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Logic and Artificial IntelligenceLecture 25

Eric Pacuit

Currently Visiting the Center for Formal Epistemology, CMU

Center for Logic and Philosophy of ScienceTilburg University

ai.stanford.edu/∼epacuite.j.pacuit@uvt.nl

December 6, 2011

Logic and Artificial Intelligence 1/27

When are two games the same?

I Whose point-of-view? (players, modelers)

I Game-theoretic analysis should not depend on “irrelevant”mathematical details

I Different perspectives: transformations, structural, agent

Logic and Artificial Intelligence 2/27

When are two games the same?

I Whose point-of-view? (players, modelers)

I Game-theoretic analysis should not depend on “irrelevant”mathematical details

I Different perspectives: transformations, structural, agent

Logic and Artificial Intelligence 2/27

The same decision problem

A

A

o1 o2 o3

D1

A

o1 o2 o3

D2

Logic and Artificial Intelligence 3/27

Thompson Transformations

Game-theoretic analysis should not depend on “irrelevant” featuresof the (mathematical) description of the game.

F. B. Thompson. Equivalence of Games in Extensive Form. Classics in GameTheory, pgs 36 - 45, 1952.

(Osborne and Rubinstein, pgs. 203 - 212)

Logic and Artificial Intelligence 4/27

A

A

B

o1 o2 o3 o4 o5 o6

A A

A

B

A A

o1 o2 o1 o2

BBB

A AAAAA

o3 o4 o5 o6

Addition of Superfluous Move

Logic and Artificial Intelligence 5/27

A

A

B

o1 o2 o3 o4 o5 o6

A A

A

o1 o2

B

A AAA

o3 o4 o5 o6

Coalescing of moves

Logic and Artificial Intelligence 6/27

A

A

B

o1 o2 o3 o4 o5 o6

A A

A

B

o1 o2 o3 o4 o5 o6

A A A

Inflation/deflation

Logic and Artificial Intelligence 7/27

A

A

B

o1 o2 o3 o4 o5 o6

A A

A

A

o1 o2 o3 o5 o4 o6

A B B

Interchange of moves

Logic and Artificial Intelligence 8/27

Theorem (Thompson) Each of the previous transformationspreserves the reduced strategic form of the game. In finiteextensive games (without uncertainty between subhistories), if anytwo games have the same reduced normal form then one can beobtained from the other by a sequence of the four transformations.

Logic and Artificial Intelligence 9/27

Other transformations/game forms

Kohlberg and Mertens. On Strategic Stability of Equilibria. Econometrica(1986).

Elmes and Reny. On The Strategic Equivalence of Extensive Form Games. Jour-nal of Economic Theory (1994).

G. Bonanno. Set-Theoretic Equivalence of Extensive-Form Games. IJGT (1992).

Logic and Artificial Intelligence 10/27

Games as Processes

I When are two processes the same?

I Extensive games are natural process models which supportmany familiar modal logics such as bisimulation.

I From this point-of-view, “When are two games the same?”goes tandem with asking “what are appropriate languages forgames”

J. van Benthem. Extensive Games as Process Models. IJGT, 2001.

Logic and Artificial Intelligence 11/27

Games as Processes

I When are two processes the same?

I Extensive games are natural process models which supportmany familiar modal logics such as bisimulation.

I From this point-of-view, “When are two games the same?”goes tandem with asking “what are appropriate languages forgames”

J. van Benthem. Extensive Games as Process Models. IJGT, 2001.

Logic and Artificial Intelligence 11/27

Games as Processes

I When are two processes the same?

I Extensive games are natural process models which supportmany familiar modal logics such as bisimulation.

I From this point-of-view, “When are two games the same?”goes tandem with asking “what are appropriate languages forgames”

J. van Benthem. Extensive Games as Process Models. IJGT, 2001.

Logic and Artificial Intelligence 11/27

Games as Processes

I When are two processes the same?

I Extensive games are natural process models which supportmany familiar modal logics such as bisimulation.

I From this point-of-view, “When are two games the same?”goes tandem with asking “what are appropriate languages forgames”

J. van Benthem. Extensive Games as Process Models. IJGT, 2001.

Logic and Artificial Intelligence 11/27

Game Algebra

Definition Two games γ1 and γ2 are equivalent providedEγ1 = Eγ2 in all models (iff 〈γ1〉p ↔ 〈γ2〉p is valid for a p whichoccurs neither in γ1 nor in γ2. )

Logic and Artificial Intelligence 12/27

Game Algebra

Definition Two games γ1 and γ2 are equivalent providedEγ1 = Eγ2 in all models (iff 〈γ1〉p ↔ 〈γ2〉p is valid for a p whichoccurs neither in γ1 nor in γ2. )

Logic and Artificial Intelligence 12/27

Game Algebra

Game Boards: Given a set of states or positions B, for each gameg and each player i there is an associated relation E i

g ⊆ B × 2B :

pE igT holds if in position p, i can force that the outcome of g will

be a position in T .

I (monotonicity) if pE igT and T ⊆ U then pE i

gU

I (consistency) if pE igT then not pE 1−i

g (B − T )

Given a game board (a set B with relations E ig for each game and

player), we say that two games g , h (g ≈ h) are equivalent ifE ig = E i

h for each i .

Logic and Artificial Intelligence 12/27

Game Algebra

1. Standard Laws of Boolean Algebras

2. (x ; y); z ≈ x ; (y ; z)

3. (x ∨ y); z ≈ (x ; z) ∨ (y ; z), (x ∧ y); z ≈ (x ; z) ∧ (y ; z)

4. −x ;−y ≈ −(x ; y)

5. y � z ⇒ x ; y � x ; z

Logic and Artificial Intelligence 13/27

Game Algebra

1. Standard Laws of Boolean Algebras

2. (x ; y); z ≈ x ; (y ; z)

3. (x ∨ y); z ≈ (x ; z) ∨ (y ; z), (x ∧ y); z ≈ (x ; z) ∧ (y ; z)

4. −x ;−y ≈ −(x ; y)

5. y � z ⇒ x ; y � x ; z

Logic and Artificial Intelligence 13/27

Game Algebra

1. Standard Laws of Boolean Algebras

2. (x ; y); z ≈ x ; (y ; z)

3. (x ∨ y); z ≈ (x ; z) ∨ (y ; z), (x ∧ y); z ≈ (x ; z) ∧ (y ; z)

4. −x ;−y ≈ −(x ; y)

5. y � z ⇒ x ; y � x ; z

Logic and Artificial Intelligence 13/27

Game Algebra

1. Standard Laws of Boolean Algebras

2. (x ; y); z ≈ x ; (y ; z)

3. (x ∨ y); z ≈ (x ; z) ∨ (y ; z), (x ∧ y); z ≈ (x ; z) ∧ (y ; z)

4. −x ;−y ≈ −(x ; y)

5. y � z ⇒ x ; y � x ; z

Logic and Artificial Intelligence 13/27

Game Algebra

1. Standard Laws of Boolean Algebras

2. (x ; y); z ≈ x ; (y ; z)

3. (x ∨ y); z ≈ (x ; z) ∨ (y ; z), (x ∧ y); z ≈ (x ; z) ∧ (y ; z)

4. −x ;−y ≈ −(x ; y)

5. y � z ⇒ x ; y � x ; z

Logic and Artificial Intelligence 13/27

Game Algebra

1. Standard Laws of Boolean Algebras

2. (x ; y); z ≈ x ; (y ; z)

3. (x ∨ y); z ≈ (x ; z) ∨ (y ; z), (x ∧ y); z ≈ (x ; z) ∧ (y ; z)

4. −x ;−y ≈ −(x ; y)

5. y � z ⇒ x ; y � x ; z

Logic and Artificial Intelligence 13/27

Theorem Sound and complete axiomatizations of (iteration free)game algebra

Y. Venema. Representing Game Algebras. Studia Logica 75 (2003).

V. Goranko. The Basic Algebra of Game Equivalences. Studia Logica 75 (2003).

Logic and Artificial Intelligence 14/27

I Reasoning about strategic/extensive games

I Reasoning with strategic/extensive games

I Reasoning in strategic/extensive games

Logic and Artificial Intelligence 15/27

Rationality in Interaction

What does it mean to be rational when the outcome of one’saction depends upon the actions of other people and everyone istrying to guess what the others will do?

In social interaction, rationality has to be enriched with furtherassumptions about individuals’ mutual knowledge and beliefs,but these assumptions are not without consequence.

C. Bicchieri. Rationality and Game Theory. Chapter 10 in Handbook of Ratio-nality.

Logic and Artificial Intelligence 16/27

Rationality in Interaction

What does it mean to be rational when the outcome of one’saction depends upon the actions of other people and everyone istrying to guess what the others will do?

In social interaction, rationality has to be enriched with furtherassumptions about individuals’ mutual knowledge and beliefs,but these assumptions are not without consequence.

C. Bicchieri. Rationality and Game Theory. Chapter 10 in Handbook of Ratio-nality.

Logic and Artificial Intelligence 16/27

Key Assumptions

CK1 The structure of the game, including players’ strategy sets andpayoff functions, is common knowledge among the players.

CK2 The players are rational (i.e., they are expected utilitymaximizers) and this is common knowledge.

Logic and Artificial Intelligence 17/27

BI Puzzle

A B A

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

(6,6)R1 r R2

D1 d D2

Logic and Artificial Intelligence 18/27

BI Puzzle

A B A

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

(6,6)R1 r R2

D1 d D2

Logic and Artificial Intelligence 18/27

BI Puzzle

A B (7,5)

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

(6,6)R1 r

D1 d

Logic and Artificial Intelligence 18/27

BI Puzzle

A B (7,5)

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

(6,6)R1 r

D1 d

Logic and Artificial Intelligence 18/27

BI Puzzle

A (1,6) (7,5)

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

(6,6)R1

D1

Logic and Artificial Intelligence 18/27

BI Puzzle

A (1,6) (7,5)

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

(6,6)R1

D1

Logic and Artificial Intelligence 18/27

BI Puzzle

A (1,6) (7,5)

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

(6,6)

D1

Logic and Artificial Intelligence 18/27

BI Puzzle

A B A

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

(6,6)R1 r R2

D1 d D2

Logic and Artificial Intelligence 18/27

But what if...

A B A

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

(6,6)R1 r R2

D1 d D2

I Are the players irrational?

I What argument leads to the BI solution?

Logic and Artificial Intelligence 19/27

But what if...

A B A

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

(6,6)R1 r R2

D1 d D2

I Are the players irrational?

I What argument leads to the BI solution?

Logic and Artificial Intelligence 19/27

R. Aumann. Backwards induction and common knowledge of rationality. Gamesand 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.

Logic and Artificial Intelligence 20/27

Models of Extensive Games

Let Γ be a non-degenerate extensive game with perfectinformation. Let Γi be the set of nodes controlled by player i .

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

Given a vertex v in Γ and strategy profile σ, σ specifies a uniquepath from v to an end-node.

M(Γ) = 〈W ,∼i , σ〉 where σ : W → Strat(Γ) and ∼i⊆W ×W isan 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

′).

Logic and Artificial Intelligence 21/27

Models of Extensive Games

Let Γ be a non-degenerate extensive game with perfectinformation. Let Γi be the set of nodes controlled by player i .

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

Given a vertex v in Γ and strategy profile σ, σ specifies a uniquepath from v to an end-node.

M(Γ) = 〈W ,∼i , σ〉 where σ : W → Strat(Γ) and ∼i⊆W ×W isan 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

′).

Logic and Artificial Intelligence 21/27

Models of Extensive Games

Let Γ be a non-degenerate extensive game with perfectinformation. Let Γi be the set of nodes controlled by player i .

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

Given a vertex v in Γ and strategy profile σ, σ specifies a uniquepath from v to an end-node.

M(Γ) = 〈W ,∼i , σ〉 where σ : W → Strat(Γ) and ∼i⊆W ×W isan 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

′).

Logic and Artificial Intelligence 21/27

Models of Extensive Games

Let Γ be a non-degenerate extensive game with perfectinformation. Let Γi be the set of nodes controlled by player i .

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

Given a vertex v in Γ and strategy profile σ, σ specifies a uniquepath from v to an end-node.

M(Γ) = 〈W ,∼i , σ〉 where σ : W → Strat(Γ) and ∼i⊆W ×W isan 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

′).

Logic and Artificial Intelligence 21/27

Models of Extensive Games

Let Γ be a non-degenerate extensive game with perfectinformation. Let Γi be the set of nodes controlled by player i .

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

Given a vertex v in Γ and strategy profile σ, σ specifies a uniquepath from v to an end-node.

M(Γ) = 〈W ,∼i , σ〉 where σ : W → Strat(Γ) and ∼i⊆W ×W isan 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

′).

Logic and Artificial Intelligence 21/27

Models of Extensive Games

Let Γ be a non-degenerate extensive game with perfectinformation. Let Γi be the set of nodes controlled by player i .

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

Given a vertex v in Γ and strategy profile σ, σ specifies a uniquepath from v to an end-node.

M(Γ) = 〈W ,∼i , σ〉 where σ : W → Strat(Γ) and ∼i⊆W ×W isan 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

′).

Logic and Artificial Intelligence 21/27

Rationality

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

i is rational at v in w provided for all strategies si 6= σi (w),hvi (σ(w ′)) ≥ hvi ((σ−i (w

′), si )) for some w ′ ∈ [w ]i .

Logic and Artificial Intelligence 22/27

Rationality

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

i is rational at v in w provided for all strategies si 6= σi (w),hvi (σ(w ′)) ≥ hvi ((σ−i (w

′), si )) for some w ′ ∈ [w ]i .

Logic and Artificial Intelligence 22/27

Substantive Rationality

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

Logic and Artificial Intelligence 23/27

Stalnaker Rationality

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

f : W × Γi →W , f (w , v) = w ′, then w ′ is the “closest state to wwhere the vertex v is reached.

(F1) v is reached in f (w , v) (i.e., v is on the path determined byσ(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

Logic and Artificial Intelligence 24/27

Stalnaker Rationality

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

f : W × Γi →W , f (w , v) = w ′, then w ′ is the “closest state to wwhere the vertex v is reached.

(F1) v is reached in f (w , v) (i.e., v is on the path determined byσ(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

Logic and Artificial Intelligence 24/27

Stalnaker Rationality

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

f : W × Γi →W , f (w , v) = w ′, then w ′ is the “closest state to wwhere the vertex v is reached.

(F1) v is reached in f (w , v) (i.e., v is on the path determined byσ(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

Logic and Artificial Intelligence 24/27

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 ) = s i

[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}

Logic and Artificial Intelligence 25/27

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 ) = s i

[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}

Logic and Artificial Intelligence 25/27

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)

I W = {w1,w2,w3,w4,w5} with σ(wi ) = s i

I [wi ]A = {wi} for i = 1, 2, 3, 4, 5

I [wi ]B = {wi} for i = 1, 4, 5 and [w2]B = [w3]B = {w2,w3}

Logic and Artificial Intelligence 25/27

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.

Logic and Artificial Intelligence 25/27

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.

Logic and Artificial Intelligence 25/27

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 addsasdf asdf add fa sdf asdf adds f asfd

Logic and Artificial Intelligence 25/27

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 addsasdf asdf add fa sdf asdf adds f asfd

Logic and Artificial Intelligence 25/27

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 commonknowledge of S-rationality at w1.

Logic and Artificial Intelligence 25/27

Aumann’s Theorem: If Γ is a non-degenerate game of perfectinformation, then in all models of Γ, we have C (A− Rat) ⊆ BI

Stalnaker’s Theorem: There exists a non-degenerate game Γ ofperfect information and an extended model of Γ in which theselection function satisfies F1-F3 such that C (S − Rat) 6⊆ BI .

Revising beliefs during play:

“Although it is common knowledge that Ann would play across ifv3 were reached, if Ann were to play across at v1, Bob wouldconsider it possible that Ann would play down at v3”

Logic and Artificial Intelligence 26/27

Aumann’s Theorem: If Γ is a non-degenerate game of perfectinformation, then in all models of Γ, we have C (A− Rat) ⊆ BI

Stalnaker’s Theorem: There exists a non-degenerate game Γ ofperfect information and an extended model of Γ in which theselection function satisfies F1-F3 such that C (S − Rat) 6⊆ BI .

Revising beliefs during play:

“Although it is common knowledge that Ann would play across ifv3 were reached, if Ann were to play across at v1, Bob wouldconsider it possible that Ann would play down at v3”

Logic and Artificial Intelligence 26/27

F4. For all players i and vertices v , if w ′ ∈ [f (w , v)]i then thereexists a state w ′′ ∈ [w ]i such that σ(w ′) and σ(w ′′) agree on thesubtree of Γ below v .

Theorem (Halpern). If Γ is a non-degenerate game of perfectinformation, then for every extended model of Γ in which theselection function satisfies F1-F4, we have C (S − Rat) ⊆ BI .Moreover, there is an extend model of Γ in which the selectionfunction satisfies F1-F4.

J. Halpern. Substantive Rationality and Backward Induction. Games and Eco-nomic Behavior, 37, pp. 425-435, 1998.

Logic and Artificial Intelligence 27/27

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