Game Theoretic Pragmatics - uni-tuebingen.de

Game Theoretic PragmaticsSession 9: Pragmatic Reasoning about Unawareness

Michael Franke, Roland Muhlenbernd & Jason Quinley

Seminar fur SprachwissenschaftEberhard Karls Universitat Tubingen

Course Overview (partly tentative)

date content

21-4 Gricean Pragmatics & Decision Theory rm, mf

28-4∗ Relevance & Implicatures rm, mf

05-5 Questions and Decision Problems mf

12-5∗ Introduction to Game Theory mf

19-5 Game Theory in Pragmatics rm, mf

26-5 Pentecost — no class

02-6∗ Neo-Gricean Pragmatics rm

09-6 ibr model 1 rm, mf

16-6∗ ibr model 2 rm, mf

23-6 Pragmatic Reasoning about Unawareness mf

30-6∗ Language Learning in Network Games rm

07-7 Politeness & the Handicap Principle jq

14-7 exam

∗ homework dates (due 1 week after)

Unawareness Games with Unawareness IBR with Unawareness Examples Enfin

Today’s Session

1 Unawareness

2 Games with Unawareness

3 IBR with Unawareness

4 Examples

3 / 35


Subjective Conceptualizations of a Strategic Situation

• traditional game theory:• modeller’s game = agents’ conceptualization thereof

• games with unawareness:• allow different/diverging conceptualizations of a game

• nb: unawareness 6= probability-zero belief• strategic impact of unawareness

Unawareness

• agent is unaware of p iff she cannot conceptualize it• no explicit beliefs about p

• unawareness may originate from:• (principled) lack of computing resources• (deliberate) economy of representation• lack of conceptual understanding• inattentiveness to contingencies

• representing unawareness:• partial representations (e.g. Heifetz et al. 2006, 2008; Li 2009)• subjective languages (e.g. Fagin and Halpern 1988; Halpern 2001)

4 / 35


Unawareness & Language (de Jager 2009)

• main contribution: semantics/pragmatics of modals and conditionals,e.g.:

• relevance of “might ϕ”• Sobel-sequences

• focus on single-agent perspective in decision problems

Pragmatic Reasoning about Unawareness

• where is language use and interpretation sensitive todiverging/impoverished context representations?

• how can we model these representations?

• how do interlocutors reason about unawareness?

5 / 35


Previous Models of Unawareness in Games

• functions from choice points to choice points (in poss. different games)• Halpern and Rego (2006), Heifetz et al. (2010)

• infinite collection of finite sequences of “views”• Feinberg (2009)

Present Approach

• similar to game models by Stalnaker (1994, 1998)• use structures borrowed from modal logic

• possible worlds & accessibility relations

• define partial representations separately• pruning: neglect, omission, oversight• lumping: indistinguishability

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Example ((Un-)awareness of Others’ (Un-)awareness)

i

ja

〈2, 2〉

c

〈3, 1〉

d

a

jb

〈1, 0〉

e

〈0, 1〉

f

b

w0 i

ja

〈2, 2〉

c

〈3, 1〉

d

a

jb

〈.2, .8〉

{e, f}

b

w1

i

ja

〈2, 2〉

c

〈3, 1〉

d

a

jb

〈1, 0〉

e

〈0, 1〉

f

b

w2 i

ja

〈2, 2〉

c

〈3, 1〉

d

a

w3

jb

ja

i

7 / 35


Definition (Dynamic Game with Imperfect Information)

A dynamic game with imperfect information is a structure

Γ =⟨H,<, N, {Ai}i∈N , P, {ui}i∈N , Pr, {Vi}i∈N

⟩where:

• 〈H,<〉 is a game tree,

•⟨N, {Ai}i∈N

⟩are labels,

•⟨P, A, {ui}i∈N , Pr, {Vi}i∈N

⟩is the labeling.

h0

h1

z1

<

z2

<

<

h2

z3

<

z4

<

<

game tree i

j

〈2, 2〉

c

〈3, 1〉

d

a

j

〈1, 0〉

e

〈0, 1〉

f

b

ornamented tree

apply labels

8 / 35


Definition (Pruning)

Γ′ is a pruning of Γ, Γ′ vp Γ, if:

• H′ ⊆ H and <′=< �H′

• labels and labeling restricted in the obvious way

• u′i(z′) = ui(z) for some z ∈ Z such that z′ ≤ z.

i

j

〈3, 1〉

d

a

〈1, 0〉

b

i

j

〈2, 2〉

c

〈3, 1〉

d

a

j

〈1, 0〉

e

〈0, 1〉

f

bwp

• nb: vp is a partial order

9 / 35


Definition (Lumping)

Γ′ is a lump of Γ, Γ′ vl Γ, iff Γ′ is derived from Γ by zero or more lumpingsteps:

• merge two identically labelled branches

• keep labeling and reattach accordingly.

hp

h1

. . .

Γh1

a

h2

. . .

Γh2

b

h3

c

hp

{h1, h2}

. . .

Γh1t Γh2

{a, b}

h3

cwl

• nb: vl is a partial order

10 / 35


Definition (Lumping (Continued))

• if {h1, . . . , hn} is a lumped branch, then there is an implicit belief whichhi is implemented that gives utilities for terminal nodes

Definition (Reduced Games)

Let v be the transitive closure of vp and vl.

If Γ′ v Γ, we call Γ′ a reduction, or a partial representation of Γ.

11 / 35


Definition (Reoccurrence)

Node h ∈ H reoccurs in H′ (or Γ′) iff either h ∈ H′ or there is some h′ ∈ H′

such that h ∈ h′.

Information state v ∈ V from Γ reoccurs in V′ (or Γ′) iff some h ∈ v reoccursin Γ′.

Definition (Projection)

Let Γ′ v Γ, and define a projection function π as a function that takeselements from {h ∈ H | h reoccurs in H′} to H′, and that takes elementsfrom {v ∈ V | v reoccurs in V′} to V′, as follows:

π(h) =

{h if h ∈ H′

h′ for some h′ ∈ H′ for which h ∈ h′

π(v) = v′ for some v′ ∈ V′ such that some h ∈⋃

v′ for some h ∈ v .

FactWhenever Γ′ v Γ, then there is a unique projection π.

12 / 35


Definition (Awareness Structure)

Fix Γ with V =⋃

i<n Vi. An awareness structure based on Γ is a tupleA(Γ) =

⟨W, w0, {Rv}v∈V , L

⟩with:

• W — set of worlds,

• w0 — actual world,

• Rv ⊆ W×W — accessibility relations for each v ∈ V,

• L : W → G — assigns each world w a game L(w) ∈ G.

that meets the following constraints:

Centering: L(w0) = Γ,

Reduction: if wRvw′, then L(w′) v L(w),

Existence: if v is an information state in game L(w), then there is aworld w′ such that wRvw′,

Relevance: whenever wRvw′ then v is an information state in L(w) thatreoccurs in L(w′),

Stationarity: for all v the relations Rv are transitive and Euclidean.

13 / 35


Example ((Un-)awareness of Others’ (Un-)awareness)

i

ja

〈2, 2〉

c

〈3, 1〉

d

a

jb

〈1, 0〉

e

〈0, 1〉

f

b

w0 i

ja

〈2, 2〉

c

〈3, 1〉

d

a

jb

〈.2, .8〉

{e, f}

b

w1

i

ja

〈2, 2〉

c

〈3, 1〉

d

a

jb

〈1, 0〉

e

〈0, 1〉

f

b

w2 i

ja

〈2, 2〉

c

〈3, 1〉

d

a

w3

jb

ja

i

14 / 35


Example (Uncertainty about (Un-)awareness)

i

ja

〈2, 2〉

c

〈3, 1〉

d

a

jb

〈1, 0〉

e

〈0, 1〉

f

b

w0 i

ja

〈2, 2〉

c

〈3, 1〉

d

a

jb

〈1, 0〉

e

〈0, 1〉

f

b

w1

i

ja

〈2, 2〉

c

〈3, 1〉

d

a

jb

〈1, 0〉

e

〈0, 1〉

f

b

w2 i

ja

〈2, 2〉

c

〈3, 1〉

d

a

jb

〈.2, .8〉

{e, f}

b

w3

i

jb

ii

15 / 35


IBR Reasoning on Unawareness Structures (Preliminaries)

[today: unawareness structures without lumping]

• for all w, L(w) is a (possibly trivial) interpretation game:

notation:⟨. . . , Tw, Pr(·)w, Mw, [[·]]w , . . .

⟩• set of viewpoints V = Tw0 ∪Mw0

• worlds accessible from w by sequence of views~v:

w[v] ={

w′ ∈ W | wRvw′}

w[~vv] =⋃

w′∈w[~v]

w′[v]

• from Stationarity and Reduction:

w′, w′′ ∈ w[v] → L(w′) = L(w′′)

notation:⟨

. . . , Tw[v], Pr(·)w[v], Mw[v], [[·]]w[v] , . . .⟩

16 / 35


IBR Reasoning on Unawareness Structures (Naıve Types)

• define types for all worlds w and all views v that occur in w

• naıve types:

Sw0(t, m) =

1

|[[m]]w| if t, m reoccur in L(w) and t ∈ [[m]]w

0 otherwise

BRR0(m, w) = arg max

a∈Aw ∑t∈Tw

Pr(t| [[m]])w ×UwR(t, m, a)

Rw0(m, a) =

1

|BRR0(m,w)| if m, a reoccur in L(w) and a ∈ BR0(m, w)

0 otherwise

17 / 35


IBR Reasoning on Unawareness Structures (Sophisticated Senders)

• behavioral beliefs of level-(k + 1) sender in world w for view t:

ρw,tk+1

(m, a) = ∑w′∈w[tm]

1

|w[tm]| × Rw′k (m, a) ; defined if w[t] 6= ∅

• nb: two sorts of “probability-0 beliefs”:• from explicit behavioral belief that m is not interpreted as t′

• from unawareness of m or a

• level-(k + 1) senders play best response to ρw,tk+1

:

BRSk+1

(t, w) = arg maxm∈Mw[t]

∑a∈Aw[m]

ρw,tk+1

(m, a)×Uw[t]S (t, m, a)

Swk+1

(t, m) =

1

|BRSk+1(t,w)| if t, m reoccur in L(w) and

m ∈ BRSk+1

(t, w)

0 otherwise

18 / 35


IBR Reasoning on Unawareness Structures (Sophisticated Receivers)

• behavioral beliefs of level-(k + 1) receiver in world w for view m:

σw,mk+1

(t, m′) = ∑w′∈w[mt]

1

|w[mt]| × Sw′k (t, m′) ; defined if w[m] 6= ∅

• posterior for level-(k + 1) receivers in world w and view m:

µw,mk+1

(t|m′) =Pr(t)w[m] × σw

k+1(t, m′)

∑t′∈Tw[m] Pr(t′)w[m] × σwk+1

(t′, m′)

• level-(k + 1) receivers play best response to µw,mk+1

:

BRRk+1(m, w) = arg max

a∈Aw[m]∑

t∈Tw[m]

µw,mk+1

(t|m)×Uw[m]R (t, m, a)

Rwk+1

(m, a) =

1

|BRRk+1(m,w)| if t, m reoccur in L(w) and

a ∈ BRRk+1(m, w)

0 otherwise

19 / 35


Example 1 (Division of Pragmatic Labor) (Horn 1984)

(1) a. Black Bart killed the sheriff. m

b. Black Bart killed the sheriff in a stereotypical way. t

(2) a. Black Bart caused the sheriff to die. m∗

b. Black Bart killed the sheriff in a non-stereotypical way. t∗

• unmarked form pairs with unmarked meaning m↔ t

• marked form pairs with marked meaning m∗ ↔ t∗

20 / 35


Example 2 (Division of Pragmatic Labor)

(3) a. Sue smiled. m

b. Sue smiled genuinely. t

(4) a. The corners of Sue’s lips turned slightly upwards. m∗

b. Sue faked a smile. t∗

Example 3 (Division of Pragmatic Labor)

(5) a. Mrs T sang ‘Home Sweet Home.’ m

b. Mrs T sang a lovely song. t

(6) a. Mrs T produced a series of sounds roughly corresponding to thescore of ‘Home Sweet Home.’ m∗

b. Mrs T sang very badly. t∗

21 / 35


Traditional Explanation: Skewed Interpretation Game

N

S

R

〈1, 1〉

a

〈0, 0〉

a∗

m

R

〈1+ε, 1〉

a

〈0+ε, 0〉

a∗

m∗

t

p > .5

S

R

〈0, 0〉

a

〈1, 1〉

a∗

m

R

〈0+ε, 0〉

a

〈1+ε, 1〉

a∗

m∗

t∗

1− p

• problem: what exactly do prior probabilities and message costs encode?

22 / 35


Unawareness Model for “Division of Pragmatic Labor”

N

S

R

〈1, 1〉

a

〈0, 0〉

a∗

m

R

〈1, 1〉

a

〈0, 0〉

a∗

m∗

t

1/2

S

R

〈0, 0〉

a

〈1, 1〉

a∗

m

R

〈0, 0〉

a

〈1, 1〉

a∗

m∗

t∗

1/2

w0

N S R 〈1, 1〉w1

t m a

StRm

23 / 35


IBR Reasoning with Unawareness

• ibr sequence for world w1:

Rw1

0=

{m 7→ am∗ 7→ undefined

}Sw1

0=

{t 7→ mt∗ 7→ undefined

}

Sw1

1=

{t 7→ mt∗ 7→ undefined

}Rw1

1=

{m 7→ am∗ 7→ undefined

}...

...

• ibr sequence for world w0:

Rw0

0=

{m 7→ am∗ 7→ a, a∗

}Sw0

0=

{t 7→ mt∗ 7→ m, m∗

}

Sw0

1=

{t 7→ mt∗ 7→ m∗

}Rw0

1=

{m 7→ am∗ 7→ a∗

}...

...

24 / 35


Example (Asymmetric Granularity of Scales) (Matsumoto 1995)

(7) a. It was warm yesterday. It was not hot yesterday.6 It was not a little bit more than warm yesterday.

b. It was warm yesterday, and today it is a little bit more than warm. It was not hot yesterday. It was not a little bit more than warm yesterday.

• normal scale:

〈warm, hot〉• contextual scale:

〈warm, a-little-bit-warmer-than-warm, hot〉

25 / 35


Example (Asymmetric Granularity of Scales)

N

S

R

1

aw0

aw+

0

ah

mwR

1

aw0

aw+

0

ah

mw+

R

1

aw0

aw+

0

ah

mh

tw

S

. . .

tw+

S

. . .

th

Γ0: Modeller’s Game

26 / 35



N

S

R

1

{aw, aw+}0

ah

mwR

1 0

mh

{tw, tw+}S

R

0 1

mwR

0 1

mh

th

Γ1: Lumped Game

27 / 35



w0 : Γ0 w1 : Γ1

Stw , Sth

Rmw , Rmh

Awareness Structure

• if R is aware of tw+ and mw+ , he will interpret mw as implicating ¬tw+

• but he will then also know that he would otherwise have interpretedmw as including tw+ (without explicit conceptualization)

28 / 35


Example (Context Coordination)

(8) a. There is pizza in the fridge.

b. If you get hungry, there is pizza in the fridge.

c. If Johnny gets hungry, there is pizza in the fridge.

• (8a) may be functionally ambiguous in context• (8b) and (8c) are not

• conditional construction used for context model accommodation(→ proper action choice)

29 / 35



N

S

R

1

ae

0

af

my→p

R

1

ae

0

af

mj→p

R

1

ae

0

af

mp

ty

S

R

0

ae

1

af

my→p

R

0

ae

1

af

mj→p

R

0

ae

1

af

mp

tj

Γ0: Modeller’s Game

30 / 35



N

S

R

1

ae

my→p

R

1

ae

mp

ty

Γ1: R’s Game after my→p

N

S

R

1

af

mj→p

R

1

af

mp

tj

Γ2: R’s Game after mj→p

31 / 35



N

S

R

mp

{ty, tj

}Γ3: R’s Game after mp

32 / 35



w0 : Γ0

w1 : Γ1 w2 : Γ2

w3 : Γ3

Rmy→p Rmj→p

Rmp

Rmp Rmp

Awareness Structure

• S uses Biscuit conditionals to help accommodate proper context model• R realizes this

33 / 35


Summary

• awareness structures• partial games from pruning & lumping• pointed, interpreted Kripke models

• ibr reasoning with unawareness• applications:

• division of pragmatic labor• granularity of scales• contextual coordinators

Conclusions & Outlook

• awareness impacts pragmatic reasoning• more applications:

• “clausal implicatures”• (conditional) excluded middle• conditionals (esp. counterfactuals)• vagueness• meaningnn

34 / 35


Homework

• read:• Michael Franke (2010) “Pragmatic Reasoning about Unawareness”,

manuscript, University of Tubingen

Next Sessions

• “Learning from Neighbors”

• Politeness & the Handicap Principle

35 / 35

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