Pragmatics & Game Theory: Branches of Game-Theoretic ... › ~roland › PGT1314 › folien › Session10.pdf · Roland Mühlenbernd Branches of Game-Theoretic Linguistics. Introduction

IntroductionThe Strategic Implicature Model

Strategic Language UseEvolutionary Game Theory

Pragmatics & Game Theory:Branches of Game-Theoretic Linguistics

Roland Mühlenbernd

WiSe 13/14

Roland Mühlenbernd Branches of Game-Theoretic Linguistics



Table of Content

1 IntroductionReviewOverview

2 The Strategic Implicature ModelThe Local GameThe Global Game

3 Strategic Language UseIndirect SpeechRelationship Negotiation

4 Evolutionary Game TheoryBehavioral StrategiesLearning Dynamics




ReviewOverview

Review

conversational implicature

4 maxims: quality, quantity, relevance, mannerparticularized & generalized implicatures

neo-Gricean pragmatics

scalar & clausal implicaturesQ- and R-based implicatures (Horn)Horn's division of pragmatic laborQ-, I- and M-based implicatures (Levinson)

game-theoretic pragmatics

situations: signaling gamespossible behavior of production/perception: strategies (pure &behavioral)actual behavior: rationalistic solution concepts (here: IBR)




ReviewOverview

Overview

Game Theory and Linguistics

Language Evolution

Signaling Games

GT in Lang. Use Pragm. Reasoning

Signaling Games

IBR model SIM




The Local GameThe Global Game

The Strategic Implicature Model

Parikh's (1991) Strategic Implicature Model was the �rstaccount to use game theory for matters in pragmatics.

Parikh deals with particularized implicatures of relevance

he distinguishes between two types of relevance:

logical relevance: a proposition is logical relevant with respectto a set of propositions, if the two together have a 'nontrivial'implication without having it individually (in spirit of Sperberand Wilson, 1986)rational relevance: a proposition is rational relevant, if itincreases the expected utility of a participant's goal





Example: B's decision problem

Situation: A and B have to attend a talk at 5 pm and B believesthe probability of it's being time for the talk now is .2 and thatthere is still time is .8.

c: continue previous action, d: depart for the talk, p: it's time for the talk

The information that it is time for the talk has rational relevancehere, since it has a positive value of information: 10 - 6 = 4.





A Relevance Implicature

Example

Situation: It is an important talk today (at 5 pm)A: 'It is 5 pm.'+> It is time for the talk.

logical relevance: the proposition (expression) is relevantbecause it is non-trivial in the given set of propositions(situation, context) by generating the implicature.

rational relevance: the proposition is rational relevant for thehearer since it increases his expected utility

note that both types of relevance are essential for Grice's ideaof the cooperation principle!





The SIM Model For The Given Example

situation s: 'important talk at 5 pm'; and empty context s ′

expression ϕ: 'It is 5 pm.'

information states t: it is 5 pm and the talk is now; and t′: it is 5 pm

literal interpretation l (�ts to t′) and pragmatic interpretation p (�ts to t)

production and interpretation cost: -1

incorrect interpretation: -2, pragmatic inference: -1

information value by rational relevance: +4

-2, -1-4,-3

-2, -1-3,-21,2-2, -1

-2, -1-4,-3-5,-4





The SIM Model's Local Game

B could reason that A could havean alternative in situation s

to use a more complex expressionµ: 'It is time for the talk now.'(extra costs: -.5)

to express meaning e: it's time forthe talk

A could also have an alternative insituation s ′

to be silent (ν: empty message)that expresses nothing (e′: emptymeaning)





The SIM Model's Local Game

Solution idea:

A's information set is {s}and {s ′}B's information set is {t, t′},{e} and {e′}there are four senderstrategies and two receiverstrategies

and therefore 8 strategypro�les

there is only one unique Nashequilibrium{(s, ϕ), (s ′, ν); ({t, t′}, p)}





The SIM Model's Global Game

the local game results after Ahas uttered ϕ

if A would utter somethingelse (e.g. ψ or µ: 'thespeaker entered the stage','the audience is waiting'...),it would emerge a di�erentlocal game

each local game has aexpected utility value

A should use the expressionthat triggers the local gamewith maximal expected utility

If LG denotes the set of all local games that result from A's choice, then〈GG , LG 〉 is called the Strategic Implicature Model.





The SIM Model

Conclusion

the SIM model is (one of) the �rst model that combinesPragmatics and Game Theory

the model deals with particularized implicatures (relevance)and therefore takes a given context/situation into account

the solution concept is a unique Nash equilibrium that depictspragmatic language use

but: there is criticism for model and solution concept:

there are a lot of numbers made up, which di�erences arecritical for the right solutionthe Nash equilibrium explains why to stay with a strategy, butnot how to come to itthere is no good solution for the case of multiple Nashequilibria




Indirect SpeechRelationship Negotiation

Overview


Language Evolution

Signaling Games

GT in Lang. Use

Indirect Speech

Pragm. Reasoning

Signaling Games

IBR model SIM





The Logic of Indirect Speech (Pinker, Nowak & Lee 2007)

Example 1

'Would you like to come up and see my etchings?'+> a sexual come-on

Example 2

'If you could pass the guacamole, that would be awesome.'+> a polite request

Example 3

'Nice store you got here. Would be a real shame if something happenedto it.' +> a threat

Example 5

'Gee, o�cer, is there some way we could take care of the ticket here?'+> a bribe





The Logic of Indirect Speech

Direct bribe: 'If you let me go without a ticket, I'll pay you 50dollar.'

Indirect bribe: 'Gee, o�cer, is there some way we could takecare of the ticket here?' (showing a �fty dollar bill)

dishonest o�cer honest o�cer

Don't bribe -100 -100Bribe -50 -500Indirect bribe -50 -100





The Logic of Indirect Speech

The directness of a proposition is probably continuous:

'If you let me go without a ticket, I'll pay you 50 dollar.''Is there some way we could take care of the ticket here?''I've learned my lesson; you don't have to worry about me doingthis again.'

a corrupt cop is more sensitive to recognize bribe than a honest cop

there is a threshold for both to recognize the bribe





Relationship Negotiation

People use indirect speech also in nonlegal situations, wherethere are no �nancial or legal payo�s and penalties

Example: bribing a maitre d' to get seated in a full restaurantdirectly without reservation

When relations are ambiguous, a divergent understandingbetween the parties can lead to the aversive emotion we callawkwardness

dishonest maitre d' honest maitre d'

Don't bribe long wait (D/D) long wait (D/D)Bribe instant seating (R/R) awkwardness (R/D)Indirect bribe instant seating (R/R) long wait (D/D)

D: dominance relationship; R: reciprocity relationship





Relationship Negotiation

There are three essential human relationship types:

boss - employee (dominance)good friends (communality)business transaction (reciprocity)

relationship mismatches impose emotional costs (awkwardness)

sometime relationship types are highly unambiguous

indirect speech can help to avoid relationship mismatches by�the bene�t of the doubt�

indirect speech merely provides shared individual knowledge,but not common knowledge





Overview


Language Evolution

Signaling Games

GT in Lang. Use

Indirect Speech

Pragm. Reasoning

Signaling Games

IBR model SIM




Behavioral StrategiesLearning Dynamics

Coordination & Signaling

R L

R 1 0L 0 1

aL aStL 1 0tS 0 1

Messages: One or two lanterns?

s1:tL m1

tS m2

s2:tL

m2tS

m1

s3:tL m1

tS m2

s4:tL

m2tS

m1

r1:m1 aL

m2 aS

r2:m1

aSm2

aL

r3:m1 aL

m2 aS

r4:m1

aSm2

aL





a signaling game is a tuple SG = 〈{S ,R},T ,Pr ,M,A,U〉a Lewis game is de�ned by:

T = {tL, tS}M = {m1,m2}A = {aL, aS}Pr(tL) = Pr(tS) = .5

U(ti , aj) =

{1 if i = j

0 else

aL aStL 1 0tS 0 1

N

S

R

1 0

R

1 0

S

R

0 1

R

0 1

.5 .5tL tS

m1 m2 m1 m2

aL aS aL aS aL aS aL aS





Pure strategies

Pure strategies are contingency plans, players act according to.

sender strategy: s : T → M

receiver strategy: r : M → A

s1:tL m1

tS m2

s2:tL

m2tS

m1

s3:tL m1

tS m2

s4:tL

m2tS

m1

r1:m1 aL

m2 aS

r2:m1

aSm2

aL

r3:m1 aL

m2 aS

r4:m1

aSm2

aL





Signaling Systems

signaling systems are combinations of pure strategies. TheLewis game has two: L1 = 〈s1, r1〉 and L2 = 〈s2, r2〉

L1:tL

tS

m1

m2

aL

aSL2:tL

tS

m1

m2

aL

aS

signaling systems are strict Nash equilibria of the EU-table:

r1 r2 r3 r4s1 1 0 .5 .5s2 0 1 .5 .5s3 .5 .5 .5 .5s4 .5 .5 .5 .5

in signaling systems messages associate states and actionsuniquely

signaling systems constitute evolutionary stable states





Behavioral Strategies

Behavioral strategies are functions that map choice points toprobability distributions over actions available in that choice point.

behavioral sender strategyσ : T → ∆(M)

behavioral receiver strategyρ : M → ∆(A)

σ =

t1 7→[m1 7→ .9m2 7→ .1

]t2 7→

[m1 7→ .5m2 7→ .5

] ρ =

m1 7→[a1 7→ .33a2 7→ .67

]m2 7→

[a1 7→ 1a2 7→ 0

]





Learning Dynamics & Signaling Games

Extensions in time:

agents play the game repeatedly

agents' decisions are in�uenced by previous encounters

application of learning dynamics like reinforcement learning

belief learning





Best Response & Expected Utility

Playing Best Response means to make a choice thatmaximizes the Expected Utility.

EUS(m|t, β) =∑a∈A

β(a|m)× U(t, a) (1)

EUR(a|m, β) =∑t∈T

β(m|t)× U(t, a) (2)

How does an agent get belief β?





Belief Learning

The belief is a result of observation

Example:

SO a1 a2m1 8 2m2 7 13

β =

m1 7→[a1 7→ .8a2 7→ .2

]m2 7→

[a1 7→ .35a2 7→ .65

]

RO t1 t2m1 6 0m2 4 4

β =

t1 7→[m1 7→ .6m2 7→ .4

]t2 7→

[m1 7→ 0m2 7→ 1

]





Best Response as Behavioural Strategy

behavioural sender strategyσ : T → ∆(M)

σ(m|t) =

{1

|BR(t)| if m ∈ BR(t)

0 else

behavioural receiver strategyρ : M → ∆(A)

ρ(a|m) =

{1

|BR(m)| if a ∈ BR(m)

0 else





Reinforcement Learning

S Rts

tg

m1

m2

as

ag

0

0

0

0the sender has an urn for eachstate t ∈ T

each urn contains balls of eachmessage m ∈ M

the sender decides by drawingfrom urn 0t

the receiver has an urn for eachmessage m ∈ M

each urn contains balls of eachaction a ∈ A

the receiver decides by drawingfrom urn 0t

successful communication → urn update

in general a signaling system emerges over timeRoland Mühlenbernd Branches of Game-Theoretic Linguistics




Behavioural & Pure Strategies

Pure strategies are a subset of behavioural strategies.

Example:

σ2:t1

m2t2

m1

ρ2:m1

a2m2

a1

σ2 =

t1 7→[m1 7→ 0m2 7→ 1

]t2 7→

[m1 7→ 1m2 7→ 0

] ρ2 =

m1 7→[a1 7→ 0a2 7→ 1

]m2 7→

[a1 7→ 1a2 7→ 0

]

Note: If an agents plays σ2 as sender and ρ2 as receiver, we say, hehas learned the signaling language L2 = 〈σ2, ρ2〉.





Conclusion

1 The Strategic Implicature Model (Parikh 1991)

is the �rst model that combines pragmatics and game theorydeals (basically) with relevance implicaturesuses the Nash equilibrium as solution conceptbut: there is criticism for model and solution concept:

2 Strategic Language Use

is a branch in the tree of research directions that use gametheory for issues in linguisticsascribes two essential functions to communication: informationtransmission and relationship negotiationintegrates social costs (awkwardness)

3 EGT and Signaling Games

gives a framework to model dynamics of language evolutionand changeintegrates e.g. learning dynamics on repeated plays


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Pragmatics & Game Theory: Branches of Game-Theoretic ... › ~roland › PGT1314 › folien › Session10.pdf · Roland Mühlenbernd Branches of Game-Theoretic Linguistics. Introduction