⇒ IESDS

◮ We show by induction on n that the if the players have n-level of

mutual belief in rationality then they do not play strategies that would be eliminated at the n + 1th round of IESDS.

◮ Basic case, n = 0. All the players are rational. We know that a

strictly dominated strategy, i.e. one that would be eliminated in the 1st round of IESDS, is never a best response. So no player is playing such a strategy.

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SLIDE 82

RCBR in the Matrix

Proof: RCBR ⇒ IESDS

◮ We show by induction on n that the if the players have n-level of

mutual belief in rationality then they do not play strategies that would be eliminated at the n + 1th round of IESDS.

◮ Basic case, n = 0. All the players are rational. We know that a

strictly dominated strategy, i.e. one that would be eliminated in the 1st round of IESDS, is never a best response. So no player is playing such a strategy.

◮ Inductive step. Suppose that it is mutual belief up to degree nth

that all players are rational.

slide-83
SLIDE 83

RCBR in the Matrix

Proof: RCBR ⇒ IESDS

◮ We show by induction on n that the if the players have n-level of

mutual belief in rationality then they do not play strategies that would be eliminated at the n + 1th round of IESDS.

◮ Basic case, n = 0. All the players are rational. We know that a

strictly dominated strategy, i.e. one that would be eliminated in the 1st round of IESDS, is never a best response. So no player is playing such a strategy.

◮ Inductive step. Suppose that it is mutual belief up to degree nth

that all players are rational. Take any strategy si of an agent i that would not survive n + 1 round of IESDS. This strategy is never a best response to a belief whose support is included in the set of states where the others play strategies that would not survive nth round of IESDS. But by our IH this is precisely the kind of belief that all i’s type have by IH, so i is not playing si either.

Eric Pacuit and Olivier Roy 32

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SLIDE 84

RCBR in the Matrix

“Converse direction” From IESDS to RCBR

Given any strategy profile that survives IESDS, there is a model in and a state in that model where this profile RCBR holds at that state.

Eric Pacuit and Olivier Roy 33

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SLIDE 85

RCBR in the Matrix

“Converse direction” From IESDS to RCBR

Given any strategy profile that survives IESDS, there is a model in and a state in that model where this profile RCBR holds at that state.

◮ Trivial? Mathematically, yes.

Eric Pacuit and Olivier Roy 33

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SLIDE 86

RCBR in the Matrix

“Converse direction” From IESDS to RCBR

Given any strategy profile that survives IESDS, there is a model in and a state in that model where this profile RCBR holds at that state.

◮ Trivial? Mathematically, yes. ◮ ... but conceptually important. One can always view or

interpret the choice of a strategy profile that would survive the iterative elimination procedure as one that results from RCBR.

Eric Pacuit and Olivier Roy 33

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SLIDE 87

RCBR in the Matrix

“Converse direction” From IESDS to RCBR

Given any strategy profile that survives IESDS, there is a model in and a state in that model where this profile RCBR holds at that state.

◮ Trivial? Mathematically, yes. ◮ ... but conceptually important. One can always view or

interpret the choice of a strategy profile that would survive the iterative elimination procedure as one that results from RCBR. Is the entire set of strategy profiles that survive IESDS always consistent with rationality and common belief in rationality? Yes.

◮ For any game G, there is a type structure for that game in

which the strategy profiles consistent with rationality and common belief in rationality is the set of strategies that survive iterative removal of strictly dominated strategies.

  • A. Friedenberg and J. Kiesler. Iterated Dominance Revisited. Working paper,

2011.

Eric Pacuit and Olivier Roy 33

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SLIDE 88

Subgames

Let H = H1, . . . , Hn, u1, . . . , un be an arbitrary strategic game.

Eric Pacuit and Olivier Roy 34

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SLIDE 89

Subgames

Let H = H1, . . . , Hn, u1, . . . , un be an arbitrary strategic game. A restriction of H is a sequence G = (G1, . . . , Gn) such that Gi ⊆ Hi for all i ∈ {1, . . . , n}. The set of all restrictions of a game H ordered by componentwise set inclusion forms a complete lattice.

Eric Pacuit and Olivier Roy 34

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SLIDE 90

Game Models

Relational models: W , Ri where Ri ⊆ W × W . Write Ri(w) = {v | wRiv}. Events: E ⊆ W Knowledge/Belief: ✷E = {w | Ri(w) ⊆ E} Common knowledge/belief: ✷1E = ✷E ✷k+1E = ✷✷kE ✷∗E = ∞

k=1 ✷kE

  • Fact. An event F is called evident provided F ⊆ ✷F. w ∈ ✷∗E

provided there is an evident event F such that w ∈ F ⊆ ✷E.

Eric Pacuit and Olivier Roy 35

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SLIDE 91

Game Models

Let G = (G1, . . . , Gn) be a restriction of a game H. A knowledge/belief model of G is a tuple W , R1, . . . , Rn, σ1, . . . , σn where W , R1, . . . , Rn is a knowledge/belief model and σi : W → Gi.

Eric Pacuit and Olivier Roy 36

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SLIDE 92

Game Models

Let G = (G1, . . . , Gn) be a restriction of a game H. A knowledge/belief model of G is a tuple W , R1, . . . , Rn, σ1, . . . , σn where W , R1, . . . , Rn is a knowledge/belief model and σi : W → Gi. Given a model W , R1, . . . , Rn, σ1, . . . σn for a restriction G and a sequence E = {E1, . . . , En} where Ei ⊆ W , GE = (σ1(E1), . . . , σn(En))

Eric Pacuit and Olivier Roy 36

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SLIDE 93

Some Lattice Theory

◮ (D, ⊆) is a lattice with largest element ⊤. T : D → D an

  • perator.

Eric Pacuit and Olivier Roy 37

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SLIDE 94

Some Lattice Theory

◮ (D, ⊆) is a lattice with largest element ⊤. T : D → D an

  • perator.

◮ T is monotonic if for all G, G ′, G ⊆ G ′ implies T(G) ⊆ T(G ′)

Eric Pacuit and Olivier Roy 37

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SLIDE 95

Some Lattice Theory

◮ (D, ⊆) is a lattice with largest element ⊤. T : D → D an

  • perator.

◮ T is monotonic if for all G, G ′, G ⊆ G ′ implies T(G) ⊆ T(G ′) ◮ G is a fixed-point if T(G) = G

Eric Pacuit and Olivier Roy 37

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SLIDE 96

Some Lattice Theory

◮ (D, ⊆) is a lattice with largest element ⊤. T : D → D an

  • perator.

◮ T is monotonic if for all G, G ′, G ⊆ G ′ implies T(G) ⊆ T(G ′) ◮ G is a fixed-point if T(G) = G ◮ νT is the largest fixed point of T

Eric Pacuit and Olivier Roy 37

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SLIDE 97

Some Lattice Theory

◮ (D, ⊆) is a lattice with largest element ⊤. T : D → D an

  • perator.

◮ T is monotonic if for all G, G ′, G ⊆ G ′ implies T(G) ⊆ T(G ′) ◮ G is a fixed-point if T(G) = G ◮ νT is the largest fixed point of T ◮ T ∞ is the “outcome of T: T 0 = ⊤, T α+1 = T(T α),

T β =

α<β T α, The outcome of iterating T is the least α

such that T α+1 = T α, denoted T ∞

Eric Pacuit and Olivier Roy 37

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SLIDE 98

Some Lattice Theory

◮ (D, ⊆) is a lattice with largest element ⊤. T : D → D an

  • perator.

◮ T is monotonic if for all G, G ′, G ⊆ G ′ implies T(G) ⊆ T(G ′) ◮ G is a fixed-point if T(G) = G ◮ νT is the largest fixed point of T ◮ T ∞ is the “outcome of T: T 0 = ⊤, T α+1 = T(T α),

T β =

α<β T α, The outcome of iterating T is the least α

such that T α+1 = T α, denoted T ∞

◮ Tarski’s Fixed-Point Theorem: Every monotonic operator

T has a (least and largest) fixed point T ∞ = νT = {G | G ⊆ T(G)}.

Eric Pacuit and Olivier Roy 37

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SLIDE 99

Some Lattice Theory

◮ (D, ⊆) is a lattice with largest element ⊤. T : D → D an

  • perator.

◮ T is monotonic if for all G, G ′, G ⊆ G ′ implies T(G) ⊆ T(G ′) ◮ G is a fixed-point if T(G) = G ◮ νT is the largest fixed point of T ◮ T ∞ is the “outcome of T: T 0 = ⊤, T α+1 = T(T α),

T β =

α<β T α, The outcome of iterating T is the least α

such that T α+1 = T α, denoted T ∞

◮ Tarski’s Fixed-Point Theorem: Every monotonic operator

T has a (least and largest) fixed point T ∞ = νT = {G | G ⊆ T(G)}.

◮ T is contracting if T(G) ⊆ G. Every contracting operator has

an outcome (T ∞ is well-defined)

Eric Pacuit and Olivier Roy 37

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SLIDE 100

Rationality Properties

ϕ(si, Gi, G−i) holds between a strategy si ∈ Hi, a set of strategies Gi for player i and strategies G−i of the opponents. Intuitively si is ϕ-optimal strategy for player i in the restricted game Gi, G−i, u1, . . . , un (where the payoffs are suitably restricted).

Eric Pacuit and Olivier Roy 38

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SLIDE 101

Rationality Properties

ϕ(si, Gi, G−i) holds between a strategy si ∈ Hi, a set of strategies Gi for player i and strategies G−i of the opponents. Intuitively si is ϕ-optimal strategy for player i in the restricted game Gi, G−i, u1, . . . , un (where the payoffs are suitably restricted). ϕi is monotonic if for all G−i, G ′

−i ⊆ H−i and si ∈ Hi

G−i ⊆ G ′

−i and ϕ(si, Hi, G−i) implies ϕ(si, Hi, G ′ −i)

Eric Pacuit and Olivier Roy 38

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SLIDE 102

Removing Strategies

If ϕ = (ϕ1, . . . , ϕn), then define Tϕ(G) = G ′ where

◮ G = (G1, . . . , Gn), G ′ = (G ′ 1, . . . , G ′ n), ◮ for all i ∈ {1, . . . , n},

G ′

i = {si ∈ Gi | ϕi(si, Hi, G−i)}

Eric Pacuit and Olivier Roy 39

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SLIDE 103

Removing Strategies

If ϕ = (ϕ1, . . . , ϕn), then define Tϕ(G) = G ′ where

◮ G = (G1, . . . , Gn), G ′ = (G ′ 1, . . . , G ′ n), ◮ for all i ∈ {1, . . . , n},

G ′

i = {si ∈ Gi | ϕi(si, Hi, G−i)}

Tϕ is contracting, so it has an outcome T ∞

ϕ

Eric Pacuit and Olivier Roy 39

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SLIDE 104

Removing Strategies

If ϕ = (ϕ1, . . . , ϕn), then define Tϕ(G) = G ′ where

◮ G = (G1, . . . , Gn), G ′ = (G ′ 1, . . . , G ′ n), ◮ for all i ∈ {1, . . . , n},

G ′

i = {si ∈ Gi | ϕi(si, Hi, G−i)}

Tϕ is contracting, so it has an outcome T ∞

ϕ

If each ϕi is monotonic, then νTϕ exists and equals T ∞

ϕ .

Eric Pacuit and Olivier Roy 39

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SLIDE 105

Rational Play

Let H = H1, . . . , Hn, u1, . . . , un a strategic game and W , R1, . . . , Rn, σ1, . . . , σn a model for H. σi(w) is the strategy player is using in state w. GRi(w) is a restriction of H giving i’s view of the game.

Eric Pacuit and Olivier Roy 40

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SLIDE 106

Rational Play

Let H = H1, . . . , Hn, u1, . . . , un a strategic game and W , R1, . . . , Rn, σ1, . . . , σn a model for H. σi(w) is the strategy player is using in state w. GRi(w) is a restriction of H giving i’s view of the game. Player i is ϕi-rational in the state w if ϕi(σi(w), Hi, (GRi(w))−i) holds.

Eric Pacuit and Olivier Roy 40

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SLIDE 107

Rational Play

Let H = H1, . . . , Hn, u1, . . . , un a strategic game and W , R1, . . . , Rn, σ1, . . . , σn a model for H. σi(w) is the strategy player is using in state w. GRi(w) is a restriction of H giving i’s view of the game. Player i is ϕi-rational in the state w if ϕi(σi(w), Hi, (GRi(w))−i) holds. Rat(ϕ) = {w ∈ W | each player is ϕi-rational in w} ✷Rat(ϕ) ✷∗Rat(ϕ)

Eric Pacuit and Olivier Roy 40

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SLIDE 108

Theorem (Apt and Zvesper).

◮ Suppose that each ϕi is monotonic. Then for all belief models

for H, GRat(ϕ)∩B∗(Rat(ϕ)) ⊆ T ∞

ϕ ◮ Suppose that each ϕi is monotonic. Then for all knowledge

models for H, GK ∗(Rat(ϕ)) ⊆ T ∞

ϕ ◮ For some standard knowledge model for H,

T ∞

ϕ ⊆ GK ∗(Rat(ϕ))

  • K. Apt and J. Zvesper. The Role of Monotonicity in the Epistemic Analysis of
  • Games. Games, 1(4), pgs. 381-394, 2010.

Eric Pacuit and Olivier Roy 41

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SLIDE 109

Claim If each ϕi is monotonic, then GRat(ϕ)∩✷∗Rat(ϕ) ⊆ T ∞

ϕ .

Eric Pacuit and Olivier Roy 42

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SLIDE 110

Claim If each ϕi is monotonic, then GRat(ϕ)∩✷∗Rat(ϕ) ⊆ T ∞

ϕ .

Let si be an element of the ith component of GRat(ϕ)∩✷∗Rat(ϕ): si = σi(w) for some w ∈ Rat(ϕ) ∩ ✷∗Rat(ϕ)

Eric Pacuit and Olivier Roy 42

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SLIDE 111

Claim If each ϕi is monotonic, then GRat(ϕ)∩✷∗Rat(ϕ) ⊆ T ∞

ϕ .

Let si be an element of the ith component of GRat(ϕ)∩✷∗Rat(ϕ): si = σi(w) for some w ∈ Rat(ϕ) ∩ ✷∗Rat(ϕ) there is an F such that F ⊆ ✷F and w ∈ F ⊆ ✷Rat(ϕ) = {v ∈ W | ∀i Ri(v) ⊆ Rat(ϕ)}

Eric Pacuit and Olivier Roy 42

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SLIDE 112

Claim If each ϕi is monotonic, then GRat(ϕ)∩✷∗Rat(ϕ) ⊆ T ∞

ϕ .

Let si be an element of the ith component of GRat(ϕ)∩✷∗Rat(ϕ): si = σi(w) for some w ∈ Rat(ϕ) ∩ ✷∗Rat(ϕ) there is an F such that F ⊆ ✷F and w ∈ F ⊆ ✷Rat(ϕ) = {v ∈ W | ∀i Ri(v) ⊆ Rat(ϕ)}

  • Claim. GF∩Rat(ϕ) is post-fixed point of Tϕ

(GF∩Rat(ϕ) ⊆ Tϕ(GF∩Rat(ϕ))).

Eric Pacuit and Olivier Roy 42

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SLIDE 113

Claim If each ϕi is monotonic, then GRat(ϕ)∩✷∗Rat(ϕ) ⊆ T ∞

ϕ .

Let si be an element of the ith component of GRat(ϕ)∩✷∗Rat(ϕ): si = σi(w) for some w ∈ Rat(ϕ) ∩ ✷∗Rat(ϕ) there is an F such that F ⊆ ✷F and w ∈ F ⊆ ✷Rat(ϕ) = {v ∈ W | ∀i Ri(v) ⊆ Rat(ϕ)}

  • Claim. GF∩Rat(ϕ) is post-fixed point of Tϕ

(GF∩Rat(ϕ) ⊆ Tϕ(GF∩Rat(ϕ))). Since each ϕi is monotonic, Tϕ is monotonic and by Tarski’s fixed-point theorem, GF∩Rat(ϕ) ⊆ T ∞

ϕ . But si = σi(w) and

w ∈ F ∩ Rat(ϕ), so si is the ith component in T ∞

ϕ .

Eric Pacuit and Olivier Roy 42

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SLIDE 114

F ⊆ ✷F and w ∈ F ⊆ ✷Rat(ϕ) = {v ∈ W | ∀i Ri(v) ⊆ Rat(ϕ)}

  • Claim. GF∩Rat(ϕ) is post-fixed point of Tϕ

(GF∩Rat(ϕ) ⊆ Tϕ(GF∩Rat(ϕ))).

Eric Pacuit and Olivier Roy 43

slide-115
SLIDE 115

F ⊆ ✷F and w ∈ F ⊆ ✷Rat(ϕ) = {v ∈ W | ∀i Ri(v) ⊆ Rat(ϕ)}

  • Claim. GF∩Rat(ϕ) is post-fixed point of Tϕ

(GF∩Rat(ϕ) ⊆ Tϕ(GF∩Rat(ϕ))). Let w′ ∈ F ∩ Rat(ϕ) and let i ∈ {1, . . . , n}.

Eric Pacuit and Olivier Roy 43

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SLIDE 116

F ⊆ ✷F and w ∈ F ⊆ ✷Rat(ϕ) = {v ∈ W | ∀i Ri(v) ⊆ Rat(ϕ)}

  • Claim. GF∩Rat(ϕ) is post-fixed point of Tϕ

(GF∩Rat(ϕ) ⊆ Tϕ(GF∩Rat(ϕ))). Let w′ ∈ F ∩ Rat(ϕ) and let i ∈ {1, . . . , n}. Since w′ ∈ Rat(ϕ), ϕi(σi(w′), Hi, (GRi(w))−i) holds.

Eric Pacuit and Olivier Roy 43

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SLIDE 117

F ⊆ ✷F and w ∈ F ⊆ ✷Rat(ϕ) = {v ∈ W | ∀i Ri(v) ⊆ Rat(ϕ)}

  • Claim. GF∩Rat(ϕ) is post-fixed point of Tϕ

(GF∩Rat(ϕ) ⊆ Tϕ(GF∩Rat(ϕ))). Let w′ ∈ F ∩ Rat(ϕ) and let i ∈ {1, . . . , n}. Since w′ ∈ Rat(ϕ), ϕi(σi(w′), Hi, (GRi(w))−i) holds. F is evident, so Ri(w′) ⊆ F. We also have Ri(w′) ⊆ Rat(ϕ).

Eric Pacuit and Olivier Roy 43

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SLIDE 118

F ⊆ ✷F and w ∈ F ⊆ ✷Rat(ϕ) = {v ∈ W | ∀i Ri(v) ⊆ Rat(ϕ)}

  • Claim. GF∩Rat(ϕ) is post-fixed point of Tϕ

(GF∩Rat(ϕ) ⊆ Tϕ(GF∩Rat(ϕ))). Let w′ ∈ F ∩ Rat(ϕ) and let i ∈ {1, . . . , n}. Since w′ ∈ Rat(ϕ), ϕi(σi(w′), Hi, (GRi(w))−i) holds. F is evident, so Ri(w′) ⊆ F. We also have Ri(w′) ⊆ Rat(ϕ). Hence, Ri(w′) ⊆ F ∩ Rat(ϕ).

Eric Pacuit and Olivier Roy 43

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SLIDE 119

F ⊆ ✷F and w ∈ F ⊆ ✷Rat(ϕ) = {v ∈ W | ∀i Ri(v) ⊆ Rat(ϕ)}

  • Claim. GF∩Rat(ϕ) is post-fixed point of Tϕ

(GF∩Rat(ϕ) ⊆ Tϕ(GF∩Rat(ϕ))). Let w′ ∈ F ∩ Rat(ϕ) and let i ∈ {1, . . . , n}. Since w′ ∈ Rat(ϕ), ϕi(σi(w′), Hi, (GRi(w))−i) holds. F is evident, so Ri(w′) ⊆ F. We also have Ri(w′) ⊆ Rat(ϕ). Hence, Ri(w′) ⊆ F ∩ Rat(ϕ). This implies (GRi(w′)) ⊆ (GF∩Rat(ϕ))−i, and so by monotonicity of ϕi, ϕi(si, Hi, (GF∩Rat(ϕ))−i) holds.

Eric Pacuit and Olivier Roy 43

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SLIDE 120

F ⊆ ✷F and w ∈ F ⊆ ✷Rat(ϕ) = {v ∈ W | ∀i Ri(v) ⊆ Rat(ϕ)}

  • Claim. GF∩Rat(ϕ) is post-fixed point of Tϕ

(GF∩Rat(ϕ) ⊆ Tϕ(GF∩Rat(ϕ))). Let w′ ∈ F ∩ Rat(ϕ) and let i ∈ {1, . . . , n}. Since w′ ∈ Rat(ϕ), ϕi(σi(w′), Hi, (GRi(w))−i) holds. F is evident, so Ri(w′) ⊆ F. We also have Ri(w′) ⊆ Rat(ϕ). Hence, Ri(w′) ⊆ F ∩ Rat(ϕ). This implies (GRi(w′)) ⊆ (GF∩Rat(ϕ))−i, and so by monotonicity of ϕi, ϕi(si, Hi, (GF∩Rat(ϕ))−i) holds. This means GF∩Rat(ϕ) ⊆ Tϕ(GF∩Rat(ϕ))

Eric Pacuit and Olivier Roy 43

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SLIDE 121

sdi(si, Gi, G−i) is ¬∃s′

i ∈ Gi, ∀s−i ∈ G−iui(s′ i, s−i) > ui(si, s−i)

Eric Pacuit and Olivier Roy 44

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SLIDE 122

sdi(si, Gi, G−i) is ¬∃s′

i ∈ Gi, ∀s−i ∈ G−iui(s′ i, s−i) > ui(si, s−i)

bri(si, Gi, G−i) is ∃µi ∈ Bi(G−i)∀s′

i ∈ Gi, Ui(si, µi) ≥ Ui(s′ i, µi).

Eric Pacuit and Olivier Roy 44

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SLIDE 123

sdi(si, Gi, G−i) is ¬∃s′

i ∈ Gi, ∀s−i ∈ G−iui(s′ i, s−i) > ui(si, s−i)

bri(si, Gi, G−i) is ∃µi ∈ Bi(G−i)∀s′

i ∈ Gi, Ui(si, µi) ≥ Ui(s′ i, µi).

Uϕ(G) = G ′ where G ′

i = {si ∈ Gi | ϕi(si, Gi, G−i)}.

Eric Pacuit and Olivier Roy 44

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SLIDE 124

sdi(si, Gi, G−i) is ¬∃s′

i ∈ Gi, ∀s−i ∈ G−iui(s′ i, s−i) > ui(si, s−i)

bri(si, Gi, G−i) is ∃µi ∈ Bi(G−i)∀s′

i ∈ Gi, Ui(si, µi) ≥ Ui(s′ i, µi).

Uϕ(G) = G ′ where G ′

i = {si ∈ Gi | ϕi(si, Gi, G−i)}.

Note: Uϕ is not monotonic.

Eric Pacuit and Olivier Roy 44

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SLIDE 125
  • Corollary. For all belief models, GRat(br)∩✷∗Rat(br) ⊆ U∞

sd . For all

G, we have Tbr(G) ⊆ Tsd(G) Tsd(G) ⊆ Usd(G) Then, T ∞

sd ⊆ U∞ sd .

Eric Pacuit and Olivier Roy 45

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SLIDE 126
  • Corollary. For all belief models, GRat(br)∩✷∗Rat(br) ⊆ U∞

sd . For all

G, we have Tbr(G) ⊆ Tsd(G) Tsd(G) ⊆ Usd(G) Then, T ∞

sd ⊆ U∞ sd .

  • Fact. Consider two operators T1, T2 on (D, ⊆) such that,

◮ for all G, T1(G) ⊆ T2(G) ◮ T1 is monotonic ◮ T2 is contracting

Then, T ∞

1

⊆ T ∞

2 .

Eric Pacuit and Olivier Roy 45

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SLIDE 127

This analysis does not work for weak dominance...

Eric Pacuit and Olivier Roy 46

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SLIDE 128

Common knowledge of rationality (CKR) in the tree.

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RCK in the tree

Backwards Induction

Invented by Zermelo, Backwards Induction is an iterative algorithm for “solving” and extensive game.

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(1, 0) (2, 3) (1, 5) A (3, 1) (4, 4) B B A

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SLIDE 131

RCK in the tree

(1, 0) (2, 3) (1, 5) A (3, 1) (4, 4) B B A

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SLIDE 132

RCK in the tree

(1, 0) (2, 3) (1, 5) (4, 4) (3, 1) (4, 4) B B A

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SLIDE 133

RCK in the tree

(1, 0) (2, 3) (1, 5) (4, 4) (3, 1) (4, 4) B B A

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RCK in the tree

(1, 0) (2, 3) (1, 5) (4, 4) (3, 1) (4, 4) (2, 3) B A

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SLIDE 135

RCK in the tree

(1, 0) (2, 3) (1, 5) (4, 4) (3, 1) (4, 4) (2, 3) B A

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SLIDE 136

RCK in the tree

(1, 0) (2, 3) (1, 5) (4, 4) (3, 1) (4, 4) (2, 3) (1, 5) A

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(1, 0) (2, 3) (1, 5) (4, 4) (3, 1) (4, 4) (2, 3) (1, 5) A

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SLIDE 138

RCK in the tree

(1, 0) (2, 3) (1, 5) (4, 4) (3, 1) (4, 4) (2, 3) (1, 5) (2, 3)

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SLIDE 139

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BI Puzzle

A B A

(2,1) (1,6) (7,5) (6,6) R1 r R2 D1 d D2

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SLIDE 140

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BI Puzzle

A B A

(2,1) (1,6) (7,5) (6,6) R1 r R2 D1 d D2

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SLIDE 141

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BI Puzzle

A B

(7,5) (2,1) (1,6) (7,5) (6,6) R1 r D1 d

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SLIDE 142

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BI Puzzle

A B

(7,5) (2,1) (1,6) (7,5) (6,6) R1 r D1 d

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BI Puzzle

A

(1,6) (7,5) (2,1) (1,6) (7,5) (6,6) R1 D1

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SLIDE 144

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BI Puzzle

A

(1,6) (7,5) (2,1) (1,6) (7,5) (6,6) R1 D1

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RCK in the tree

BI Puzzle

A

(1,6) (7,5) (2,1) (1,6) (7,5) (6,6) D1

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RCK in the tree

BI Puzzle

A B A

(2,1) (1,6) (7,5) (6,6) R1 r R2 D1 d D2

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RCK in the tree

But what if Bob has to move?

A B A

(2,1) (1,6) (7,5) (6,6) R1 r R2 D1 d D2

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But what if Bob has to move?

A B A

(2,1) (1,6) (7,5) (6,6) R1 r R2 D1 d D2 What should Bob thinks of Ann?

◮ Either she doesn’t believe that he is rational and that he

believes that she would choose R2.

◮ Or Ann made a “mistake” (= irrational move) at the first

turn. Either way, rationality is not “common knowledge”.

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  • R. Aumann. Backwards induction and common knowledge of rationality. Games

and Economic Behavior, 8, pgs. 6 - 19, 1995.

  • R. Stalnaker. Knowledge, belief and counterfactual reasoning in games. Eco-

nomics and Philosophy, 12, pgs. 133 - 163, 1996.

  • J. Halpern. Substantive Rationality and Backward Induction. Games and Eco-

nomic Behavior, 37, pp. 425-435, 1998.

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Models of Extensive Games

Let Γ be a non-degenerate extensive game with perfect

  • information. Let Γi be the set of nodes controlled by player i.

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Models of Extensive Games

Let Γ be a non-degenerate extensive game with perfect

  • information. Let Γi be the set of nodes controlled by player i.

A strategy profile σ describes the choice for each player i at all vertices where i can choose.

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Models of Extensive Games

Let Γ be a non-degenerate extensive game with perfect

  • information. Let Γi be the set of nodes controlled by player i.

A strategy profile σ describes the choice for each player i at all vertices where i can choose. Given a vertex v in Γ and strategy profile σ, σ specifies a unique path from v to an end-node.

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SLIDE 153

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Models of Extensive Games

Let Γ be a non-degenerate extensive game with perfect

  • information. Let Γi be the set of nodes controlled by player i.

A strategy profile σ describes the choice for each player i at all vertices where i can choose. Given a vertex v in Γ and strategy profile σ, σ specifies a unique path from v to an end-node. M(Γ) = W , ∼i, σ where σ : W → Strat(Γ) and ∼i⊆ W × W is an equivalence relation.

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SLIDE 154

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Models of Extensive Games

Let Γ be a non-degenerate extensive game with perfect

  • information. Let Γi be the set of nodes controlled by player i.

A strategy profile σ describes the choice for each player i at all vertices where i can choose. Given a vertex v in Γ and strategy profile σ, σ specifies a unique path from v to an end-node. M(Γ) = W , ∼i, σ where σ : W → Strat(Γ) and ∼i⊆ W × W is an equivalence relation. If σ(w) = σ, then σi(w) = σi and σ−i(w) = σ−i

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Models of Extensive Games

Let Γ be a non-degenerate extensive game with perfect

  • information. Let Γi be the set of nodes controlled by player i.

A strategy profile σ describes the choice for each player i at all vertices where i can choose. Given a vertex v in Γ and strategy profile σ, σ specifies a unique path from v to an end-node. M(Γ) = W , ∼i, σ where σ : W → Strat(Γ) and ∼i⊆ W × W is an equivalence relation. If σ(w) = σ, then σi(w) = σi and σ−i(w) = σ−i (A1) If w ∼i w′ then σi(w) = σi(w′).

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Rationality

hv

i (σ) denote “i’s payoff if σ is followed from node v”

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Rationality

hv

i (σ) denote “i’s payoff if σ is followed from node v”

i is rational at v in w provided for all strategies si = σi(w), hv

i (σ(w′)) ≥ hv i ((σ−i(w′), si)) for some w′ ∈ [w]i.

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Substantive Rationality

i is substantively rational in state w if i is rational at a vertex v in w of every vertex in v ∈ Γi

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Stalnaker Rationality

For every vertex v ∈ Γi, if i were to actually reach v, then what he would do in that case would be rational.

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Stalnaker Rationality

For every vertex v ∈ Γi, if i were to actually reach v, then what he would do in that case would be rational. f : W × Γi → W , f (w, v) = w′, then w′ is the “closest state to w where the vertex v is reached.

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Stalnaker Rationality

For every vertex v ∈ Γi, if i were to actually reach v, then what he would do in that case would be rational. f : W × Γi → W , f (w, v) = w′, then w′ is the “closest state to w where the vertex v is reached. (F1) v is reached in f (w, v) (i.e., v is on the path determined by σ(f (w, v))) (F2) If v is reached in w, then f (w, v) = w (F3) σ(f (w, v)) and σ(w) agree on the subtree of Γ below v

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RCK in the tree

A B A (3, 3) (2, 2) (1, 1) (0, 0) a a a d d d s1 = (da, d), s2 = (aa, d), s3 = (ad, d), s4 = (aa, a), s5 = (ad, a) W = {w1, w2, w3, w4, w5} with σ(wi) = si [wi]A = {wi} for i = 1, 2, 3, 4, 5 [wi]B = {wi} for i = 1, 4, 5 and [w2]B = [w3]B = {w2, w3}

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A B A (3, 3) (2, 2) (1, 1) (0, 0) a a a d d d s1 = (da, d), s2 = (aa, d), s3 = (ad, d), s4 = (aa, a), s5 = (ad, a) W = {w1, w2, w3, w4, w5} with σ(wi) = si [wi]A = {wi} for i = 1, 2, 3, 4, 5 [wi]B = {wi} for i = 1, 4, 5 and [w2]B = [w3]B = {w2, w3}

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RCK in the tree

A B A (3, 3) (2, 2) (1, 1) (0, 0) a a a d d d s1 = (da, d), s2 = (aa, d), s3 = (ad, d), s4 = (aa, a), s5 = (ad, a)

◮ W = {w1, w2, w3, w4, w5} with σ(wi) = si ◮ [wi]A = {wi} for i = 1, 2, 3, 4, 5 ◮ [wi]B = {wi} for i = 1, 4, 5 and [w2]B = [w3]B = {w2, w3}

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A B A (3, 3) (2, 2) (1, 1) (0, 0) a a a d d d s1 = (da, d), s2 = (aa, d), s3 = (ad, d), s4 = (aa, a), s5 = (ad, a) w1 w2 w3 w4 w5 It is common knowledge at w1 that if vertex v2 were reached, Bob would play down.

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A B v2 A (3, 3) (2, 2) (1, 1) (0, 0) a a a d d d s1 = (da, d), s2 = (aa, d), s3 = (ad, d), s4 = (aa, a), s5 = (ad, a) w1 w2 w3 w4 w5 It is common knowledge at w1 that if vertex v2 were reached, Bob would play down.

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A B v2 A (3, 3) (2, 2) (1, 1) (0, 0) a a a d d d s1 = (da, d), s2 = (aa, d), s3 = (ad, d), s4 = (aa, a), s5 = (ad, a) w1 w2 w3 w4 w5 Bob is not rational at v2 in w1 add asdf a def add fa sdf asdfa adds asdf asdf add fa sdf asdf adds f asfd

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A B v2 A (3, 3) (2, 2) (1, 1) (0, 0) a a a d d d s1 = (da, d), s2 = (aa, d), s3 = (ad, d), s4 = (aa, a), s5 = (ad, a) w1 w2 w3 w4 w5 Bob is rational at v2 in w2 add asdf a def add fa sdf asdfa adds asdf asdf add fa sdf asdf adds f asfd

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A B v2 A v3 (3, 3) (2, 2) (1, 1) (0, 0) a a a d d d s1 = (da, d), s2 = (aa, d), s3 = (ad, d), s4 = (aa, a), s5 = (ad, a) w1 w2 w3 w4 w5 Note that f (w1, v2) = w2 and f (w1, v3) = w4, so there is common knowledge of S-rationality at w1.

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Aumann’s Theorem: If Γ is a non-degenerate game of perfect information, then in all models of Γ, we have C(A − Rat) ⊆ BI Stalnaker’s Theorem: There exists a non-degenerate game Γ of perfect information and an extended model of Γ in which the selection function satisfies F1-F3 such that C(S − Rat) ⊆ BI.

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Aumann’s Theorem: If Γ is a non-degenerate game of perfect information, then in all models of Γ, we have C(A − Rat) ⊆ BI Stalnaker’s Theorem: There exists a non-degenerate game Γ of perfect information and an extended model of Γ in which the selection function satisfies F1-F3 such that C(S − Rat) ⊆ BI. Revising beliefs during play: “Although it is common knowledge that Ann would play across if v3 were reached, if Ann were to play across at v1, Bob would consider it possible that Ann would play down at v3”

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  • F4. For all players i and vertices v, if w′ ∈ [f (w, v)]i then there

exists a state w′′ ∈ [w]i such that σ(w′) and σ(w′′) agree on the subtree of Γ below v. Theorem (Halpern). If Γ is a non-degenerate game of perfect information, then for every extended model of Γ in which the selection function satisfies F1-F4, we have C(S − Rat) ⊆ BI. Moreover, there is an extend model of Γ in which the selection function satisfies F1-F4.

  • J. Halpern. Substantive Rationality and Backward Induction. Games and Eco-

nomic Behavior, 37, pp. 425-435, 1998.

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Proof of Halpern’s Theorem

(1, 0) (2, 3) (1, 5) A (3, 1) (4, 4) B B A

◮ Suppose w ∈ C(S − Rat). We show by induction on k that

for all w′ reachable from w by a finite path along the union of the relations ∼i, if v is at most k moves away from a leaf, then σi(w) is i’s backward induction move at w′.

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Proof of Halpern’s Theorem

(1, 0) (2, 3) (1, 5) A (3, 1) (4, 4) B B A

◮ Base case: we are at most 1 move away from a leaf. Suppose

w ∈ C(S − Rat). Take any w′ reachable from w.

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Proof of Halpern’s Theorem

(1, 0) (2, 3) (1, 5) A (3, 1) (4, 4) B B A

◮ Base case: we are at most 1 move away from a leaf. Suppose

w ∈ C(S − Rat). Take any w′ reachable from w. Since w ∈ C(S − Rat), we know that w′ ∈ C(S − Rat).

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Proof of Halpern’s Theorem

(1, 0) (2, 3) (1, 5) A (3, 1) (4, 4) B B A

◮ Base case: we are at most 1 move away from a leaf. Suppose

w ∈ C(S − Rat). Take any w′ reachable from w. Since w ∈ C(S − Rat), we know that w′ ∈ C(S − Rat). So i must play her BI move at f (w′, v).

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Proof of Halpern’s Theorem

(1, 0) (2, 3) (1, 5) A (3, 1) (4, 4) B B A

◮ Base case: we are at most 1 move away from a leaf. Suppose

w ∈ C(S − Rat). Take any w′ reachable from w. Since w ∈ C(S − Rat), we know that w′ ∈ C(S − Rat). So i must play her BI move at f (w′, v). But then by F3 this must also be the case at (w′, v).

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Proof of Halpern’s Theorem

(1, 0) (2, 3) (1, 5) A (3, 1) (4, 4) B B A

◮ Base case: we are at most 1 move away from a leaf. Suppose

w ∈ C(S − Rat). Take any w′ reachable from w. Since w ∈ C(S − Rat), we know that w′ ∈ C(S − Rat). So i must play her BI move at f (w′, v). But then by F3 this must also be the case at (w′, v).

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Proof of Halpern’s Theorem

(1, 0) (2, 3) (1, 5) A (3, 1) (4, 4) B B A

w’ w′ w′′

∗ ◮ Suppose w ∈ C(S − Rat). Take any w′ reachable from w.

Assume, towards contradiction, that σ(w)i(v) = a is not the BI move for player i. By the same argument as before, i must be rational at w′′ = f (w′, v). Furthermore, by F3 all players play according to the BI solution after v at (w′, v). Furthermore, by IH, at all vertices below v the players must play their BI moves.

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Proof of Halpern’s Theorem

(1, 0) (2, 3) (1, 5) A (3, 1) (4, 4) B B A

w’ w′ w′′ w3

∗

f

◮ Induction step. Suppose w ∈ C(S − Rat). Take any w′

reachable from w. Assume, towards contradiction, that σ(w)i(v) = a is not the BI move for player i. Since w is also in C(S − Rat), we know by definition i must be rational at w′′ = f (w′, v). But then, by F3 and our IH, all players play according to the BI solution after v at w′′.

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Proof of Halpern’s Theorem

(1, 0) (2, 3) (1, 5) A (3, 1) (4, 4) B B A

w’ w′ w′′ w3

∗

f i

◮ i’s rationality at w′′ means, in particular, that there is a

w3 ∈ [w′′]i such that hv

i (σi(w′′), σ−i(w3)) ≥ hv i ((bii, σ−i(w3)))

for bii i’s backward induction strategy.

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Proof of Halpern’s Theorem

(1, 0) (2, 3) (1, 5) A (3, 1) (4, 4) B B A

w’ w′ w′′ w3 w4

∗

f i i

◮ But then by F4 there must exists w4 ∈ [w]i such that σ(w4)

σ(w3) at the same in the sub-tree starting at v.

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Proof of Halpern’s Theorem

(1, 0) (2, 3) (1, 5) A (3, 1) (4, 4) B B A

w’ w′ w′′ w3 w4

∗

f i i

◮ But then by F4 there must exists w4 ∈ [w]i such that σ(w4)

σ(w3) at the same in the sub-tree starting at v. Since w4 is reachable from w, in that state all players play according to the backward induction after v, and so this is also true of w3.

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Proof of Halpern’s Theorem

(1, 0) (2, 3) (1, 5) A (3, 1) (4, 4) B B A

w’ w′ w′′ w3 w4

∗

f i i

◮ But then by F4 there must exists w4 ∈ [w]i such that σ(w4)

σ(w3) at the same in the sub-tree starting at v. Since w4 is reachable from w, in that state all players play according to the backward induction after v, and so this is also true of w3. But then since the game is non-degenerate, playing something else than bii must make i strictly worst off at that state, a contradiction.

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