Logic and Arti cial Intelligenceweb.pacuit.io/classes/logicai-cmu/logai-lec25.pdf · When are two games the same? I Whose point-of-view? (players, modelers) I Game-theoretic analysis

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/∼[email protected]

December 6, 2011

Logic and Artificial Intelligence 1/27

http://ai.stanford.edu/~epacuit

mailto:[email protected]

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


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


The same decision problem

A

A

o1 o2 o3

D1

A

o1 o2 o3

D2


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)


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


A

A

B

o1 o2 o3 o4 o5 o6

A A

A

o1 o2

B

A AAA

o3 o4 o5 o6

Coalescing of moves


A

A

B

o1 o2 o3 o4 o5 o6

A A

A

B

o1 o2 o3 o4 o5 o6

A A A

Inflation/deflation


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


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.


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).


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.


Games as Processes






Games as Processes






Games as Processes






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. )


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. )


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 .


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


Game Algebra


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


Game Algebra


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


Game Algebra


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


Game Algebra


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


Game Algebra


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


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).


I Reasoning about strategic/extensive games

I Reasoning with strategic/extensive games

I Reasoning in strategic/extensive games


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.


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.


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.


BI Puzzle

A B A

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

(6,6)R1 r R2

D1 d D2


BI Puzzle

A B A

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

(6,6)R1 r R2

D1 d D2


BI Puzzle

A B (7,5)

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

(6,6)R1 r

D1 d


BI Puzzle

A B (7,5)

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

(6,6)R1 r

D1 d


BI Puzzle

A (1,6) (7,5)

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

(6,6)R1

D1


BI Puzzle

A (1,6) (7,5)

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

(6,6)R1

D1


BI Puzzle

A (1,6) (7,5)

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

(6,6)

D1


BI Puzzle

A B A

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

(6,6)R1 r R2

D1 d D2


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?


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?


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.


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

′).









′).









′).









′).









′).









′).


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 .


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 .


Substantive Rationality

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


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
















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}


A B A(3, 3)

(2, 2) (1, 1) (0, 0)

a a 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}


A B A(3, 3)

(2, 2) (1, 1) (0, 0)

a a 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}


A B A(3, 3)

(2, 2) (1, 1) (0, 0)

a a a


w1 w2 w3

w4 w5

It is common knowledge at w1 that if vertex v2 were reached,Bob would play down.


A B

v2

A(3, 3)

(2, 2) (1, 1) (0, 0)

a a a


w1 w2 w3

w4 w5

It is common knowledge at w1 that if vertex v2 were reached,Bob would play down.


A B

v2

A(3, 3)

(2, 2) (1, 1) (0, 0)

a a 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


A B

v2

A(3, 3)

(2, 2) (1, 1) (0, 0)

a a 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


A B

v2

A

v3(3, 3)

(2, 2) (1, 1) (0, 0)

a a 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.


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”


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”


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.


Documents

Logic and Arti cial Intelligenceweb.pacuit.io/classes/logicai-cmu/logai-lec25.pdf · When are two games the same? I Whose point-of-view? (players, modelers) I Game-theoretic analysis