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=4cGame Theory and Gricean PragmaticsLesson III""CAnton Benz
Zentrum fr Allgemeine Sprachwissenschaften
ZAS Berlin .PP9P9&7eCourse OverviewLesson 1: Introduction
From Grice to Lewis
Relevance Scale Approaches
Lesson 2: Signalling Games
Lewis Signalling Conventions
Parikh s Radical Underspecification Model
Lesson 3: The Optimal Answer Approach I
Lesson 4: The Optimal Answer Approach II
Comparison with Relevance Scale Approaches
Decision Contexts with Multiple ObjectivesZ0ZZIZQZVZ0H&()Vo
VcOptimal Answer Approach 0Lesson III April, 5th ( > dOverview of Lesson III
&Natural Information and Conversational Implicatures
An Example: Scalar Implicatures
Natural Information and Conversational Implicatures
Calculating Implicatures in Signalling Games
Optimal Answers
Core Examples
Optimal Answers in Support Problems
Examples
Support Problems and Signalling Games 4ZZZ;Z'Z4;%"
'
The AgendaPutting Grice on Lewisean feet! (f4Natural Information and Conversational Implicatures 55.
PJ3Explanation of Implicatures Optimal Answer Approach$4* PStart with a signalling game where the hearer interprets forms by their literal meaning.
Impose pragmatic constraints and calculate equilibria that solve this game.
Implicature F +> j is explained if for all solutions (S,H):
S1(F) = jfnn+O
ContrastIn an information based approach:
Implicatures emerge from indicated meaning (in the sense of Lewis).
Implicatures are not initial candidate interpretations.
Speaker does not maximise relevance.
No diachronic process.X"ZZ;+1,8$hAssumption: speaker and hearer use language according to a semantic convention.
Goal: Explain how implicatures can emerge out of semantic language use.
Nonreductionist perspective..iRepresentation of AssumptionSemantics defines interpretation of forms.
Let [F] denote the semantic meaning.
Hence, assumption: H(F)=[F], i.e.:
H(F) is the semantic meaning of F
Semantic meaning @ Lewis imperative signal.VtZ"ZZZ8K Background (Repetition)
Lewis (IV.4,1996) distinguishes between
indicative signals
imperative signals
Two possible definitions of meaning:
Indicative:
[F] = M :iff S1(F)=M
Imperative:
[F] = M :iff H(F)=M(P&P(PPP
PPN
, Q
An ExamplexWe consider the standard example:
Some of the boys came to the party.
said: at least two came
implicated: not all came\"% 1HSThe GameUThe Solved GameV2The hearer can infer after receiving A(some) that:33(%[3Natural Information and Conversational Implicatures443 VNatural and NonNatural Meaning (mGrice distinguished between
natural meaning
nonnatural meaning
Communicated meaning is nonnatural meaning.
8$nWExample$I show Mr. X a photograph of Mr. Y displaying undue familiarity to Mrs. X.
I draw a picture of Mr. Y behaving in this manner and show it to Mr. X.
The photograph naturally means that Mr. Y was unduly familiar to Mrs. X
The picture nonnaturally means that Mr. Y was unduly familiar to Mrs. XnZnZZ <
/X
eTaking a photo of a scene necessarily entails that the scene is real.
Every branch which contains a showing of a photo must contain a situation which is depicted by it.
The showing of the photo means naturally that there was a situation where Mr. Y was unduly familiar with Mrs. X.
The drawing of a picture does not imply that the depicted scene is real.BGZZIZGIYNatural Information of SignalsLet G be a signalling game.
Let S be a set of strategy pairs (S,H).
We identify the natural information of a form F in G with respect to S with:
The set of all branches of G where the speaker chooses F. ;h<<VZInformation coincides with S1(F) in case of simple Lewisean signalling games.
Generalises to arbitrary games which contain semantic interpretation games in embedded form.
Conversational Implicatures are implied by the natural information of an utterance.<S4!The Standard Example reconsidered
"V
Some of the boys came to the party.
said: at least two came
implicated: not all cameF& 1& UX"The game defined by pure semantics##(YXThe game after optimising speaker s strategy(\The possible worldsw1: 100% of the boys came to the party.
w2: More than 50% of the boys came to the party.
w3: Less than 50% of the boys came to the party.H'0/^&The possible Branches of the Game Tree''(`5The unique signalling strategy that solves this game:66(Z4The Natural Information carried by utterance A(some)55(NThe branches allowed by strategy S:
w1,A(all), {w1}
w2,A(most), {w1,w2}
w3,A(some), {w1,w2,w3}
Natural information carried by A(some):
{w3,A(some), {w1,w2,w3}}$ZAZ(ZPZ2$
(
b Implicatures in Signalling Games!!(A special case
c!As Signalling Game (Repetition)&
A signalling game is a tuple:
N,, p, (A1,A2), (u1, u2)
N: Set of two players S,H.
: Set of types representing the speakers private information.
p: A probability measure over representing the hearer s expectations about the speaker s type. "
"d"p(A1,A2): the speaker s and hearer s action sets:
A1 is a set of forms F / meanings M.
A2 is a set of actions.
(u1,u2): the speaker s and hearer s payoff functions with
ui: A1A2 RP1>:+5 e#Strategies in a Signalling GameLet [ ] : F M be a given semantics.
The speaker s strategies are of the form:
S : A1 such that
S() = F [F]
i.e. if the speaker says F, then he knows that F is true (Maxim of Quality).Pu
+
@NfDf$(Definition of Implicature(special case))*Given a signalling game as before, then an implicature
F +> y
is explained iff the following set is a subset of [y] = {w  w = y}:V8 H =><L,h&Preview
Later, we apply this criterion to calculating implicatures of answers.
The definition depends on the method of finding solutions.n[First we need a method for calculating optimal answers.
The resulting signalling and interpretation strategies are then the solutions which we use as imput for calculating implicatures. *'<8Ng%Optimal Answersi'
Core Examplesj(Italian NewspaperSomewhere in the streets of Amsterdam...
J: Where can I buy an Italian newspaper?
E: At the station and at the Palace but nowhere else. (SE)
E: At the station. (A) / At the Palace. (B)
.)n)k)The answer (SE) is called strongly exhaustive.
The answers (A) and (B) are called mention some answers.
A and B are as good as SE or as A B:
E: There are Italian newspapers at the station but none at the Palace./9@)@Gn#$Gl*Partial AnswersIf E knows only that A, then A is an optimal answer:
E: There are no Italian newspapers at the station.
If E only knows that the Palace sells foreign newspapers, then this is an optimal answer:
E: The Palace has foreign newspapers.^74nZ&n74Z&m+Partial answers may also arise in situations where speaker E has full knowledge:
I: I need patrol for my car. Where can I get it?
E: There is a garage round the corner.
J: Where can I buy an Italian newspaper?
E: There is a news shop round the corner.NQ@1'@ )* n,#Optimal Answers in Support Problems$$#
The Framework
oSupport ProblemDefinition: A support problem is a five tuple (,PE,PI,A,u) such that
(, PE) and (, PI) are finite probability spaces,
(,PI,A, u) is a decision problem.
Let K:= {wW PE(w) > 0 } (E s knowledge set).
Then, we assume in addition:
for all A : PE(A) = PI(AK) .G3$@M!n#"
"
"
!"
=
x/&"np.Support Problemq/,I s Decision Situation,I optimises expected utilities of actions:
,r0I will choose an action aA that optimises expected utility, i.e. for all actions b
EU(b,A) EU(aA,A)
Given answer A, H(A) = aA.
For simplicity we assume that I s choice aA is commonly known.HT ?6+t: s1,E s Decision Situation*E optimises expected utilities of answers:++t2
^(Quality): The speaker can only say what he thinks to be true.
(Quality) restricts answers to:$_>$ u3Examples The Italian Newspaper Examplesv5Italian NewspaperSomewhere in the streets of Amsterdam...
J: Where can I buy an Italian newspaper?
E: At the station and at the Palace but nowhere else. (SE)
E: At the station. (A) / At the Palace. (B)
.)n)w4"Possible Worlds (equally probable)##(x6Actions and AnswersI s actions:
a: going to station;
b: going to Palace;
Answers:
A: at the station (A = {w1,w2})
B: at the Palace (B = {w1,w3})
* ?
* _Let utilities be such that they only distinguish between success (value 1) and failure (value 0).
Let s consider answer A = {w1,w2}.
Assume that the speaker knows that A, i.e. there are Italian newspapers at the station.
6~]`The CalculationIf hearing A induces hearer to choose a (i.e. aA=a going to station ):
If hearing A induces hearer to choose b (i.e. aA=b going to Palace ):
If PE(B) = 1, then EUE(A) = EUE(b) = 1.
PE(B) < 1 leads to a contradiction.L# >.H3+bPE(B) < 1 leads to a contradiction:
aA = b implies EUI(bA) EUI(aA) = 1.
Hence, EUI(bA) = vA PI(v) u(v,b) =1.
Therefore PI(BA) =1, hence PI(BA) = PI(A), hence PI(A\B)=0.
PE(A\B)=0, due to wellbehavedness.
PE(BA)=PE(A)=1, hence PE(B) = 1.>%ZnZ %
[$a=Case: Speaker knows that Italian newspaper are at both places>>(fCalculation showed that EUE(A) = 1.
Expected utility cannot be higher than 1 (due to assumptions).
Similar: EUE(B) = 1; EUE(AB) = 1.
Hence, all these answers are equally optimal.O5y7
More Cases,E knows that A and B:
EUE(A) = EUE(B) = EUE(AB)
E knows that A and B:
EUE(A) = EUE(A B)
E knows only that A:
For all admissible C: EUE(C) EUE(A)` &
The following example shows how the method of finding optimal answers in support problems interacts with the general theory of implicatures in signalling games.$
DHip Hop at Roter SalonJohn loves to dance to Salsa music and he loves to dance to Hip Hop but he can t stand it if a club mixes both styles.
J: I want to dance tonight. Is the Music in Roter Salon ok?
E: Tonight they play Hip Hop at the Roter Salon.
+> They play only Hip Hop.0wnw,/#]FA game tree for the situation where both Salsa and Hip Hop are playingGG G+After the first step of backward induction:,,(^,After the second step of backward induction:(If we say that a proposition is the more relevant the higher the expected utility after learning it, then relevance scale approaches predict that Hip Hop implicates that both, Salsa and Hip Hop, are playing.
Worst case compatible with what was said!(*(2$EHip Hop at Roter SalonAbbreviations:
Good(x) := HAssumptions hEqual Probabilities
Independence: X,Y{H,S,Good}T5n&
ILearning H(x) or S(x) raises expected utility of going to salon x:
EUI(goingtox) < EUI(stayhome) < EUI(goingtoxH(x))
EUI(goingtox) < EUI(stayhome) < EUI(goingtoxS(x)) Cnrn ,
/ KViolating Assumptions IIThe Roter Salon and the Grner Salon share two DJs. One of them only plays Salsa, the other one mainly plays Hip Hop but mixes into it some Salsa. There are only these two Djs, and if one of them is at the Roter Salon, then the other one is at the Grner Salon. John loves to dance to Salsa music and he loves to dance to Hip Hop but he can t stand it if a club mixes both styles.
J: I want to dance tonight. Is the Music in Roter Salon ok?
E: Tonight they play Hip Hop at the Roter Salon.^~Zn Z"" % 0%Support Problems and Signalling Games%In our model, the speaker finds an optimal answer by backward induction in support problems.
This is not a standard method for solving coordination problems in signalling games.H$~Signalling Game
A signalling game is a tuple:
N,, p, (A1,A2), (u1, u2)
N: Set of two players S,H.
: Set of types representing the speakers private information
p: A probability measure over representing the hearer s expectations about S type. "
"Solution to a Signalling GameThe standard solution concept for Signalling games is that of a perfect Bayesian equilibrium!
(S,H) strategies:
S : A1
H : A1 A2q @$"$$$
$,$$
$"$$$
$$_
"Perfect Bayesian equilibrium (S,H)6#((((" S() argmaxF u1(F,H(F),)
"F H(F) argmaxM m(F)u2(F,M,)
where m is defined by
m(F) = 0 if S()F
m(F) = p() / p(S1[F]) if S()=F
if p(S1[F]) > 0, else m(F) is arbitrary.rExx:x,x H$
%Structural Descriptions with Contexts&&(
4A structural description with contexts is a tuple , F, M, Gen with:
a set of possible worlds,
F a set of forms,
M a set of meanings,
Gen F M:HS2 ! ! *3eTask
zGiven:
a set of support problems S with fixed decision problem (,PI,A,u) for a
structural description , F, M, Gen.
Wanted:
Representation as signalling game:
N,, p, (AE,AI), (uE, uI)ZpZZ#ZZ! + T>YConstruction
(
^Let s=(,PE,PI,A,u) be a given support problem.
Remember: there is a common prior P on such that:
PE(X) = PI(XKs) for Ks:= {wW PE(w) > 0}
Add Ks to (i.e. = {Ks sS})
The speaker s action set AE is identical with the set of forms F.
The hearer s action set is identical to the action set of s.:dZ,Z!ZZ"
E
"
""""*#"#<"
0(<
The game is a game of pure coordination with respect to joint payoff functions
ui: F AI R
uI(A,a,K) := EUI(aK)
uE(A,a,K) := EUE(aK) (= EUI(aK))
On; O !
RN>
(p is arbitrary.
Forms F have to be interpreted by their semantic meaning [F].
The speaker has to conform to the maxim of quality, i.e. S(Ks) Adms
n ` (MFResult
pThe strategy pair defined by:
S(Ks) Ops, H(A) = aA
is a Perfect Bayesian Equilibrium of the associated signalling game.
it (weakly) Pareto dominates all other strategy pairs (S ,H ).!
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=cGame Theory and Gricean PragmaticsLesson III""CAnton Benz
Zentrum fr Allgemeine Sprachwissenschaften
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From Grice to Lewis
Relevance Scale Approaches
Lesson 2: Signalling Games
Lewis Signalling Conventions
Parikh s Radical Underspecification Model
Lesson 3: The Optimal Answer Approach I
Lesson 4: The Optimal Answer Approach II
Comparison with Relevance Scale Approaches
Decision Contexts with Multiple ObjectivesZ0ZZIZQZVZ0H&()Vo
VcOptimal Answer Approach 0Lesson III April, 5th ( > dOverview of Lesson III
&Natural Information and Conversational Implicatures
An Example: Scalar Implicatures
Natural Information and Conversational Implicatures
Calculating Implicatures in Signalling Games
Optimal Answers
Core Examples
Optimal Answers in Support Problems
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Support Problems and Signalling Games 4ZZZ;Z'Z4;%"
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The AgendaPutting Grice on Lewisean feet! (f4Natural Information and Conversational Implicatures 55.
PJ3Explanation of Implicatures Optimal Answer Approach$4* PStart with a signalling game where the hearer interprets forms by their literal meaning.
Impose pragmatic constraints and calculate equilibria that solve this game.
Implicature F +> j is explained if for all solutions (S,H):
S1(F) = jfnn+O
ContrastIn an information based approach:
Implicatures emerge from indicated meaning (in the sense of Lewis).
Implicatures are not initial candidate interpretations.
Speaker does not maximise relevance.
No diachronic process.X"ZZ;+1,8$hAssumption: speaker and hearer use language according to a semantic convention.
Goal: Explain how implicatures can emerge out of semantic language use.
Nonreductionist perspective..iRepresentation of AssumptionSemantics defines interpretation of forms.
Let [F] denote the semantic meaning.
Hence, assumption: H(F)=[F], i.e.:
H(F) is the semantic meaning of F
Semantic meaning @ Lewis imperative signal.VtZ"ZZZ8K Background (Repetition)
Lewis (IV.4,1996) distinguishes between
indicative signals
imperative signals
Two possible definitions of meaning:
Indicative:
[F] = M :iff S1(F)=M
Imperative:
[F] = M :iff H(F)=M(P&P(PPP
PPN
, Q
An ExamplexWe consider the standard example:
Some of the boys came to the party.
said: at least two came
implicated: not all came\"% 1HSThe GameUThe Solved GameV2The hearer can infer after receiving A(some) that:33(%[3Natural Information and Conversational Implicatures443 VNatural and NonNatural Meaning (mGrice distinguished between
natural meaning
nonnatural meaning
Communicated meaning is nonnatural meaning.
8$nWExample$I show Mr. X a photograph of Mr. Y displaying undue familiarity to Mrs. X.
I draw a picture of Mr. Y behaving in this manner and show it to Mr. X.
The photograph naturally means that Mr. Y was unduly f
!"#$%&'()*+,./0123456789:;<?@ABCEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxz{}~amiliar to Mrs. X
The picture nonnaturally means that Mr. Y was unduly familiar to Mrs. XnZnZZ <
/X
eTaking a photo of a scene necessarily entails that the scene is real.
Every branch which contains a showing of a photo must contain a situation which is depicted by it.
The showing of the photo means naturally that there was a situation where Mr. Y was unduly familiar with Mrs. X.
The drawing of a picture does not imply that the depicted scene is real.BGZZIZGIYNatural Information of SignalsLet G be a signalling game.
Let S be a set of strategy pairs (S,H).
We identify the natural information of a form F in G with respect to S with:
The set of all branches of G where the speaker chooses F. ;h<<VZInformation coincides with S1(F) in case of simple Lewisean signalling games.
Generalises to arbitrary games which contain semantic interpretation games in embedded form.
Conversational Implicatures are implied by the natural information of an utterance.<S4!The Standard Example reconsidered
"V
Some of the boys came to the party.
said: at least two came
implicated: not all cameF& 1& UX"The game defined by pure semantics##(YXThe game after optimising speaker s strategy(\The possible worldsw1: 100% of the boys came to the party.
w2: More than 50% of the boys came to the party.
w3: Less than 50% of the boys came to the party.H'0/^&The possible Branches of the Game Tree''(`5The unique signalling strategy that solves this game:66(Z4The Natural Information carried by utterance A(some)55(NThe branches allowed by strategy S:
w1,A(all), {w1}
w2,A(most), {w1,w2}
w3,A(some), {w1,w2,w3}
Natural information carried by A(some):
{w3,A(some), {w1,w2,w3}}$ZAZ(ZPZ2$
(
b Implicatures in Signalling Games!!(A special case
c!As Signalling Game (Repetition)&
A signalling game is a tuple:
N,, p, (A1,A2), (u1, u2)
N: Set of two players S,H.
: Set of types representing the speakers private information.
p: A probability measure over representing the hearer s expectations about the speaker s type. "
"d"p(A1,A2): the speaker s and hearer s action sets:
A1 is a set of forms F / meanings M.
A2 is a set of actions.
(u1,u2): the speaker s and hearer s payoff functions with
ui: A1A2 RP1>:+5 e#Strategies in a Signalling GameLet [ ] : F M be a given semantics.
The speaker s strategies are of the form:
S : A1 such that
S() = F [F]
i.e. if the speaker says F, then he knows that F is true (Maxim of Quality).Pu
+
@NfDf$(Definition of Implicature(special case))*Given a signalling game as before, then an implicature
F +> y
is explained iff the following set is a subset of [y] = {w  w = y}:V8 H =><L,h&Preview
Later, we apply this criterion to calculating implicatures of answers.
The definition depends on the method of finding solutions.n[First we need a method for calculating optimal answers.
The resulting signalling and interpretation strategies are then the solutions which we use as imput for calculating implicatures. *'<8Ng%Optimal Answersi'
Core Examplesj(Italian NewspaperSomewhere in the streets of Amsterdam...
J: Where can I buy an Italian newspaper?
E: At the station and at the Palace but nowhere else. (SE)
E: At the station. (A) / At the Palace. (B)
.)n)k)The answer (SE) is called strongly exhaustive.
The answers (A) and (B) are called mention some answers.
A and B are as good as SE or as A B:
E: There are Italian newspapers at the station but none at the Palace./9@)@Gn#$Gl*Partial AnswersIf E knows only that A, then A is an optimal answer:
E: There are no Italian newspapers at the station.
If E only knows that the Palace sells foreign newspapers, then this is an optimal answer:
E: The Palace has foreign newspapers.^74nZ&n74Z&m+Partial answers may also arise in situations where speaker E has full knowledge:
I: I need patrol for my car. Where can I get it?
E: There is a garage round the corner.
J: Where can I buy an Italian newspaper?
E: There is a news shop round the corner.NQ@1'@ )* n,#Optimal Answers in Support Problems$$#
The Framework
oSupport ProblemDefinition: A support problem is a five tuple (,PE,PI,A,u) such that
(, PE) and (, PI) are finite probability spaces,
(,PI,A, u) is a decision problem.
Let K:= {wW PE(w) > 0 } (E s knowledge set).
Then, we assume in addition:
for all A : PE(A) = PI(AK) .G3$@M!n#"
"
"
!"
=
x/&"np.Support Problemq/,I s Decision Situation,I optimises expected utilities of actions:
,r0I will choose an action aA that optimises expected utility, i.e. for all actions b
EU(b,A) EU(aA,A)
Given answer A, H(A) = aA.
For simplicity we assume that I s choice aA is commonly known.HT ?6+t: s1,E s Decision Situation*E optimises expected utilities of answers:++t2
^(Quality): The speaker can only say what he thinks to be true.
(Quality) restricts answers to:$_>$ u3Examples The Italian Newspaper Examplesv5Italian NewspaperSomewhere in the streets of Amsterdam...
J: Where can I buy an Italian newspaper?
E: At the station and at the Palace but nowhere else. (SE)
E: At the station. (A) / At the Palace. (B)
.)n)w4"Possible Worlds (equally probable)##(x6Actions and AnswersI s actions:
a: going to station;
b: going to Palace;
Answers:
A: at the station (A = {w1,w2})
B: at the Palace (B = {w1,w3})
* ?
* _Let utilities be such that they only distinguish between success (value 1) and failure (value 0).
Let s consider answer A = {w1,w2}.
Assume that the speaker knows that A, i.e. there are Italian newspapers at the station.
6~]`The CalculationIf hearing A induces hearer to choose a (i.e. aA=a going to station ):
If hearing A induces hearer to choose b (i.e. aA=b going to Palace ):
If PE(B) = 1, then EUE(A) = EUE(b) = 1.
PE(B) < 1 leads to a contradiction.L# >.H3+bPE(B) < 1 leads to a contradiction:
aA = b implies EUI(bA) EUI(aA) = 1.
Hence, EUI(bA) = vA PI(v) u(v,b) =1.
Therefore PI(BA) =1, hence PI(BA) = PI(A), hence PI(A\B)=0.
PE(A\B)=0, due to wellbehavedness.
PE(BA)=PE(A)=1, hence PE(B) = 1.>%ZnZ %
[$a=Case: Speaker knows that Italian newspaper are at both places>>(fCalculation showed that EUE(A) = 1.
Expected utility cannot be higher than 1 (due to assumptions).
Similar: EUE(B) = 1; EUE(AB) = 1.
Hence, all these answers are equally optimal.O5y7
More Cases,E knows that A and B:
EUE(A) = EUE(B) = EUE(AB)
E knows that A and B:
EUE(A) = EUE(A B)
E knows only that A:
For all admissible C: EUE(C) EUE(A)` &
The following example shows how the method of finding optimal answers in support problems interacts with the general theory of implicatures in signalling games.$
DHip Hop at Roter SalonJohn loves to dance to Salsa music and he loves to dance to Hip Hop but he can t stand it if a club mixes both styles.
J: I want to dance tonight. Is the Music in Roter Salon ok?
E: Tonight they play Hip Hop at the Roter Salon.
+> They play only Hip Hop.0wnw,/#]FA game tree for the situation where both Salsa and Hip Hop are playingGG G+After the first step of backward induction:,,(^,After the second step of backward induction:(If we say that a proposition is the more relevant the higher the expected utility after learning it, then relevance scale approaches predict that Hip Hop implicates that both, Salsa and Hip Hop, are playing.
Worst case compatible with what was said!(*(2$EHip Hop at Roter SalonAbbreviations:
Good(x) := HAssumptions hEqual Probabilities
Independence: X,Y{H,S,Good}T5n&
ILearning H(x) or S(x) raises expected utility of going to salon x:
EUI(goingtox) < EUI(stayhome) < EUI(goingtoxH(x))
EUI(goingtox) < EUI(stayhome) < EUI(goingtoxS(x)) Cnrn ,
/ KViolating Assumptions IIThe Roter Salon and the Grner Salon share two DJs. One of them only plays Salsa, the other one mainly plays Hip Hop but mixes into it some Salsa. There are only these two Djs, and if one of them is at the Roter Salon, then the other one is at the Grner Salon. John loves to dance to Salsa music and he loves to dance to Hip Hop but he can t stand it if a club mixes both styles.
J: I want to dance tonight. Is the Music in Roter Salon ok?
E: Tonight they play Hip Hop at the Roter Salon.^~Zn Z"" % 0%Support Problems and Signalling Games%In our model, the speaker finds an optimal answer by backward induction in support problems.
This is not a standard method for solving coordination problems in signalling games.H$~Signalling Game
A signalling game is a tuple:
N,, p, (A1,A2), (u1, u2)
N: Set of two players S,H.
: Set of types representing the speakers private information
p: A probability measure over representing the hearer s expectations about S type. "
"Solution to a Signalling GameThe standard solution concept for Signalling games is that of a perfect Bayesian equilibrium!
(S,H) strategies:
S : A1
H : A1 A2q @$"$$$
$,$$
$"$$$
$$_
"Perfect Bayesian equilibrium (S,H)6#((((" S() argmaxF u1(F,H(F),)
"F H(F) argmaxM m(F)u2(F,M,)
where m is defined by
m(F) = 0 if S()F
m(F) = p() / p(S1[F]) if S()=F
if p(S1[F]) > 0, else m(F) is arbitrary.rExx:x,x H$
%Structural Descriptions with Contexts&&(
4A structural description with contexts is a tuple , F, M, Gen with:
a set of possible worlds,
F a set of forms,
M a set of meanings,
Gen F M:HS2 ! ! *3eTask
zGiven:
a set of support problems S with fixed decision problem (,PI,A,u) for a
structural description , F, M, Gen.
Wanted:
Representation as signalling game:
N,, p, (AE,AI), (uE, uI)ZpZZ#ZZ! + T>YConstruction
(
^Let s=(,PE,PI,A,u) be a given support problem.
Remember: there is a common prior P on such that:
PE(X) = PI(XKs) for Ks:= {wW PE(w) > 0}
Add Ks to (i.e. = {Ks sS})
The speaker s action set AE is identical with the set of forms F.
The hearer s action set is identical to the action set of s.:dZ,Z!ZZ"
E
"
""""*#"#<"
0(<
The game is a game of pure coordination with respect to joint payoff functions
ui: F AI R
uI(A,a,K) := EUI(aK)
uE(A,a,K) := EUE(aK) (= EUI(aK))
On; O !
RN>
(p is arbitrary.
Forms F have to be interpreted by their semantic meaning [F].
The speaker has to conform to the maxim of quality, i.e. S(Ks) Adms
n ` *MFResult
pThe strategy pairs defined by:
S(Ks) Ops, H(A) = aA
are Perfect Bayesian Equilibria of the associated signalling game.
they (weakly) Pareto dominate all other strategy pairs (S ,H )."
/p
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BildschirmprsentationSDUSM
TArialTimes New Roman
WingdingsArial BlackSymbolScript MT BoldPixel.Game Theory and Gricean Pragmatics Lesson IIICourse OverviewOptimal Answer ApproachOverview of Lesson IIIThe Agenda5Natural Information and Conversational Implicatures 4Explanation of Implicatures Optimal Answer Approach ContrastPowerPointPrsentationRepresentation of AssumptionBackground (Repetition) An Example The GameThe Solved Game3The hearer can infer after receiving A(some) that:4Natural Information and Conversational Implicatures Natural and NonNatural MeaningExamplePowerPointPrsentationNatural Information of SignalsPowerPointPrsentation"The Standard Example reconsidered#The game defined by pure semanticsThe game after optimising speakers strategyThe possible worlds'The possible Branches of the Game Tree6The unique signalling strategy that solves this game:5The Natural Information carried by utterance A(some)!Implicatures in Signalling Games As Signalling Game (Repetition)PowerPointPrsentation Strategies in a Signalling Game)Definition of Implicature (special case)PreviewPowerPointPrsentationOptimal AnswersCore ExamplesItalian NewspaperPowerPointPrsentationPartial AnswersPowerPointPrsentation$Optimal Answers in Support ProblemsSupport ProblemSupport ProblemIs Decision SituationPowerPointPrsentationEs Decision SituationPowerPointPrsentation ExamplesItalian Newspaper#Possible Worlds (equally probable)Actions and AnswersPowerPointPrsentationThe CalculationPowerPointPrsentation>Case: Speaker knows that Italian newspaper are at both placesMore CasesPowerPointPrsentationHip Hop at Roter SalonGA game tree for the situation where both Salsa and Hip Hop are playing,After the first step of backward induction:After the second step of backward induction:PowerPointPrsentationHip Hop at Roter Salon
Assumptions PowerPointPrsentationViolating Assumptions II&Support Problems and Signalling GamesPowerPointPrsentationSignalling GameSolution to a Signalling Game#Perfect Bayesian equilibrium (S,H)Task
ConstructionPowerPointPrsentation ResultVerwendete SchriftartenEntwurfsvorlageFolientitelMFolientitelMgeFolie"_/
Anton BenzAnton Benz"_
Anton BenzAnton BenzAnton Benz
Zentrum fr Allgemeine Sprachwissenschaften
ZAS Berlin .PP9P9&7eCourse OverviewLesson 1: Introduction
From Grice to Lewis
Relevance Scale Approaches
Lesson 2: Signalling Games
Lewis Signalling Conventions
Parikh s Radical Underspecification Model
Lesson 3: The Optimal Answer Approach I
Lesson 4: The Optimal Answer Approach II
Comparison with Relevance Scale Approaches
Decision Contexts with Multiple ObjectivesZ0ZZIZQZVZ0H&()Vo
VcOptimal Answer Approach 0Lesson III April, 5th ( > dOverview of Lesson III
&Natural Information and Conversational Implicatures
An Example: Scalar Implicatures
Natural Information and Conversational Implicatures
Calculating Implicatures in Signalling Games
Optimal Answers
Core Examples
Optimal Answers in Support Problems
Examples
Support Problems and Signalling Games 4ZZZ;Z'Z4;%"
'
The AgendaPutting Grice on Lewisean feet! (f4Natural Information and Conversational Implicatures 55.
PJ3Explanation of Implicatures Optimal Answer Approach$4* PStart with a signalling game where the hearer interprets forms by their literal meaning.
Impose pragmatic constraints and calculate equilibria that solve this game.
Implicature F +> j is explained if for all solutions (S,H):
S1(F) = jfnn+O
ContrastIn an information based approach:
Implicatures emerge from indicated meaning (in the sense of Lewis).
Implicatures are not initial candidate interpretations.
Speaker does not maximise relevance.
No diachronic process.X"ZZ;+1,8$hAssumption: speaker and hearer use language according to a semantic convention.
Goal: Explain how implicatures can emerge out of semantic language use.
Nonreductionist perspective..iRepresentation of AssumptionSemantics defines interpretation of forms.
Let [F] denote the semantic meaning.
Hence, assumption: H(F)=[F], i.e.:
H(F) is the semantic meaning of F
Semantic meaning @ Lewis imperative signal.VtZ"ZZZ8K Background (Repetition)
Lewis (IV.4,1996) distinguishes between
indicative signals
imperative signals
Two possible definitions of meaning:
Indicative:
[F] = M :iff S1(F)=M
Imperative:
[F] = M :iff H(F)=M(P&P(PPP
PPN
, Q
An ExamplexWe consider the standard example:
Some of the boys came to the party.
said: at least two came
implicated: not all came\"% 1HSThe GameUThe Solved GameV2The hearer can infer after receiving A(some) that:33(%[3Natural Information and Conversational Implicatures443 VNatural and NonNatural Meaning (mGrice distinguished between
natural meaning
nonnatural meaning
Communicated meaning is nonnatural meaning.
8$nWExample$I show Mr. X a photograph of Mr. Y displaying undue familiarity to Mrs. X.
I draw a picture of Mr. Y behaving in this manner and show it to Mr. X.
The photograph naturally means that Mr. Y was unduly familiar to Mrs. X
The picture nonnaturally means that Mr. Y was unduly familiar to Mrs. XnZnZZ <
/X
eTaking a photo of a scene necessarily entails that the scene is real.
Every branch which contains a showing of a photo must contain a situation which is depicted by it.
The showing of the photo means naturally that there was a situation where Mr. Y was unduly familiar with Mrs. X.
The drawing of a picture does not imply that the depicted scene is real.BGZZIZGIYNatural Information of SignalsLet G be a signalling game.
Let S be a set of strategy pairs (S,H).
We identify the natural information of a form F in G with respect to S with:
The set of all branches of G where the speaker chooses F. ;h<<VZInformation coincides with S1(F) in case of simple Lewisean signalling games.
Generalises to arbitrary games which contain semantic interpretation games in embedded form.
Conversational Implicatures are implied by the natural information of an utterance.<S4!The Standard Example reconsidered
"V
Some of the boys came to the party.
said: at least two came
implicated: not all cameF& 1& UX"The game defined by pure semantics##(YXThe game after optimising speaker s strategy(\The possible worldsw1: 100% of the boys came to the party.
w2: More than 50% of the boys came to the party.
w3: Less than 50% of the boys came to the party.H'0/^&The possible Branches of the Game Tree''(`5The unique signalling strategy that solves this game:66(Z4The Natural Information carried by utterance A(some)55(NThe branches allowed by strategy S:
w1,A(all), {w1}
w2,A(most), {w1,w2}
w3,A(some), {w1,w2,w3}
Natural information carried by A(some):
{w3,A(some), {w1,w2,w3}}$ZAZ(ZPZ2$
(
b Implicatures in Signalling Games!!(A special case
c!As Signalling Game (Repetition)&
A signalling game is a tuple:
N,, p, (A1,A2), (u1, u2)
N: Set of two players S,H.
: Set of types representing the speakers private information.
p: A probability measure over representing the hearer s expectations about the speaker s type. "
"d"p(A1,A2): the speaker s and hearer s action sets:
A1 is a set of forms F / meanings M.
A2 is a set of actions.
(u1,u2): the speaker s and hearer s payoff functions with
ui: A1A2 RP1>:+5 e#Strategies in a Signalling GameLet [ ] : F M be a given semantics.
The speaker s strategies are of the form:
S : A1 such that
S() = F [F]
i.e. if the speaker says F, then he knows that F is true (Maxim of Quality).Pu
+
@NfDf$(Definition of Implicature(special case))*Given a signalling game as before, then an implicature
F +> y
is explained iff the following set is a subset of [y] = {w  w = y}:V8 H =><L,h&Preview
Later, we apply this criterion to calculating implicatures of answers.
The definition depends on the method of finding solutions.n[First we need a method for calculating optimal answers.
The resulting signalling and interpretation strategies are then the solutions which we use as imput for calculating implicatures. *'<8Ng%Optimal Answersi'
Core Examplesj(Italian NewspaperSomewhere in the streets of Amsterdam...
J: Where can I buy an Italian newspaper?
E: At the station and at the Palace but nowhere else. (SE)
E: At the station. (A) / At the Palace. (B)
.)n)k)The answer (SE) is called strongly exhaustive.
The answers (A) and (B) are called mention some answers.
A and B are as good as SE or as A B:
E: There are Italian newspapers at the station but none at the Palace./9@)@Gn#$Gl*Partial AnswersIf E knows only that A, then A is an optimal answer:
E: There are no Italian newspapers at the station.
If E only knows that the Palace sells foreign newspapers, then this is an optimal answer:
E: The Palace has foreign newspapers.^74nZ&n74Z&m+Partial answers may also arise in situations where speaker E has full knowledge:
I: I need patrol for my car. Where can I get it?
E: There is a garage round the corner.
J: Where can I buy an Italian newspaper?
E: There is a news shop round the corner.NQ@1'@ )* n,#Optimal Answers in Support Problems$$#
The Framework
oSupport ProblemDefinition: A support problem is a five tuple (,PE,PI,A,u) such that
(, PE) and (, PI) are finite probability spaces,
(,PI,A, u) is a decision problem.
Let K:= {wW PE(w) > 0 } (E s knowledge set).
Then, we assume in addition:
for all A : PE(A) = PI(AK) .G3$@M!n#"
"
"
!"
=
x/&"np.Support Problemq/,I s Decision Situation,I optimises expected utilities of actions:
,r0I will choose an action aA that optimises expected utility, i.e. for all actions b
EU(b,A) EU(aA,A)
Given answer A, H(A) = aA.
For simplicity we assume that I s choice aA is commonly known.HT ?6+t: s1,E s Decision Situation*E optimises expected utilities of answers:++t2
^(Quality): The speaker can only say what he thinks to be true.
(Quality) restricts answers to:$_>$ u3Examples The Italian Newspaper Examplesv5Italian NewspaperSomewhere in the streets of Amsterdam...
J: Where can I buy an Italian newspaper?
E: At the station and at the Palace but nowhere else. (SE)
E: At the station. (A) / At the Palace. (B)
.)n)w4"Possible Worlds (equally probable)##(x6Actions and AnswersI s actions:
a: going to station;
b: going to Palace;
Answers:
A: at the station (A = {w1,w2})
B: at the Palace (B = {w1,w3})
* ?
* _Let utilities be such that they only distinguish between success (value 1) and failure (value 0).
Let s consider answer A = {w1,w2}.
Assume that the speaker knows that A, i.e. there are Italian newspapers at the station.
6~]`The CalculationIf hearing A induces hearer to choose a (i.e. aA=a going to station ):
If hearing A induces hearer to choose b (i.e. aA=b going to Palace ):
If PE(B) = 1, then EUE(A) = EUE(B) = 1.
PE(B) < 1 leads to a contradiction.L# J.H3+bPE(B) < 1 leads to a contradiction:
aA = b implies EUI(bA) EUI(aA) = 1.
Hence, EUI(bA) = vA PI(v) u(v,b) =1.
Therefore PI(BA) =1, hence PI(BA) = PI(A), hence PI(A\B)=0.
PE(A\B)=0, due to wellbehavedness.
PE(BA)=PE(A)=1, hence PE(B) = 1.>%ZnZ %
[$a=Case: Speaker knows that Italian newspaper are at both places>>(fCalculation showed that EUE(A) = 1.
Expected utility cannot be higher than 1 (due to assumptions).
Similar: EUE(B) = 1; EUE(AB) = 1.
Hence, all these answers are equally optimal.O5y7
More Cases,E knows that A and B:
EUE(A) = EUE(B) = EUE(AB)
E knows that A and B:
EUE(A) = EUE(A B)
E knows only that A:
For all admissible C: EUE(C) EUE(A)` &
The following example shows how the method of finding optimal answers in support problems interacts with the general theory of implicatures in signalling games.$
DHip Hop at Roter SalonJohn loves to dance to Salsa music and he loves to dance to Hip Hop but he can t stand it if a club mixes both styles.
J: I want to dance tonight. Is the Music in Roter Salon ok?
E: Tonight they play Hip Hop at the Roter Salon.
+> They play only Hip Hop.0wnw,/#]FA game tree for the situation where both Salsa and Hip Hop are playingGG G+After the first step of backward induction:,,(^,After the second step of backward induction:(If we say that a proposition is the more relevant the higher the expected utility after learning it, then relevance scale approaches predict that Hip Hop implicates that both, Salsa and Hip Hop, are playing.
Worst case compatible with what was said!(*(2$EHip Hop at Roter SalonAbbreviations:
Good(x) := HAssumptions hEqual Probabilities
Independence: X,Y{H,S,Good}T5n&
ILearning H(x) or S(x) raises expected utility of going to salon x:
EUI(goingtox) < EUI(stayhome) < EUI(goingtoxH(x))
EUI(goingtox) < EUI(stayhome) < EUI(goingtoxS(x)) Cnrn ,
/ KViolating Assumptions IIThe Roter Salon and the Grner Salon share two DJs. One of them only plays Salsa, the other one mainly plays Hip Hop but mixes into it some Salsa. There are only these two Djs, and if one of them is at the Roter Salon, then the other one is at the Grner Salon. John loves to dance to Salsa music and he loves to dance to Hip Hop but he can t stand it if a club mixes both styles.
J: I want to dance tonight. Is the Music in Roter Salon ok?
E: Tonight they play Hip Hop at the Roter Salon.^~Zn Z"" % 0%Support Problems and Signalling Games%In our model, the speaker finds an optimal answer by backward induction in support problems.
This is not a standard method for solving coordination problems in signalling games.H$~Signalling Game
A signalling game is a tuple:
N,, p, (A1,A2), (u1, u2)
N: Set of two players S,H.
: Set of types representing the speakers private information
p: A probability measure over representing the hearer s expectations about S type. "
"Solution to a Signalling GameThe standard solution concept for Signalling games is that of a perfect Bayesian equilibrium!
(S,H) strategies:
S : A1
H : A1 A2q @$"$$$
$,$$
$"$$$
$$_
"Perfect Bayesian equilibrium (S,H)6#((((" S() argmaxF u1(F,H(F),)
"F H(F) argmaxM m(F)u2(F,M,)
where m is defined by
m(F) = 0 if S()F
m(F) = p() / p(S1[F]) if S()=F
if p(S1[F]) > 0, else m(F) is arbitrary.rExx:x,x H$
%Structural Descriptions with Contexts&&(
4A structural description with contexts is a tuple , F, M, Gen with:
a set of possible worlds,
F a set of forms,
M a set of meanings,
Gen F M:HS2 ! ! *3eTask
zGiven:
a set of support problems S with fixed decision problem (,PI,A,u) for a
structural description , F, M, Gen.
Wanted:
Representation as signalling game:
N,, p, (AE,AI), (uE, uI)ZpZZ#ZZ! + T>YConstruction
(
^Let s=(,PE,PI,A,u) be a given support problem.
Remember: there is a common prior P on such that:
PE(X) = PI(XKs) for Ks:= {wW PE(w) > 0}
Add Ks to (i.e. = {Ks sS})
The speaker s action set AE is identical with the set of forms F.
The hearer s action set is identical to the action set of s.:dZ,Z!ZZ"
E
"
""""*#"#<"
0(<
The game is a game of pure coordination with respect to joint payoff functions
ui: F AI R
uI(A,a,K) := EUI(aK)
uE(A,a,K) := EUE(aK) (= EUI(aK))
On; O !
RN>
(p is arbitrary.
Forms F have to be interpreted by their semantic meaning [F].
The speaker has to conform to the maxim of quality, i.e. S(Ks) Adms
n ` *MFResult
pThe strategy pairs defined by:
S(Ks) Ops, H(A) = aA
are Perfect Bayesian Equilibria of the associated signalling game.
they (weakly) Pareto dominate all other strategy pairs (S ,H )."
/p
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=cGame Theory and Gricean PragmaticsLesson III""CAnton Benz
Zentrum fr Allgemeine Sprachwissenschaften
ZAS Berlin .PP9P9&7eCourse OverviewLesson 1: Introduction
From Grice to Lewis
Relevance Scale Approaches
Lesson 2: Signalling Games
Lewis Signalling Conventions
Parikh s Radical Underspecification Model
Lesson 3: The Optimal Answer Approach I
Lesson 4: The Optimal Answer Approach II
Comparison with Relevance Scale Approaches
Decision Contexts with Multiple ObjectivesZ0ZZIZQZVZ0H&()Vo
VcOptimal Answer Approach 0Lesson III April, 5th ( > dOverview of Lesson III
&Natural Information and Conversational Implicatures
An Example: Scalar Implicatures
Natural Information and Conversational Implicatures
Calculating Implicatures in Signalling Games
Optimal Answers
Core Examples
Optimal Answers in Support Problems
Examples
Support Problems and Signalling Games 4ZZZ;Z'Z4;%"
'
The AgendaPutting Grice on Lewisean feet! (f4Natural Information and Conversational Implicatures 55.
PJ3Explanation of Implicatures Optimal Answer Approach$4* PStart with a signalling game where the hearer interprets forms by their literal meaning.
Impose pragmatic constraints and calculate equilibria that solve this game.
Implicature F +> j is explained if for all solutions (S,H):
S1(F) = jfnn+O
ContrastIn an information based approach:
Implicatures emerge from indicated meaning (in the sense of Lewis).
Implicatures are not initial candidate interpretations.
Speaker does not maximise relevance.
No diachronic process.X"ZZ;+1,8$hAssumption: speaker and hearer use language according to a semantic convention.
Goal: Explain how implicatures can emerge out of semantic language use.
Nonreductionist perspective..iRepresentation of AssumptionSemantics defines interpretation of forms.
Let [F] denote the semantic meaning.
Hence, assumption: H(F)=[F], i.e.:
H(F) is the semantic meaning of F
Semantic meaning @ Lewis imperative signal.VtZ"ZZZ8K Background (Repetition)
Lewis (IV.4,1996) distinguishes between
indicative signals
imperative signals
Two possible definitions of meaning:
Indicative:
[F] = M :iff S1(F)=M
Imperative:
[F] = M :iff H(F)=M(P&P(PPP
PPN
, Q
An ExamplexWe consider the standard example:
Some of the boys came to the party.
said: at least two came
implicated: not all came\"% 1HSThe GameUThe Solved GameV2The hearer can infer after receiving A(some) that:33(%[3Natural Information and Conversational Implicatures443 VNatural and NonNatural Meaning (mGrice distinguished between
natural meaning
nonnatural meaning
Communicated meaning is nonnatural meaning.
8$nWExample$I show Mr. X a photograph of Mr. Y displaying undue familiarity to Mrs. X.
I draw a picture of Mr. Y behaving in this manner and show it to Mr. X.
The photograph naturally means that Mr. Y was unduly familiar to Mrs. X
The picture nonnaturally means that Mr. Y was unduly familiar to Mrs. XnZnZZ <
/X
eTaking a photo of a scene necessarily entails that the scene is real.
Every branch which contains a showing of a photo must contain a situation which is depicted by it.
The showing of the photo means naturally that there was a situation where Mr. Y was unduly familiar with Mrs. X.
The drawing of a picture does not imply that the depicted scene is real.BGZZIZGIYNatural Information of SignalsLet G be a signalling game.
Let S be a set of strategy pairs (S,H).
We identify the natural information of a form F in G with respect to S with:
The set of all branches of G where the speaker chooses F. ;h<<VZInformation coincides with S1(F) in case of simple Lewisean signalling games.
Generalises to arbitrary games which contain semantic interpretation games in embedded form.
Conversational Implicatures are implied by the natural information of an utterance.<S4!The Standard Example reconsidered
"V
Some of the boys came to the party.
said: at least two came
implicated: not all cameF& 1& UX"The game defined by pure semantics##(YXThe game after optimising speaker s strategy(\The possible worldsw1: 100% of the boys came to the party.
w2: More than 50% of the boys came to the party.
w3: Less than 50% of the boys came to the party.H'0/^&The possible Branches of the Game Tree''(`5The unique signalling strategy that solves this game:66(Z4The Natural Information carried by utterance A(some)55(NThe branches allowed by strategy S:
w1,A(all), {w1}
w2,A(most), {w1,w2}
w3,A(some), {w1,w2,w3}
Natural information carried by A(some):
{w3,A(some), {w1,w2,w3}}$ZAZ(ZPZ2$
(
b Implicatures in Signalling Games!!(A special case
c!As Signalling Game (Repetition)&
A signalling game is a tuple:
N,, p, (A1,A2), (u1, u2)
N: Set of two players S,H.
: Set of types representing the speakers private information.
p: A probability measure over representing the hearer s expectations about the speaker s type. "
"d"p(A1,A2): the speaker s and hearer s action sets:
A1 is a set of forms F / meanings M.
A2 is a set of actions.
(u1,u2): the speaker s and hearer s payoff functions with
ui: A1A2 RP1>:+5 e#Strategies in a Signalling GameLet [ ] : F M be a given semantics.
The speaker s strategies are of the form:
S : A1 such that
S() = F [F]
i.e. if the speaker says F, then he knows that F is true (Maxim of Quality).Pu
+
@NfDf$(Definition of Implicature(special case))*Given a signalling game as before, then an implicature
F +> y
is explained iff the following set is a subset of [y] = {w  w = y}:V8 H =><L,h&Preview
Later, we apply this criterion to calculating implicatures of answers.
The definition depends on the method of finding solutions.n[First we need a method for calculating optimal answers.
The resulting signalling and interpretation strategies are then the solutions which we use as imput for calculating implicatures. *'<8Ng%Optimal Answersi'
Core Examplesj(Italian NewspaperSomewhere in the streets of Amsterdam...
J: Where can I buy an Italian newspaper?
E: At the station and at the Palace but nowhere else. (SE)
E: At the station. (A) / At the Palace. (B)
.)n)k)The answer (SE) is called strongly exhaustive.
The answers (A) and (B) are called mention some answers.
A and B are as good as SE or as A B:
E: There are Italian newspapers at the station but none at the Palace./9@)@Gn#$Gl*Partial AnswersIf E knows only that A, then A is an optimal answer:
E: There are no Italian newspapers at the station.
If E only knows that the Palace sells foreign newspapers, then this is an optimal answer:
E: The Palace has foreign newspapers.^74nZ&n74Z&m+Partial answers may also arise in situations where speaker E has full knowledge:
I: I need patrol for my car. Where can I get it?
E: There is a garage round the corner.
J: Where can I buy an Italian newspaper?
E: There is a news shop round the corner.NQ@1'@ )* n,#Optimal Answers in Support Problems$$#
The Framework
oSupport ProblemDefinition: A support problem is a five tuple (,PE,PI,A,u) such that
(, PE) and (, PI) are finite probability spaces,
(,PI,A, u) is a decision problem.
Let K:= {wW PE(w) > 0 } (E s knowledge set).
Then, we assume in addition:
for all A : PE(A) = PI(AK) .G3$@M!n#"
"
"
!"
=
x/&"np.Support Problemq/,I s Decision Situation,I optimises expected utilities of actions:
,r0I will choose an action aA that optimises expected utility, i.e. for all actions b
EU(b,A) EU(aA,A)
Given answer A, H(A) = aA.
For simplicity we assume that I s choice aA is commonly known.HT ?6+t: s1,E s Decision Situation*E optimises expected utilities of answers:++t2
^(Quality): The speaker can only say what he thinks to be true.
(Quality) restricts answers to:$_>$ u3Examples The Italian Newspaper Examplesv5Italian NewspaperSomewhere in the streets of Amsterdam...
J: Where can I buy an Italian newspaper?
E: At the station and at the Palace but nowhere else. (SE)
E: At the station. (A) / At the Palace. (B)
.)n)w4"Possible Worlds (equally probable)##(x6Actions and AnswersI s actions:
a: going to station;
b: going to Palace;
Answers:
A: at the station (A = {w1,w2})
B: at the Palace (B = {w1,w3})
* ?
* _Let utilities be such that they only distinguish between success (value 1) and failure (value 0).
Let s consider answer A = {w1,w2}.
Assume that the speaker knows that A, i.e. there are Italian newspapers at the station.
6~]`The CalculationIf hearing A induces hearer to choose a (i.e. aA=a going to station ):
If hearing A induces hearer to choose b (i.e. aA=b going to Palace ):
If PE(B) = 1, then EUE(A) = EUE(B) = 1.
PE(B) < 1 leads to a contradiction.L# J.H3+bPE(B) < 1 leads to a contradiction:
aA = b implies EUI(bA) EUI(aA) = 1.
Hence, EUI(bA) = vA PI(v) u(v,b) =1.
Therefore PI(BA) =1, hence PI(BA) = PI(A), hence PI(A\B)=0.
PE(A\B)=0, because $K: PE(X)= PI(XK).
PE(BA)=PE(A)=1, hence PE(B) = 1.%ZnZ
%
O"a=Case: Speaker knows that Italian newspaper are at both places>>(fCalculation showed that EUE(A) = 1.
Expected utility cannot be higher than 1 (due to assumptions).
Similar: EUE(B) = 1; EUE(AB) = 1.
Hence, all these answers are equally optimal.O5y7
More Cases,E knows that A and B:
EUE(A) = EUE(B) = EUE(AB)
E knows that A and B:
EUE(A) = EUE(A B)
E knows only that A:
For all admissible C: EUE(C) EUE(A)` &
The following example shows how the method of finding optimal answers in support problems interacts with the general theory of implicatures in signalling games.$
DHip Hop at Roter SalonJohn loves to dance to Salsa music and he loves to dance to Hip Hop but he can t stand it if a club mixes both styles.
J: I want to dance tonight. Is the Music in Roter Salon ok?
E: Tonight they play Hip Hop at the Roter Salon.
+> They play only Hip Hop.0wnw,/#]FA game tree for the situation where both Salsa and Hip Hop are playingGG G+After the first step of backward induction:,,(^,After the second step of backward induction:(If we say that a proposition is the more relevant the higher the expected utility after learning it, then relevance scale approaches predict that Hip Hop implicates that both, Salsa and Hip Hop, are playing.
Worst case compatible with what was said!(*(2$EHip Hop at Roter SalonAbbreviations:
Good(x) := HAssumptions hEqual Probabilities
Independence: X,Y{H,S,Good}T5n&
ILearning H(x) or S(x) raises expected utility of going to salon x:
EUI(goingtox) < EUI(stayhome) < EUI(goingtoxH(x))
EUI(goingtox) < EUI(stayhome) < EUI(goingtoxS(x)) Cnrn ,
/ KViolating Assumptions IIThe Roter Salon and the Grner Salon share two DJs. One of them only plays Salsa, the other one mainly plays Hip Hop but mixes into it some Salsa. There are only these two Djs, and if one of them is at the Roter Salon, then the other one is at the Grner Salon. John loves to dance to Salsa music and he loves to dance to Hip Hop but he can t stand it if a club mixes both styles.
J: I want to dance tonight. Is the Music in Roter Salon ok?
E: Tonight they play Hip Hop at the Roter Salon.^~Zn Z"" % 0%Support Problems and Signalling Games%In our model, the speaker finds an optimal answer by backward induction in support problems.
This is not a standard method for solving coordination problems in signalling games.H$~Signalling Game
A signalling game is a tuple:
N,, p, (A1,A2), (u1, u2)
N: Set of two players S,H.
: Set of types representing the speakers private information
p: A probability measure over representing the hearer s expectations about S type. "
"Solution to a Signalling GameThe standard solution concept for Signalling games is that of a perfect Bayesian equilibrium!
(S,H) strategies:
S : A1
H : A1 A2q @$"$$$
$,$$
$"$$$
$$_
"Perfect Bayesian equilibrium (S,H)6#((((" S() argmaxF u1(F,H(F),)
"F H(F) argmaxM m(F)u2(F,M,)
where m is defined by
m(F) = 0 if S()F
m(F) = p() / p(S1[F]) if S()=F
if p(S1[F]) > 0, else m(F) is arbitrary.rExx:x,x H$
%Structural Descriptions with Contexts&&(
4A structural description with contexts is a tuple , F, M, Gen with:
a set of possible worlds,
F a set of forms,
M a set of meanings,
Gen F M:HS2 ! ! *3eTask
zGiven:
a set of support problems S with fixed decision problem (,PI,A,u) for a
structural description , F, M, Gen.
Wanted:
Representation as signalling game:
N,, p, (AE,AI), (uE, uI)ZpZZ#ZZ! + T>YConstruction
(
^Let s=(,PE,PI,A,u) be a given support problem.
Remember: there is a common prior P on such that:
PE(X) = PI(XKs) for Ks:= {wW PE(w) > 0}
Add Ks to (i.e. = {Ks sS})
The speaker s action set AE is identical with the set of forms F.
The hearer s action set is identical to the action set of s.:dZ,Z!ZZ"
E
"
""""*#"#<"
0(<
The game is a game of pure coordination with respect to joint payoff functions
ui: F AI R
uI(A,a,K) := EUI(aK)
uE(A,a,K) := EUE(aK) (= EUI(aK))
On; O !
RN>
(p is arbitrary.
Forms F have to be interpreted by their semantic meaning [F].
The speaker has to conform to the maxim of quality, i.e. S(Ks) Adms
n ` *MFResult
pThe strategy pairs defined by:
S(Ks) Ops, H(A) = aA
are Perfect Bayesian Equilibria of the associated signalling game.
they (weakly) Pareto dominate all other strategy pairs (S ,H )."
/p:
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=cGame Theory and Gricean PragmaticsLesson III""CAnton Benz
Zentrum fr Allgemeine Sprachwissenschaften
ZAS Berlin .PP9P9&7eCourse OverviewLesson 1: Introduction
From Grice to Lewis
Relevance Scale Approaches
Lesson 2: Signalling Games
Lewis Signalling Conventions
Parikh s Radical Underspecification Model
Lesson 3: The Optimal Answer Approach I
Lesson 4: The Optimal Answer Approach II
Comparison with Relevance Scale Approaches
Decision Contexts with Multiple ObjectivesZ0ZZIZQZVZ0H&()Vo
VcOptimal Answer Approach 0Lesson III April, 5th ( > dOverview of Lesson III
&Natural Information and Conversational Implicatures
An Example: Scalar Implicatures
Natural Information and Conversational Implicatures
Calculating Implicatures in Signalling Games
Optimal Answers
Core Examples
Optimal Answers in Support Problems
Examples
Support Problems and Signalling Games 4ZZZ;Z'Z4;%"
'
The AgendaPutting Grice on Lewisean feet! (f4Natural Information and Conversational Implicatures 55.
PJ3Explanation of Implicatures Optimal Answer Approach$4* PStart with a signalling game where the hearer interprets forms by their literal meaning.
Impose pragmatic constraints and calculate equilibria that solve this game.
Implicature F +> j is explained if for all solutions (S,H):
S1(F) = jfnn+O
ContrastIn an information based approach:
Implicatures emerge from indicated meaning (in the sense of Lewis).
Implicatures are not initial candidate interpretations.
Speaker does not maximise relevance.
No diachronic process.X"ZZ;+1,8$hAssumption: speaker and hearer use language according to a semantic convention.
Goal: Explain how implicatures can emerge out of semantic language use.
Nonreductionist perspective..iRepresentation of AssumptionSemantics defines interpretation of forms.
Let [F] denote the semantic meaning.
Hence, assumption: H(F)=[F], i.e.:
H(F) is the semantic meaning of F
Semantic meaning @ Lewis imperative signal.VtZ"ZZZ8K Background (Repetition)
Lewis (IV.4,1996) distinguishes between
indicative signals
imperative signals
Two possible definitions of meaning:
Indicative:
[F] = M :iff S1(F)=M
Imperative:
[F] = M :iff H(F)=M(P&P(PPP
PPN
, Q
An ExamplexWe consider the standard example:
Some of the boys came to the party.
said: at least two came
implicated: not all came\"% 1HSThe GameUThe Solved GameV2The hearer can infer after receiving A(some) that:33(%[3Natural Information and Conversational Implicatures443 VNatural and NonNatural Meaning (mGrice distinguished between
natural meaning
nonnatural meaning
Communicated meaning is nonnatural meaning.
8$nWExample$I show Mr. X a photograph of Mr. Y displaying undue familiarity to Mrs. X.
I draw a picture of Mr. Y behaving in this manner and show it to Mr. X.
The photograph naturally means that Mr. Y was unduly familiar to Mrs. X
The picture nonnaturally means that Mr. Y was unduly familiar to Mrs. XnZnZZ <
/X
eTaking a photo of a scene necessarily entails that the scene is real.
Every branch which contains a showing of a photo must contain a situation which is depicted by it.
The showing of the photo means naturally that there was a situation where Mr. Y was unduly familiar with Mrs. X.
The drawing of a picture does not imply that the depicted scene is real.BGZZIZGIYNatural Information of SignalsLet G be a signalling game.
Let S be a set of strategy pairs (S,H).
We identify the natural information of a form F in G with respect to S with:
The set of all branches of G where the speaker chooses F. ;h<<VZInformation coincides with S1(F) in case of simple Lewisean signalling games.
Generalises to arbitrary games which contain semantic interpretation games in embedded form.
Conversational Implicatures are implied by the natural information of an utterance.<S4!The Standard Example reconsidered
"V
Some of the boys came to the party.
said: at least two came
implicated: not all cameF& 1& UX"The game defined by pure semantics##(YXThe game after optimising speaker s strategy(\The possible worldsw1: 100% of the boys came to the party.
w2: More than 50% of the boys came to the party.
w3: Less than 50% of the boys came to the party.H'0/^&The possible Branches of the Game Tree''(`5The unique signalling strategy that solves this game:66(Z4The Natural Information carried by utterance A(some)55(NThe branches allowed by strategy S:
w1,A(all), {w1}
w2,A(most), {w1,w2}
w3,A(some), {w1,w2,w3}
Natural information carried by A(some):
{w3,A(some), {w1,w2,w3}}$ZAZ(ZPZ2$
(
b Implicatures in Signalling Games!!(A special case
c!As Signalling Game (Repetition)&
A signalling game is a tuple:
N,, p, (A1,A2), (u1, u2)
N: Set of two players S,H.
: Set of types representing the speakers private information.
p: A probability measure over representing the hearer s expectations about the speaker s type. "
"d"p(A1,A2): the speaker s and hearer s action sets:
A1 is a set of forms F / meanings M.
A2 is a set of actions.
(u1,u2): the speaker s and hearer s payoff functions with
ui: A1A2 RP1>:+5 e#Strategies in a Signalling GameLet [ ] : F M be a given semantics.
The speaker s strategies are of the form:
S : A1 such that
S() = F [F]
i.e. if the speaker says F, then he knows that F is true (Maxim of Quality).Pu
+
@NfDf$(Definition of Implicature(special case))*Given a signalling game as before, then an implicature
F +> y
is explained iff the following set is a subset of [y] = {w  w = y}:V8 H =><L,h&Preview
Later, we apply this criterion to calculating implicatures of answers.
The definition depends on the method of finding solutions.n[First we need a method for calculating optimal answers.
The resulting signalling and interpretation strategies are then the solutions which we use as imput for calculating implicatures. *'<8Ng%Optimal Answersi'
Core Examplesj(Italian NewspaperSomewhere in the streets of Amsterdam...
J: Where can I buy an Italian newspaper?
E: At the station and at the Palace but nowhere else. (SE)
E: At the station. (A) / At the Palace. (B)
.)n)k)The answer (SE) is called strongly exhaustive.
The answers (A) and (B) are called mention some answers.
A and B are as good as SE or as A B:
E: There are Italian newspapers at the station but none at the Palace./9@)@Gn#$Gl*Partial AnswersIf E knows only that A, then A is an optimal answer:
E: There are no Italian newspapers at the station.
If E only knows that the Palace sells foreign newspapers, then this is an optimal answer:
E: The Palace has foreign newspapers.^74nZ&n74Z&m+Partial answers may also arise in situations where speaker E has full knowledge:
I: I need patrol for my car. Where can I get it?
E: There is a garage round the corner.
J: Where can I buy an Italian newspaper?
E: There is a news shop round the corner.NQ@1'@ )* n,#Optimal Answers in Support Problems$$#
The Framework
oSupport ProblemDefinition: A support problem is a five tuple (,PE,PI,A,u) such that
(, PE) and (, PI) are finite probability spaces,
(,PI,A, u) is a decision problem.
Let K:= {wW PE(w) > 0 } (E s knowledge set).
Then, we assume in addition:
for all A : PE(A) = PI(AK) .G3$@M!n#"
"
"
!"
=
x/&"np.Support Problemq/,I s Decision Situation,I optimises expected utilities of actions:
,r0I will choose an action aA that optimises expected utility, i.e. for all actions b
EU(b,A) EU(aA,A)
Given answer A, H(A) = aA.
For simplicity we assume that I s choice aA is commonly known.HT ?6+t: s1,E s Decision Situation*E optimises expected utilities of answers:++t2
^(Quality): The speaker can only say what he thinks to be true.
(Quality) restricts answers to:$_>$ u3Examples The Italian Newspaper Examplesv5Italian NewspaperSomewhere in the streets of Amsterdam...
J: Where can I buy an Italian newspaper?
E: At the station and at the Palace but nowhere else. (SE)
E: At the station. (A) / At the Palace. (B)
.)n)w4"Possible Worlds (equally probable)##(x6Actions and AnswersI s actions:
a: going to station;
b: going to Palace;
Answers:
A: at the station (A = {w1,w2})
B: at the Palace (B = {w1,w3})
* ?
* _Let utilities be such that they only distinguish between success (value 1) and failure (value 0).
Let s consider answer A = {w1,w2}.
Assume that the speaker knows that A, i.e. there are Italian newspapers at the station.
6~]`The CalculationIf hearing A induces hearer to choose a (i.e. aA=a going to station ):
If hearing A induces hearer to choose b (i.e. aA=b going to Palace ):
If PE(B) = 1, then EUE(A) = EUE(B) = 1.
PE(B) < 1 leads to a contradiction.L# J.H3+bPE(B) < 1 leads to a contradiction:
aA = b implies EUI(bA) EUI(aA) = 1.
Hence, EUI(bA) = vA PI(v) u(v,b) =1.
Therefore PI(BA) =1, hence PI(BA) = PI(A), hence PI(A\B)=0.
PE(A\B)=0, because $K: PE(X)= PI(XK).
PE(BA)=PE(A)=1, hence PE(B) = 1.%ZnZ
%
O"a=Case: Speaker knows that Italian newspaper are at both places>>(fCalculation showed that EUE(A) = 1.
Expected utility cannot be higher than 1 (due to assumptions).
Similar: EUE(B) = 1; EUE(AB) = 1.
Hence, all these answers are equally optimal.O5y7
More Cases,E knows that A and B:
EUE(A) = EUE(B) = EUE(AB)
E knows that A and B:
EUE(A) = EUE(A B)
E knows only that A:
For all admissible C: EUE(C) EUE(A)` &
The following example shows how the method of finding optimal answers in support problems interacts with the general theory of implicatures in signalling games.$
DHip Hop at Roter SalonJohn loves to dance to Salsa music and he loves to dance to Hip Hop but he can t stand it if a club mixes both styles.
J: I want to dance tonight. Is the Music in Roter Salon ok?
E: Tonight they play Hip Hop at the Roter Salon.
+> They play only Hip Hop.0wnw,/#]FA game tree for the situation where both Salsa and Hip Hop are playingGG G+After the first step of backward induction:,,(^,After the second step of backward induction:(If we say that a proposition is the more relevant the higher the expected utility after learning it, then relevance scale approaches predict that Hip Hop implicates that both, Salsa and Hip Hop, are playing.
Worst case compatible with what was said!(*(2$EHip Hop at Roter SalonAbbreviations:
Good(x) := HAssumptions hEqual Probabilities
Independence: X,Y{H,S,Good}T5n&
ILearning H(x) or S(x) raises expected utility of going to salon x:
EUI(goingtox) < EUI(stayhome) < EUI(goingtoxH(x))
EUI(goingtox) < EUI(stayhome) < EUI(goingtoxS(x)) Cnrn ,
/ KViolating Assumptions IIThe Roter Salon and the Grner Salon share two DJs. One of them only plays Salsa, the other one mainly plays Hip Hop but mixes into it some Salsa. There are only these two Djs, and if one of them is at the Roter Salon, then the other one is at the Grner Salon. John loves to dance to Salsa music and he loves to dance to Hip Hop but he can t stand it if a club mixes both styles.
J: I want to dance tonight. Is the Music in Roter Salon ok?
E: Tonight they play Hip Hop at the Roter Salon.^~Zn Z"" % 0%Support Problems and Signalling Games%In our model, the speaker finds an optimal answer by backward induction in support problems.
This is not a standard method for solving coordination problems in signalling games.H$~Signalling Game
A signalling game is a tuple:
N,, p, (A1,A2), (u1, u2)
N: Set of two players S,H.
: Set of types representing the speakers private information
p: A probability measure over representing the hearer s expectations about S type. "
"Solution to a Signalling GameThe standard solution concept for Signalling games is that of a perfect Bayesian equilibrium!
(S,H) strategies:
S : A1
H : A1 A2q @$"$$$
$,$$
$"$$$
$$_
"Perfect Bayesian equilibrium (S,H)6#((((" S() argmaxF u1(F,H(F),)
"F H(F) argmaxM m(F)u2(F,M,)
where m is defined by
m(F) = 0 if S()F
m(F) = p() / p(S1[F]) if S()=F
if p(S1[F]) > 0, else m(F) is arbitrary.rExx:x,x H$
%Structural Descriptions with Contexts&&(
4A structural description with contexts is a tuple , F, M, Gen with:
a set of possible worlds,
F a set of forms,
M a set of meanings,
Gen F M:HS2 ! ! *3eTask
zGiven:
a set of support problems S with fixed decision problem (,PI,A,u) for a
structural description , F, M, Gen.
Wanted:
Representation as signalling game:
N,, p, (AE,AI), (uE, uI)ZpZZ#ZZ! + T>YConstruction
(
^Let s=(,PE,PI,A,u) be a given support problem.
Remember: there is a common prior P on such that:
PE(X) = PI(XKs) for Ks:= {wW PE(w) > 0}
Add Ks to (i.e. = {Ks sS})
The speaker s action set AE is identical with the set of forms F.
The hearer s action set is identical to the action set of s.:dZ,Z!ZZ"
E
"
""""*#"#<"
0(<
The game is a game of pure coordination with respect to joint payoff functions
ui: F AI R
uI(A,a,K) := EUI(aK)
uE(A,a,K) := EUE(aK) (= EUI(aK))
On; O !
RN>
(p is arbitrary.
Forms F have to be interpreted by their semantic meaning [F].
The speaker has to conform to the maxim of quality, i.e. S(Ks) Adms
n ` *MFResult
pThe strategy pairs defined by:
S(Ks) Ops, H(A) = aA
are Perfect Bayesian Equilibria of the associated signalling game.
they (weakly) Pareto dominate all other strategy pairs (S ,H )."
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=kcGame Theory and Gricean PragmaticsLesson III""CAnton Benz
Zentrum fr Allgemeine Sprachwissenschaften
ZAS Berlin .PP9P9&7eCourse OverviewLesson 1: Introduction
From Grice to Lewis
Relevance Scale Approaches
Lesson 2: Signalling Games
Lewis Signalling Conventions
Parikh s Radical Underspecification Model
Lesson 3: The Optimal Answer Approach I
Lesson 4: The Optimal Answer Approach II
Comparison with Relevance Scale Approaches
Decision Contexts with Multiple ObjectivesZ0ZZIZQZVZ0H&()Vo
VcOptimal Answer Approach 0Lesson III April, 5th ( > dOverview of Lesson III
&Natural Information and Conversational Implicatures
An Example: Scalar Implicatures
Natural Information and Conversational Implicatures
Calculating Implicatures in Signalling Games
Optimal Answers
Core Examples
Optimal Answers in Support Problems
Examples
Support Problems and Signalling Games 4ZZZ;Z'Z4;%"
'
The AgendaPutting Grice on Lewisean feet! (f4Natural Information and Conversational Implicatures 55.
PJ3Explanation of Implicatures Optimal Answer Approach$4* PStart with a signalling game where the hearer interprets forms by their literal meaning.
Impose pragmatic constraints and calculate equilibria that solve this game.
Implicature F +> j is explained if for all solutions (S,H):
S1(F) = jfnn+O
ContrastIn an information based approach:
Implicatures emerge from indicated meaning (in the sense of Lewis).
Implicatures are not initial candidate interpretations.
Speaker does not maximise relevance.
No diachronic process.X"ZZ;+1,8$hAssumption: speaker and hearer use language according to a semantic convention.
Goal: Explain how implicatures can emerge out of semantic language use.
Nonreductionist perspective..iRepresentation of AssumptionSemantics defines interpretation of forms.
Let [F] denote the semantic meaning.
Hence, assumption: H(F)=[F], i.e.:
H(F) is the semantic meaning of F
Semantic meaning @ Lewis imperative signal.VtZ"ZZZ8K Background (Repetition)
Lewis (IV.4,1996) distinguishes between
indicative signals
imperative signals
Two possible definitions of meaning:
Indicative:
[F] = M :iff S1(F)=M
Imperative:
[F] = M :iff H(F)=M(P&P(PPP
PPN
, Q
An ExamplexWe consider the standard example:
Some of the boys came to the party.
said: at least two came
implicated: not all came\"% 1HSThe GameUThe Solved GameV2The hearer can infer after receiving A(some) that:33(%[3Natural Information and Conversational Implicatures443 VNatural and NonNatural Meaning (mGrice distinguished between
natural meaning
nonnatural meaning
Communicated meaning is nonnatural meaning.
8$nWExample$I show Mr. X a photograph of Mr. Y displaying undue familiarity to Mrs. X.
I draw a picture of Mr. Y behaving in this manner and show it to Mr. X.
The photograph naturally means that Mr. Y was unduly familiar to Mrs. X
The picture nonnaturally means that Mr. Y was unduly familiar to Mrs. XnZnZZ <
/X
eTaking a photo of a scene necessarily entails that the scene is real.
Every branch which contains a showing of a photo must contain a situation which is depicted by it.
The showing of the photo means naturally that there was a situation where Mr. Y was unduly familiar with Mrs. X.
The drawing of a picture does not imply that the depicted scene is real.BGZZIZGIYNatural Information of SignalsLet G be a signalling game.
Let S be a set of strategy pairs (S,H).
We identify the natural information of a form F in G with respect to S with:
The set of all branches of G where the speaker chooses F. ;h<<VZInformation coincides with S1(F) in case of simple Lewisean signalling games.
Generalises to arbitrary games which contain semantic interpretation games in embedded form.
Conversational Implicatures are implied by the natural information of an utterance.<S4!The Standard Example reconsidered
"V
Some of the boys came to the party.
said: at least two came
implicated: not all cameF& 1& UX"The game defined by pure semantics##(YXThe game after optimising speaker s strategy(\The possible worldsw1: 100% of the boys came to the party.
w2: More than 50% of the boys came to the party.
w3: Less than 50% of the boys came to the party.H'0/^&The possible Branches of the Game Tree''(`5The unique signalling strategy that solves this game:66(Z4The Natural Information carried by utterance A(some)55(NThe branches allowed by strategy S:
w1,A(all), {w1}
w2,A(most), {w1,w2}
w3,A(some), {w1,w2,w3}
Natural information carried by A(some):
{w3,A(some), {w1,w2,w3}}$ZAZ(ZPZ2$
(
b Implicatures in Signalling Games!!(A special case
c!As Signalling Game (Repetition)&
A signalling game is a tuple:
N,, p, (A1,A2), (u1, u2)
N: Set of two players S,H.
: Set of types representing the speakers private information.
p: A probability measure over representing the hearer s expectations about the speaker s type. "
"d"p(A1,A2): the speaker s and hearer s action sets:
A1 is a set of forms F / meanings M.
A2 is a set of actions.
(u1,u2): the speaker s and hearer s payoff functions with
ui: A1A2 RP1>:+5 e#Strategies in a Signalling GameLet [ ] : F M be a given semantics.
The speaker s strategies are of the form:
S : A1 such that
S() = F [F]
i.e. if the speaker says F, then he knows that F is true (Maxim of Quality).Pu
+
@NfDf$(Definition of Implicature(special case))*Given a signalling game as before, then an implicature
F +> y
is explained iff the following set is a subset of [y] = {w  w = y}:V8 H =><L,h&Preview
Later, we apply this criterion to calculating implicatures of answers.
The definition depends on the method of finding solutions.n[First we need a method for calculating optimal answers.
The resulting signalling and interpretation strategies are then the solutions which we use as imput for calculating implicatures. *'<8Ng%Optimal Answersi'
Core Examplesj(Italian NewspaperSomewhere in the streets of Amsterdam...
J: Where can I buy an Italian newspaper?
E: At the station and at the Palace but nowhere else. (SE)
E: At the station. (A) / At the Palace. (B)
.)n)k)The answer (SE) is called strongly exhaustive.
The answers (A) and (B) are called mention some answers.
A and B are as good as SE or as A B:
E: There are Italian newspapers at the station but none at the Palace./9@)@Gn#$Gl*Partial AnswersIf E knows only that A, then A is an optimal answer:
E: There are no Italian newspapers at the station.
If E only knows that the Palace sells foreign newspapers, then this is an optimal answer:
E: The Palace has foreign newspapers.^74nZ&n74Z&m+Partial answers may also arise in situations where speaker E has full knowledge:
I: I need patrol for my car. Where can I get it?
E: There is a garage round the corner.
J: Where can I buy an Italian newspaper?
E: There is a news shop round the corner.NQ@1'@ )* n,#Optimal Answers in Support Problems$$#
The Framework
oSupport ProblemDefinition: A support problem is a five tuple (,PE,PI,A,u) such that
(, PE) and (, PI) are finite probability spaces,
(,PI,A, u) is a decision problem.
Let K:= {wW PE(w) > 0 } (E s knowledge set).
Then, we assume in addition:
for all A : PE(A) = PI(AK) .G3$@M!n#"
"
"
!"
=
x/&"np.Support Problemq/,I s Decision Situation,I optimises expected utilities of actions:
,r0I will choose an action aA that optimises expected utility, i.e. for all actions b
EU(b,A) EU(aA,A)
Given answer A, H(A) = aA.
For simplicity we assume that I s choice aA is commonly known.HT ?6+t: s1,E s Decision Situation*E optimises expected utilities of answers:++t2
^(Quality): The speaker can only say what he thinks to be true.
(Quality) restricts answers to:$_>$ u3Examples The Italian Newspaper Examplesv5Italian NewspaperSomewhere in the streets of Amsterdam...
J: Where can I buy an Italian newspaper?
E: At the station and at the Palace but nowhere else. (SE)
E: At the station. (A) / At the Palace. (B)
.)n)w4"Possible Worlds (equally probable)##(x6Actions and AnswersI s actions:
a: going to station;
b: going to Palace;
Answers:
A: at the station (A = {w1,w2})
B: at the Palace (B = {w1,w3})
* ?
* _Let utilities be such that they only distinguish between success (value 1) and failure (value 0).
Let s consider answer A = {w1,w2}.
Assume that the speaker knows that A, i.e. there are Italian newspapers at the station.
6~]`The CalculationIf hearing A induces hearer to choose a (i.e. aA=a going to station ):
If hearing A induces hearer to choose b (i.e. aA=b going to Palace ):
If PE(B) = 1, then EUE(A) = EUE(B) = 1.
PE(B) < 1 leads to a contradiction.L# J.H3+bPE(B) < 1 leads to a contradiction:
aA = b implies EUI(bA) EUI(aA) = 1.
Hence, EUI(bA) = vA PI(v) u(v,b) =1.
Therefore PI(BA) =1, hence PI(BA) = PI(A), hence PI(A\B)=0.
PE(A\B)=0, because $K: PE(X)= PI(XK).
PE(BA)=PE(A)=1, hence PE(B) = 1.%ZnZ
%
O"a=Case: Speaker knows that Italian newspaper are at both places>>(fCalculation showed that EUE(A) = 1.
Expected utility cannot be higher than 1 (due to assumptions).
Similar: EUE(B) = 1; EUE(AB) = 1.
Hence, all these answers are equally optimal.O5y7
More Cases,E knows that A and B:
EUE(A) = EUE(B) = EUE(AB)
E knows that A and B:
EUE(A) = EUE(A B)
E knows only that A:
For all admissible C: EUE(C) EUE(A)` &
The following example shows how the method of finding optimal answers in support problems interacts with the general theory of implicatures in signalling games.$
DHip Hop at Roter SalonJohn loves to dance to Salsa music and he loves to dance to Hip Hop but he can t stand it if a club mixes both styles.
J: I want to dance tonight. Is the Music in Roter Salon ok?
E: Tonight they play Hip Hop at the Roter Salon.
+> They play only Hip Hop.0wnw,/#]FA game tree for the situation where both Salsa and Hip Hop are playingGG G+After the first step of backward induction:,,(^,After the second step of backward induction:(If we say that a proposition is the more relevant the higher the expected utility after learning it, then relevance scale approaches predict that Hip Hop implicates that both, Salsa and Hip Hop, are playing.
Worst case compatible with what was said!(*(2$EHip Hop at Roter SalonAbbreviations:
Good(x) := HAssumptions hEqual Probabilities
Independence: X,Y{H,S,Good}T5n&
ILearning H(x) or S(x) raises expected utility of going to salon x:
EUI(goingtox) < EUI(stayhome) < EUI(goingtoxH(x))
EUI(goingtox) < EUI(stayhome) < EUI(goingtoxS(x)) Cnrn ,
/ KViolating Assumptions IIThe Roter Salon and the Grner Salon share two DJs. One of them only plays Salsa, the other one mainly plays Hip Hop but mixes into it some Salsa. There are only these two Djs, and if one of them is at the Roter Salon, then the other one is at the Grner Salon. John loves to dance to Salsa music and he loves to dance to Hip Hop but he can t stand it if a club mixes both styles.
J: I want to dance tonight. Is the Music in Roter Salon ok?
E: Tonight they play Hip Hop at the Roter Salon.^~Zn Z"" % 0%Support Problems and Signalling Games%In our model, the speaker finds an optimal answer by backward induction in support problems.
This is not a standard method for solving coordination problems in signalling games.H$~Signalling Game
A signalling game is a tuple:
N,, p, (A1,A2), (u1, u2)
N: Set of two players S,H.
: Set of types representing the speakers private information
p: A probability measure over representing the hearer s expectations about S type. "
"Solution to a Signalling GameThe standard solution concept for Signalling games is that of a perfect Bayesian equilibrium!
(S,H) strategies:
S : A1
H : A1 A2q @$"$$$
$,$$
$"$$$
$$_
"Perfect Bayesian equilibrium (S,H)6#((((" S() argmaxF u1(F,H(F),)
"F H(F) argmaxM m(F)u2(F,M,)
where m is defined by
m(F) = 0 if S()F
m(F) = p() / p(S1[F]) if S()=F
if p(S1[F]) > 0, else m(F) is arbitrary.rExx:x,x H$
Task
,Given:
a set of support problems S with fixed decision problem (,PI,A,u) for a
Wanted:
Representation as signalling game:
N,, p, (AE,AI), (uE, uI)I#! 3 T>2Construction
(
tLet s=(,PE,PI,A,u) be a given support problem.
Remember: there is a common prior P on such that:
PE(X) = PI(XKs) for Ks:= {wW PE(w) > 0}
Add Ks to (i.e. = {Ks sS})
The speaker s action set AE is identical with a set of forms F / meanings M.
The hearer s action set is identical to the action set of s.dZ,Z!ZZ"
E
"
""""*!"#
bbcbb:"\
=(<
The game is a game of pure coordination with respect to joint payoff functions
ui: F AI R
uI(A,a,K) := EUI(aK)
uE(A,a,K) := EUE(aK) (= EUI(aK))
On; O !
RN>
(p is arbitrary.
Forms F have to be interpreted by their semantic meaning [F].
The speaker has to conform to the maxim of quality, i.e. S(Ks) Adms
n ` *MFResult
pThe strategy pairs defined by:
S(Ks) Ops, H(A) = aA
are Perfect Bayesian Equilibria of the associated signalling game.
they (weakly) Pareto dominate all other strategy pairs (S ,H )."
/p
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%O
=cGame Theory and Gricean PragmaticsLesson III""CAnton Benz
Zentrum fr Allgemeine Sprachwissenschaften
ZAS Berlin .PP9P9&7eCourse OverviewLesson 1: Introduction
From Grice to Lewis
Relevance Scale Approaches
Lesson 2: Signalling Games
Lewis Signalling Conventions
Parikh s Radical Underspecification Model
Lesson 3: The Optimal Answer Approach I
Lesson 4: The Optimal Answer Approach II
Comparison with Relevance Scale Approaches
Decision Contexts with Multiple ObjectivesZ0ZZIZQZVZ0H&()Vo
VcOptimal Answer Approach 0Lesson III April, 5th ( > dOverview of Lesson III
&Natural Information and Conversational Implicatures
An Example: Scalar Implicatures
Natural Information and Conversational Implicatures
Calculating Implicatures in Signalling Games
Optimal Answers
Core Examples
Optimal Answers in Support Problems
Examples
Support Problems and Signalling Games 4ZZZ;Z'Z4;%"
'
The AgendaPutting Grice on Lewisean feet! (f4Natural Information and Conversational Implicatures 55.
PJ3Explanation of Implicatures Optimal Answer Approach$4* PStart with a signalling game where the hearer interprets forms by their literal meaning.
Impose pragmatic constraints and calculate equilibria that solve this game.
Implicature F +> j is explained if for all solutions (S,H):
S1(F) = jfnn+O
ContrastIn an information based approach:
Implicatures emerge from indicated meaning (in the sense of Lewis).
Implicatures are not initial candidate interpretations.
Speaker does not maximise relevance.
No diachronic process.X"ZZ;+1,8$hAssumption: speaker and hearer use language according to a semantic convention.
Goal: Explain how implicatures can emerge out of semantic language use.
Nonreductionist perspective..iRepresentation of AssumptionSemantics defines interpretation of forms.
Let [F] denote the semantic meaning.
Hence, assumption: H(F)=[F], i.e.:
H(F) is the semantic meaning of F
Semantic meaning @ Lewis imperative signal.VtZ"ZZZ8K Background (Repetition)
Lewis (IV.4,1996) distinguishes between
indicative signals
imperative signals
Two possible definitions of meaning:
Indicative:
[F] = M :iff S1(F)=M
Imperative:
[F] = M :iff H(F)=M(P&P(PPP
PPN
, Q
An ExamplexWe consider the standard example:
Some of the boys came to the party.
said: at least two came
implicated: not all came\"% 1HSThe GameUThe Solved GameV2The hearer can infer after receiving A(some) that:33(%[3Natural Information and Conversational Implicatures443 VNatural and NonNatural Meaning (mGrice distinguished between
natural meaning
nonnatural meaning
Communicated meaning is nonnatural meaning.
8$nWExample$I show Mr. X a photograph of Mr. Y displaying undue familiarity to Mrs. X.
I draw a picture of Mr. Y behaving in this manner and show it to Mr. X.
The photograph naturally means that Mr. Y was unduly familiar to Mrs. X
The picture nonnaturally means that Mr. Y was unduly familiar to Mrs. XnZnZZ <
/X
eTaking a photo of a scene necessarily entails that the scene is real.
Every branch which contains a showing of a photo must contain a situation which is depicted by it.
The showing of the photo means naturally that there was a situation where Mr. Y was unduly familiar with Mrs. X.
The drawing of a picture does not imply that the depicted scene is real.BGZZIZGIYNatural Information of SignalsLet G be a signalling game.
Let S be a set of strategy pairs (S,H).
We identify the natural information of a form F in G with respect to S with:
The set of all branches of G where the speaker chooses F. ;h<<VZInformation coincides with S1(F) in case of simple Lewisean signalling games.
Generalises to arbitrary games which contain semantic interpretation games in embedded form.
Conversational Implicatures are implied by the natural information of an utterance.<S4!The Standard Example reconsidered
"V
Some of the boys came to the party.
said: at least two came
implicated: not all cameF& 1& UX"The game defined by pure semantics##(YXThe game after optimising speaker s strategy(\The possible worldsw1: 100% of the boys came to the party.
w2: More than 50% of the boys came to the party.
w3: Less than 50% of the boys came to the party.H'0/^&The possible Branches of the Game Tree''(`5The unique signalling strategy that solves this game:66(Z4The Natural Information carried by utterance A(some)55(NThe branches allowed by strategy S:
w1,A(all), {w1}
w2,A(most), {w1,w2}
w3,A(some), {w1,w2,w3}
Natural information carried by A(some):
{w3,A(some), {w1,w2,w3}}$ZAZ(ZPZ2$
(
b Implicatures in Signalling Games!!(A special case
c!As Signalling Game (Repetition)&
A signalling game is a tuple:
N,, p, (A1,A2), (u1, u2)
N: Set of two players S,H.
: Set of types representing the speakers private information.
p: A probability measure over representing the hearer s expectations about the speaker s type. "
"d"p(A1,A2): the speaker s and hearer s action sets:
A1 is a set of forms F / meanings M.
A2 is a set of actions.
(u1,u2): the speaker s and hearer s payoff functions with
ui: A1A2 RP1>:+5 e#Strategies in a Signalling GameLet [ ] : F M be a given semantics.
The speaker s strategies are of the form:
S : A1 such that
S() = F [F]
i.e. if the speaker says F, then he knows that F is true (Maxim of Quality).Pu
+
@NfDf$(Definition of Implicature(special case))*Given a signalling game as before, then an implicature
F +> y
is explained iff the following set is a subset of [y] = {w  w = y}:V8 H =><L,h&Preview
Later, we apply this criterion to calculating implicatures of answers.
The definition depends on the method of finding solutions.n[First we need a method for calculating optimal answers.
The resulting signalling and interpretation strategies are then the solutions which we use as imput for calculating implicatures. *'<8Ng%Optimal Answersi'
Core Examplesj(Italian NewspaperSomewhere in the streets of Amsterdam...
J: Where can I buy an Italian newspaper?
E: At the station and at the Palace but nowhere else. (SE)
E: At the station. (A) / At the Palace. (B)
.)n)k)The answer (SE) is called strongly exhaustive.
The answers (A) and (B) are called mention some answers.
A and B are as good as SE or as A B:
E: There are Italian newspapers at the station but none at the Palace./9@)@Gn#$Gl*Partial AnswersIf E knows only that A, then A is an optimal answer:
E: There are no Italian newspapers at the station.
If E only knows that the Palace sells foreign newspapers, then this is an optimal answer:
E: The Palace has foreign newspapers.^74nZ&n74Z&m+Partial answers may also arise in situations where speaker E has full knowledge:
I: I need patrol for my car. Where can I get it?
E: There is a garage round the corner.
J: Where can I buy an Italian newspaper?
E: There is a news shop round the corner.NQ@1'@ )* n,#Optimal Answers in Support Problems$$#
The Framework
oSupport ProblemDefinition: A support problem is a five tuple (,PE,PI,A,u) such that
(, PE) and (, PI) are finite probability spaces,
(,PI,A, u) is a decision problem.
Let K:= {wW PE(w) > 0 } (E s knowledge set).
Then, we assume in addition:
for all A : PE(A) = PI(AK) .G3$@M!n#"
"
"
!"
=
x/&"np.Support Problemq/,I s Decision Situation,I optimises expected utilities of actions:
,r0I will choose an action aA that optimises expected utility, i.e. for all actions b
EU(b,A) EU(aA,A)
Given answer A, H(A) = aA.
For simplicity we assume that I s choice aA is commonly known.HT ?6+t: s1,E s Decision Situation*E optimises expected utilities of answers:++t2
^(Quality): The speaker can only say what he thinks to be true.
(Quality) restricts answers to:$_>$ u3Examples The Italian Newspaper Examplesv5Italian NewspaperSomewhere in the streets of Amsterdam...
J: Where can I buy an Italian newspaper?
E: At the station and at the Palace but nowhere else. (SE)
E: At the station. (A) / At the Palace. (B)
.)n)w4"Possible Worlds (equally probable)##(x6Actions and AnswersI s actions:
a: going to station;
b: going to Palace;
Answers:
A: at the station (A = {w1,w2})
B: at the Palace (B = {w1,w3})
* ?
* _Let utilities be such that they only distinguish between success (value 1) and failure (value 0).
Let s consider answer A = {w1,w2}.
Assume that the speaker knows that A, i.e. there are Italian newspapers at the station.
6~]`The CalculationIf hearing A induces hearer to choose a (i.e. aA=a going to station ):
If hearing A induces hearer to choose b (i.e. aA=b going to Palace ):
If PE(B) = 1, then EUE(A) = EUE(B) = 1.
PE(B) < 1 leads to a contradiction.L# J.H3+bPE(B) < 1 leads to a contradiction:
aA = b implies EUI(bA) EUI(aA) = 1.
Hence, EUI(bA) = vA PI(v) u(v,b) =1.
Therefore PI(BA) =1, hence PI(BA) = PI(A), hence PI(A\B)=0.
PE(A\B)=0, because $K: PE(X)= PI(XK).
PE(BA)=PE(A)=1, hence PE(B) = 1.%ZnZ
%
O"a=Case: Speaker knows that Italian newspaper are at both places>>(fCalculation showed that EUE(A) = 1.
Expected utility cannot be higher than 1 (due to assumptions).
Similar: EUE(B) = 1; EUE(AB) = 1.
Hence, all these answers are equally optimal.O5y7
More Cases,E knows that A and B:
EUE(A) = EUE(B) = EUE(AB)
E knows that A and B:
EUE(A) = EUE(A B)
E knows only that A:
For all admissible C: EUE(C) EUE(A)` &
The following example shows how the method of finding optimal answers in support problems interacts with the general theory of implicatures in signalling games.$
DHip Hop at Roter SalonJohn loves to dance to Salsa music and he loves to dance to Hip Hop but he can t stand it if a club mixes both styles.
J: I want to dance tonight. Is the Music in Roter Salon ok?
E: Tonight they play Hip Hop at the Roter Salon.
+> They play only Hip
!"#$%&'()*+,./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{}~Hop.0wnw,/#]FA game tree for the situation where both Salsa and Hip Hop are playingGG G+After the first step of backward induction:,,(^,After the second step of backward induction:(If we say that a proposition is the more relevant the higher the expected utility after learning it, then relevance scale approaches predict that Hip Hop implicates that both, Salsa and Hip Hop, are playing.
Worst case compatible with what was said!(*(2$EHip Hop at Roter SalonAbbreviations:
Good(x) := HAssumptions hEqual Probabilities
Independence: X,Y{H,S,Good}T5n&
ILearning H(x) or S(x) raises expected utility of going to salon x:
EUI(goingtox) < EUI(stayhome) < EUI(goingtoxH(x))
EUI(goingtox) < EUI(stayhome) < EUI(goingtoxS(x)) Cnrn ,
/ KViolating Assumptions IIThe Roter Salon and the Grner Salon share two DJs. One of them only plays Salsa, the other one mainly plays Hip Hop but mixes into it some Salsa. There are only these two Djs, and if one of them is at the Roter Salon, then the other one is at the Grner Salon. John loves to dance to Salsa music and he loves to dance to Hip Hop but he can t stand it if a club mixes both styles.
J: I want to dance tonight. Is the Music in Roter Salon ok?
E: Tonight they play Hip Hop at the Roter Salon.^~Zn Z"" % 0%Support Problems and Signalling Games%In our model, the speaker finds an optimal answer by backward induction in support problems.
This is not a standard method for solving coordination problems in signalling games.H$~Signalling Game
A signalling game is a tuple:
N,, p, (A1,A2), (u1, u2)
N: Set of two players S,H.
: Set of types representing the speakers private information
p: A probability measure over representing the hearer s expectations about S type. "
"Solution to a Signalling GameThe standard solution concept for Signalling games is that of a perfect Bayesian equilibrium!
(S,H) strategies:
S : A1
H : A1 A2q @$"$$$
$,$$
$"$$$
$$_
"Perfect Bayesian equilibrium (S,H)6#((((" S() argmaxF u1(F,H(F),)
"F H(F) argmaxM m(F)u2(F,M,)
where m is defined by
m(F) = 0 if S()F
m(F) = p() / p(S1[F]) if S()=F
if p(S1[F]) > 0, else m(F) is arbitrary.rExx:x,x H$
Task
,Given:
a set of support problems S with fixed decision problem (,PI,A,u) for a
Wanted:
Representation as signalling game:
N,, p, (AE,AI), (uE, uI)I#! 3 T>2Construction
(
tLet s=(,PE,PI,A,u) be a given support problem.
Remember: there is a common prior P on such that:
PE(X) = PI(XKs) for Ks:= {wW PE(w) > 0}
Add Ks to (i.e. = {Ks sS})
The speaker s action set AE is identical with a set of forms F / meanings M.
The hearer s action set is identical to the action set of s.dZ,Z!ZZ"
E
"
""""*!"#
bbcbb:"\
=(<
The game is a game of pure coordination with respect to joint payoff functions
ui: F AI R
uI(A,a,K) := EUI(aK)
uE(A,a,K) := EUE(aK) (= EUI(aK))
On; O !
RN>
dp is arbitrary (as long as p(q)>0 for q ).
Forms F have to be interpreted by their semantic meaning [F].
The speaker has to conform to the maxim of quality, i.e. S(Ks) Adms
.n ` ^=FResult
pThe strategy pairs defined by:
S(Ks) Ops, H(A) = aA
are Perfect Bayesian Equilibria of the associated signalling game.
they (weakly) Pareto dominate all other strategy pairs (S ,H )."
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=cGame Theory and Gricean PragmaticsLesson III""CAnton Benz
Zentrum fr Allgemeine Sprachwissenschaften
ZAS Berlin .PP9P9&7eCourse OverviewLesson 1: Introduction
From Grice to Lewis
Relevance Scale Approaches
Lesson 2: Signalling Games
Lewis Signalling Conventions
Parikh s Radical Underspecification Model
Lesson 3: The Optimal Answer Approach I
Lesson 4: The Optimal Answer Approach II
Comparison with Relevance Scale Approaches
Decision Contexts with Multiple ObjectivesZ0ZZIZQZVZ0H&()Vo
VcOptimal Answer Approach 0Lesson III April, 5th ( > dOverview of Lesson III
&Natural Information and Conversational Implicatures
An Example: Scalar Implicatures
Natural Information and Conversational Implicatures
Calculating Implicatures in Signalling Games
Optimal Answers
Core Examples
Optimal Answers in Support Problems
Examples
Support Problems and Signalling Games 4ZZZ;Z'Z4;%"
'
The AgendaPutting Grice on Lewisean feet! (f4Natural Information and Conversational Implicatures 55.
PJ3Explanation of Implicatures Optimal Answer Approach$4* PStart with a signalling game where the hearer interprets forms by their literal meaning.
Impose pragmatic constraints and calculate equilibria that solve this game.
Implicature F +> j is explained if for all solutions (S,H):
S1(F) = jfnn+O
ContrastIn an information based approach:
Implicatures emerge from indicated meaning (in the sense of Lewis).
Implicatures are not initial candidate interpretations.
Speaker does not maximise relevance.
No diachronic process.X"ZZ;+1,8$hAssumption: speaker and hearer use language according to a semantic convention.
Goal: Explain how implicatures can emerge out of semantic language use.
Nonreductionist perspective..iRepresentation of AssumptionSemantics defines interpretation of forms.
Let [F] denote the semantic meaning.
Hence, assumption: H(F)=[F], i.e.:
H(F) is the semantic meaning of F
Semantic meaning @ Lewis imperative signal.VtZ"ZZZ8K Background (Repetition)
Lewis (IV.4,1996) distinguishes between
indicative signals
imperative signals
Two possible definitions of meaning:
Indicative:
[F] = M :iff S1(F)=M
Imperative:
[F] = M :iff H(F)=M(P&P(PPP
PPN
, Q
An ExamplexWe consider the standard example:
Some of the boys came to the party.
said: at least two came
implicated: not all came\"% 1HSThe GameUThe Solved GameV2The hearer can infer after receiving A(some) that:33(%[3Natural Information and Conversational Implicatures443 VNatural and NonNatural Meaning (mGrice distinguished between
natural meaning
nonnatural meaning
Communicated meaning is nonnatural meaning.
8$nWExample$I show Mr. X a photograph of Mr. Y displaying undue familiarity to Mrs. X.
I draw a picture of Mr. Y behaving in this manner and show it to Mr. X.
The photograph naturally means that Mr. Y was unduly familiar to Mrs. X
The picture nonnaturally means that Mr. Y was unduly familiar to Mrs. XnZnZZ <
/X
eTaking a photo of a scene necessarily entails that the scene is real.
Every branch which contains a showing of a photo must contain a situation which is depicted by it.
The showing of the photo means naturally that there was a situation where Mr. Y was unduly familiar with Mrs. X.
The drawing of a picture does not imply that the depicted scene is real.BGZZIZGIYNatural Information of SignalsLet G be a signalling game.
Let S be a set of strategy pairs (S,H).
We identify the natural information of a form F in G with respect to S with:
The set of all branches of G where the speaker chooses F. ;h<<VZInformation coincides with S1(F) in case of simple Lewisean signalling games.
Generalises to arbitrary games which contain semantic interpretation games in embedded form.
Conversational Implicatures are implied by the natural information of an utterance.<S4!The Standard Example reconsidered
"V
Some of the boys came to the party.
said: at least two came
implicated: not all cameF& 1& UX"The game defined by pure semantics##(YXThe game after optimising speaker s strategy(\The possible worldsw1: 100% of the boys came to the party.
w2: More than 50% of the boys came to the party.
w3: Less than 50% of the boys came to the party.H'0/^&The possible Branches of the Game Tree''(`5The unique signalling strategy that solves this game:66(Z4The Natural Information carried by utterance A(some)55(NThe branches allowed by strategy S:
w1,A(all), {w1}
w2,A(most), {w1,w2}
w3,A(some), {w1,w2,w3}
Natural information carried by A(some):
{w3,A(some), {w1,w2,w3}}$ZAZ(ZPZ2$
(
b Implicatures in Signalling Games!!(A special case
c!As Signalling Game (Repetition)&
A signalling game is a tuple:
N,, p, (A1,A2), (u1, u2)
N: Set of two players S,H.
: Set of types representing the speakers private information.
p: A probability measure over representing the hearer s expectations about the speaker s type. "
"d"p(A1,A2): the speaker s and hearer s action sets:
A1 is a set of forms F / meanings M.
A2 is a set of actions.
(u1,u2): the speaker s and hearer s payoff functions with
ui: A1A2 RP1>:+5 e#Strategies in a Signalling GameLet [ ] : F M be a given semantics.
The speaker s strategies are of the form:
S : A1 such that
S() = F [F]
i.e. if the speaker says F, then he knows that F is true (Maxim of Quality).Pu
+
@NfDf$(Definition of Implicature(special case))*Given a signalling game as before, then an implicature
F +> y
is explained iff the following set is a subset of [y] = {w  w = y}:V8 H =><L,h&Preview
Later, we apply this criterion to calculating implicatures of answers.
The definition depends on the method of finding solutions.n[First we need a method for calculating optimal answers.
The resulting signalling and interpretation strategies are then the solutions which we use as imput for calculating implicatures. *'<8Ng%Optimal Answersi'
Core Examplesj(Italian NewspaperSomewhere in the streets of Amsterdam...
J: Where can I buy an Italian newspaper?
E: At the station and at the Palace but nowhere else. (SE)
E: At the station. (A) / At the Palace. (B)
.)n)k)The answer (SE) is called strongly exhaustive.
The answers (A) and (B) are called mention some answers.
A and B are as good as SE or as A B:
E: There are Italian newspapers at the station but none at the Palace./9@)@Gn#$Gl*Partial AnswersIf E knows only that A, then A is an optimal answer:
E: There are no Italian newspapers at the station.
If E only knows that the Palace sells foreign newspapers, then this is an optimal answer:
E: The Palace has foreign newspapers.^74nZ&n74Z&m+Partial answers may also arise in situations where speaker E has full knowledge:
I: I need patrol for my car. Where can I get it?
E: There is a garage round the corner.
J: Where can I buy an Italian newspaper?
E: There is a news shop round the corner.NQ@1'@ )* n,#Optimal Answers in Support Problems$$#
The Framework
oSupport ProblemDefinition: A support problem is a five tuple (,PE,PI,A,u) such that
(, PE) and (, PI) are finite probability spaces,
(,PI,A, u) is a decision problem.
Let K:= {wW PE(w) > 0 } (E s knowledge set).
Then, we assume in addition:
for all A : PE(A) = PI(AK) .G3$@M!n#"
"
"
!"
=
x/&"np.Support Problemq/,I s Decision Situation,I optimises expected utilities of actions:
,r0I will choose an action aA that optimises expected utility, i.e. for all actions b
EU(b,A) EU(aA,A)
Given answer A, H(A) = aA.
For simplicity we assume that I s choice aA is commonly known.HT ?6+t: s1,E s Decision Situation*E optimises expected utilities of answers:++t2
^(Quality): The speaker can only say what he thinks to be true.
(Quality) restricts answers to:$_>$ u3Examples The Italian Newspaper Examplesv5Italian NewspaperSomewhere in the streets of Amsterdam...
J: Where can I buy an Italian newspaper?
E: At the station and at the Palace but nowhere else. (SE)
E: At the station. (A) / At the Palace. (B)
.)n)w4"Possible Worlds (equally probable)##(x6Actions and AnswersI s actions:
a: going to station;
b: going to Palace;
Answers:
A: at the station (A = {w1,w2})
B: at the Palace (B = {w1,w3})
* ?
* _Let utilities be such that they only distinguish between success (value 1) and failure (value 0).
Let s consider answer A = {w1,w2}.
Assume that the speaker knows that A, i.e. there are Italian newspapers at the station.
6~]`The CalculationIf hearing A induces hearer to choose a (i.e. aA=a going to station ):
If hearing A induces hearer to choose b (i.e. aA=b going to Palace ):
If PE(B) = 1, then EUE(A) = EUE(B) = 1.
PE(B) < 1 leads to a contradiction.L# J.H3+bPE(B) < 1 leads to a contradiction:
aA = b implies EUI(bA) EUI(aA) = 1.
Hence, EUI(bA) = vA PI(v) u(v,b) =1.
Therefore PI(BA) =1, hence PI(BA) = PI(A), hence PI(A\B)=0.
PE(A\B)=0, because $K: PE(X)= PI(XK).
PE(BA)=PE(A)=1, hence PE(B) = 1.%ZnZ
%
O"a=Case: Speaker knows that Italian newspaper are at both places>>(fCalculation showed that EUE(A) = 1.
Expected utility cannot be higher than 1 (due to assumptions).
Similar: EUE(B) = 1; EUE(AB) = 1.
Hence, all these answers are equally optimal.O5y7
More Cases,E knows that A and B:
EUE(A) = EUE(B) = EUE(AB)
E knows that A and B:
EUE(A) = EUE(A B)
E knows only that A:
For all admissible C: EUE(C) EUE(A)` &
The following example shows how the method of finding optimal answers in support problems interacts with the general theory of implicatures in signalling games.$
DHip Hop at Roter SalonJohn loves to dance to Salsa music and he loves to dance to Hip Hop but he can t stand it if a club mixes both styles.
J: I want to dance tonight. Is the Music in Roter Salon ok?
E: Tonight they play Hip Hop at the Roter Salon.
+> They play only Hip Hop.0wnw,/#]FA game tree for the situation where both Salsa and Hip Hop are playingGG G+After the first step of backward induction:,,(^,After the second step of backward induction:(If we say that a proposition is the more relevant the higher the expected utility after learning it, then relevance scale approaches predict that Hip Hop implicates that both, Salsa and Hip Hop, are playing.
Worst case compatible with what was said!(*(2$EHip Hop at Roter SalonAbbreviations:
Good(x) := HAssumptions hEqual Probabilities
Independence: X,Y{H,S,Good}T5n&
ILearning H(x) or S(x) raises expected utility of going to salon x:
EUI(goingtox) < EUI(stayhome) < EUI(goingtoxH(x))
EUI(goingtox) < EUI(stayhome) < EUI(goingtoxS(x)) Cnrn ,
/ KViolating Assumptions IIThe Roter Salon and the Grner Salon share two DJs. One of them only plays Salsa, the other one mainly plays Hip Hop but mixes into it some Salsa. There are only these two Djs, and if one of them is at the Roter Salon, then the other one is at the Grner Salon. John loves to dance to Salsa music and he loves to dance to Hip Hop but he can t stand it if a club mixes both styles.
J: I want to dance tonight. Is the Music in Roter Salon ok?
E: Tonight they play Hip Hop at the Roter Salon.^~Zn Z"" % 0%Support Problems and Signalling Games%In our model, the speaker finds an optimal answer by backward induction in support problems.
This is not a standard method for solving coordination problems in signalling games.H$~Signalling Game
A signalling game is a tuple:
N,, p, (A1,A2), (u1, u2)
N: Set of two players S,H.
: Set of types representing the speakers private information
p: A probability measure over representing the hearer s expectations about S type. "
"Solution to a Signalling GameThe standard solution concept for Signalling games is that of a perfect Bayesian equilibrium!
(S,H) strategies:
S : A1
H : A1 A2q @$"$$$
$,$$
$"$$$
$$_
"Perfect Bayesian equilibrium (S,H)6#((((" S() argmaxF u1(F,H(F),)
"F H(F) argmaxM m(F)u2(F,M,)
where m is defined by
m(F) = 0 if S()F
m(F) = p() / p(S1[F]) if S()=F
if p(S1[F]) > 0, else m(F) is arbitrary.rExx:x,x H$
Task
,Given:
a set of support problems S with fixed decision problem (,PI,A,u) for a
Wanted:
Representation as signalling game:
N,, p, (AE,AI), (uE, uI)I#! 3 T>2Construction
(
tLet s=(,PE,PI,A,u) be a given support problem.
Remember: there is a common prior P on such that:
PE(X) = PI(XKs) for Ks:= {wW PE(w) > 0}
Add Ks to (i.e. = {Ks sS})
The speaker s action set AE is identical with a set of forms F / meanings M.
The hearer s action set is identical to the action set of s.dZ,Z!ZZ"
E
"
""""*!"#
bbcbb:"\
=(<
The game is a game of pure coordination with respect to joint payoff functions
ui: F AI R
uI(A,a,K) := EUI(aK)
uE(A,a,K) := EUE(aK) (= EUI(aK))
On; O !
RN>
`p is arbitrary (as long as p(q)>0 for q).
Forms F have to be interpreted by their semantic meaning [F].
The speaker has to conform to the maxim of quality, i.e. S(Ks) Adms
,n
` ^=FResult
pThe strategy pairs defined by:
S(Ks) Ops, H(A) = aA
are Perfect Bayesian Equilibria of the associated signalling game.
they (weakly) Pareto dominate all other strategy pairs (S ,H )."
/p
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=cGame Theory and Gricean PragmaticsLesson III""CAnton Benz
Zentrum fr Allgemeine Sprachwissenschaften
ZAS Berlin .PP9P9&7eCourse OverviewLesson 1: Introduction
From Grice to Lewis
Relevance Scale Approaches
Lesson 2: Signalling Games
Lewis Signalling Conventions
Parikh s Radical Underspecification Model
Lesson 3: The Optimal Answer Approach I
Lesson 4: The Optimal Answer Approach II
Comparison with Relevance Scale Approaches
Decision Contexts with Multiple ObjectivesZ0ZZIZQZVZ0H&()Vo
VcOptimal Answer Approach 0Lesson III April, 5th ( > dOverview of Lesson III
&Natural Information and Conversational Implicatures
An Example: Scalar Implicatures
Natural Information and Conversational Implicatures
Calculating Implicatures in Signalling Games
Optimal Answers
Core Examples
Optimal Answers in Support Problems
Examples
Support Problems and Signalling Games 4ZZZ;Z'Z4;%"
'
The AgendaPutting Grice on Lewisean feet! (f4Natural Information and Conversational Implicatures 55.
PJ3Explanation of Implicatures Optimal Answer Approach$4* PStart with a signalling game where the hearer interprets forms by their literal meaning.
Impose pragmatic constraints and calculate equilibria that solve this game.
Implicature F +> j is explained if for all solutions (S,H):
S1(F) = jfnn+O
ContrastIn an information based approach:
Implicatures emerge from indicated meaning (in the sense of Lewis).
Implicatures are not initial candidate interpretations.
Speaker does not maximise relevance.
No diachronic process.X"ZZ;+1,8$hAssumption: speaker and hearer use language according to a semantic convention.
Goal: Explain how implicatures can emerge out of semantic language use.
Nonreductionist perspective..iRepresentation of AssumptionSemantics defines interpretation of forms.
Let [F] denote the semantic meaning.
Hence, assumption: H(F)=[F], i.e.:
H(F) is the semantic meaning of F
Semantic meaning @ Lewis imperative signal.VtZ"ZZZ8K Background (Repetition)
Lewis (IV.4,1996) distinguishes between
indicative signals
imperative signals
Two possible definitions of meaning:
Indicative:
[F] = M :iff S1(F)=M
Imperative:
[F] = M :iff H(F)=M(P&P(PPP
PPN
, Q
An ExamplexWe consider the standard example:
Some of the boys came to the party.
said: at least two came
implicated: not all came\"% 1HSThe GameUThe Solved GameV2The hearer can infer after receiving A(some) that:33(%[3Natural Information and Conversational Implicatures443 VNatural and NonNatural Meaning (mGrice distinguished between
natural meaning
nonnatural meaning
Communicated meaning is nonnatural meaning.
8$nWExample$I show Mr. X a photograph of Mr. Y displaying undue familiarity to Mrs. X.
I draw a picture of Mr. Y behaving in this manner and show it to Mr. X.
The photograph naturally means that Mr. Y was unduly familiar to Mrs. X
The picture nonnaturally means that Mr. Y was unduly familiar to Mrs. XnZnZZ <
/X
eTaking a photo of a scene necessarily entails that the scene is real.
Every branch which contains a showing of a photo must contain a situation which is depicted by it.
The showing of the photo means naturally that there was a situation where Mr. Y was unduly familiar with Mrs. X.
The drawing of a picture does not imply that the depicted scene is real.BGZZIZGIYNatural Information of SignalsLet G be a signalling game.
Let S be a set of strategy pairs (S,H).
We identify the natural information of a form F in G with respect to S with:
The set of all branches of G where the speaker chooses F. ;h<<VZInformation coincides with S1(F) in case of simple Lewisean signalling games.
Generalises to arbitrary games which contain semantic interpretation games in embedded form.
Conversational Implicatures are implied by the natural information of an utterance.<S4!The Standard Example reconsidered
"V
Some of the boys came to the party.
said: at least two came
implicated: not all cameF& 1& UX"The game defined by pure semantics##(YXThe game after optimising speaker s strategy(\The possible worldsw1: 100% of the boys came to the party.
w2: More than 50% of the boys came to the party.
w3: Less than 50% of the boys came to the party.H'0/^&The possible Branches of the Game Tree''(`5The unique signalling strategy that solves this game:66(Z4The Natural Information carried by utterance A(some)55(NThe branches allowed by strategy S:
w1,A(all), {w1}
w2,A(most), {w1,w2}
w3,A(some), {w1,w2,w3}
Natural information carried by A(some):
{w3,A(some), {w1,w2,w3}}$ZAZ(ZPZ2$
(
b Implicatures in Signalling Games!!(A special case
c!As Signalling Game (Repetition)&
A signalling game is a tuple:
N,, p, (A1,A2), (u1, u2)
N: Set of two players S,H.
: Set of types representing the speakers private information.
p: A probability measure over representing the hearer s expectations about the speaker s type. "
"d"p(A1,A2): the speaker s and hearer s action sets:
A1 is a set of forms F / meanings M.
A2 is a set of actions.
(u1,u2): the speaker s and hearer s payoff functions with
ui: A1A2 RP1>:+5 e#Strategies in a Signalling GameLet [ ] : F M be a given semantics.
The speaker s strategies are of the form:
S : A1 such that
S() = F [F]
i.e. if the speaker says F, then he knows that F is true (Maxim of Quality).Pu
+
@NfDf$(Definition of Implicature(special case))*Given a signalling game as before, then an implicature
F +> y
is explained iff the following set is a subset of [y] = {w  w = y}:V8 H =><L,h&Preview
Later, we apply this criterion to calculating implicatures of answers.
The definition depends on the method of finding solutions.n[First we need a method for calculating optimal answers.
The resulting signalling and interpretation strategies are then the solutions which we use as imput for calculating implicatures. *'<8Ng%Optimal Answersi'
Core Examplesj(Italian NewspaperSomewhere in the streets of Amsterdam...
J: Where can I buy an Italian newspaper?
E: At the station and at the Palace but nowhere else. (SE)
E: At the station. (A) / At the Palace. (B)
.)n)k)The answer (SE) is called strongly exhaustive.
The answers (A) and (B) are called mention some answers.
A and B are as good as SE or as A B:
E: There are Italian newspapers at the station but none at the Palace./9@)@Gn#$Gl*Partial AnswersIf E knows only that A, then A is an optimal answer:
E: There are no Italian newspapers at the station.
If E only knows that the Palace sells foreign newspapers, then this is an optimal answer:
E: The Palace has foreign newspapers.^74nZ&n74Z&m+Partial answers may also arise in situations where speaker E has full knowledge:
I: I need patrol for my car. Where can I get it?
E: There is a garage round the corner.
J: Where can I buy an Italian newspaper?
E: There is a news shop round the corner.NQ@1'@ )* n,#Optimal Answers in Support Problems$$#
The Framework
oSupport ProblemDefinition: A support problem is a five tuple (,PE,PI,A,u) such that
(, PE) and (, PI) are finite probability spaces,
(,PI,A, u) is a decision problem.
Let K:= {wW PE(w) > 0 } (E s knowledge set).
Then, we assume in addition:
for all A : PE(A) = PI(AK) .G3$@M!n#"
"
"
!"
=
x/&"np.Support Problemq/,I s Decision Situation,I optimises expected utilities of actions:
,r0I will choose an action aA that optimises expected utility, i.e. for all actions b
EU(b,A) EU(aA,A)
Given answer A, H(A) = aA.
For simplicity we assume that I s choice aA is commonly known.HT ?6+t: s1,E s Decision Situation*E optimises expected utilities of answers:++t2
^(Quality): The speaker can only say what he thinks to be true.
(Quality) restricts answers to:$_>$ u3Examples The Italian Newspaper Examplesv5Italian NewspaperSomewhere in the streets of Amsterdam...
J: Where can I buy an Italian newspaper?
E: At the station and at the Palace but nowhere else. (SE)
E: At the station. (A) / At the Palace. (B)
.)n)w4"Possible Worlds (equally probable)##(x6Actions and AnswersI s actions:
a: going to station;
b: going to Palace;
Answers:
A: at the station (A = {w1,w2})
B: at the Palace (B = {w1,w3})
* ?
* _Let utilities be such that they only distinguish between success (value 1) and failure (value 0).
Let s consider answer A = {w1,w2}.
Assume that the speaker knows that A, i.e. there are Italian newspapers at the station.
6~]`The CalculationIf hearing A induces hearer to choose a (i.e. aA=a going to station ):
If hearing A induces hearer to choose b (i.e. aA=b going to Palace ):
If PE(B) = 1, then EUE(A) = EUE(B) = 1.
PE(B) < 1 leads to a contradiction.L# J.H3+bPE(B) < 1 leads to a contradiction:
aA = b implies EUI(bA) EUI(aA) = 1.
Hence, EUI(bA) = vA PI(v) u(v,b) =1.
Therefore PI(BA) =1, hence PI(BA) = PI(A), hence PI(A\B)=0.
PE(A\B)=0, because $K: PE(X)= PI(XK).
PE(BA)=PE(A)=1, hence PE(B) = 1.%ZnZ
%
O"a=Case: Speaker knows that Italian newspaper are at both places>>(fCalculation showed that EUE(A) = 1.
Expected utility cannot be higher than 1 (due to assumptions).
Similar: EUE(B) = 1; EUE(AB) = 1.
Hence, all these answers are equally optimal.O5y7
More Cases,E knows that A and B:
EUE(A) = EUE(B) = EUE(AB)
E knows that A and B:
EUE(A) = EUE(A B)
E knows only that A:
For all admissible C: EUE(C) EUE(A)` &
The following example shows how the method of finding optimal answers in support problems interacts with the general theory of implicatures in signalling games.$
DHip Hop at Roter SalonJohn loves to dance to Salsa music and he loves to dance to Hip Hop but he can t stand it if a club mixes both styles.
J: I want to dance tonight. Is the Music in Roter Salon ok?
E: Tonight they play Hip Hop at the Roter Salon.
+> They play only Hip Hop.0wnw,/#]FA game tree for the situation where both Salsa and Hip Hop are playingGG G+After the first step of backward induction:,,(^,After the second step of backward induction:(If we say that a proposition is the more relevant the higher the expected utility after learning it, then relevance scale approaches predict that Hip Hop implicates that both, Salsa and Hip Hop, are playing.
Worst case compatible with what was said!(*(2$EHip Hop at Roter SalonAbbreviations:
Good(x) := HAssumptions hEqual Probabilities
Independence: X,Y{H,S,Good}T5n&
ILearning H(x) or S(x) raises expected utility of going to salon x:
EUI(goingtox) < EUI(stayhome) < EUI(goingtoxH(x))
EUI(goingtox) < EUI(stayhome) < EUI(goingtoxS(x)) Cnrn ,
/ KViolating Assumptions IIThe Roter Salon and the Grner Salon share two DJs. One of them only plays Salsa, the other one mainly plays Hip Hop but mixes into it some Salsa. There are only these two Djs, and if one of them is at the Roter Salon, then the other one is at the Grner Salon. John loves to dance to Salsa music and he loves to dance to Hip Hop but he can t stand it if a club mixes both styles.
J: I want to dance tonight. Is the Music in Roter Salon ok?
E: Tonight they play Hip Hop at the Roter Salon.^~Zn Z"" % 0%Support Problems and Signalling Games%In our model, the speaker finds an optimal answer by backward induction in support problems.
This is not a standard method for solving coordination problems in signalling games.H$~Signalling Game
A signalling game is a tuple:
N,, p, (A1,A2), (u1, u2)
N: Set of two players S,H.
: Set of types representing the speakers private information
p: A probability measure over representing the hearer s expectations about S type. "
"Solution to a Signalling GameThe standard solution concept for Signalling games is that of a perfect Bayesian equilibrium!
(S,H) strategies:
S : A1
H : A1 A2q @$"$$$
$,$$
$"$$$
$$_
"Perfect Bayesian equilibrium (S,H)6#((((" S() argmaxF u1(F,H(F),)
"F H(F) argmaxM m(F)u2(F,M,)
where m is defined by
m(F) = 0 if S()F
m(F) = p() / p(S1[F]) if S()=F
if p(S1[F]) > 0, else m(F) is arbitrary.rExx:x,x H$
Task
,Given:
a set of support problems S with fixed decision problem (,PI,A,u) for a
Wanted:
Representation as signalling game:
N,, p, (AE,AI), (uE, uI)I#! 3 T>2Construction
(
tLet s=(,PE,PI,A,u) be a given support problem.
Remember: there is a common prior P on such that:
PE(X) = PI(XKs) for Ks:= {wW PE(w) > 0}
Add Ks to (i.e. = {Ks sS})
The speaker s action set AE is identical with a set of forms F / meanings M.
The hearer s action set is identical to the action set of s.dZ,Z!ZZ"
E
"
""""*!"#
bbcbb:"\
=(<
The game is a game of pure coordination with respect to joint payoff functions
ui: F AI R
uI(A,a,K) := EUI(aK)
uE(A,a,K) := EUE(aK) (= EUI(aK))
On; O !
RN>
`p is arbitrary (as long as p(q)>0 for q).
Forms F have to be interpreted by their semantic meaning [F].
The speaker has to conform to the maxim of quality, i.e. S(Ks) Adms
,n
a ^=FResult
pThe strategy pairs defined by:
S(Ks) Ops, H(A) = aA
are Perfect Bayesian Equilibria of the associated signalling game.
they (weakly) Pareto dominate all other strategy pairs (S ,H )."
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=cGame Theory and Gricean PragmaticsLesson III""CAnton Benz
Zentrum fr Allgemeine Sprachwissenschaften
ZAS Berlin .PP9P9&7eCourse OverviewLesson 1: Introduction
From Grice to Lewis
Relevance Scale Approaches
Lesson 2: Signalling Games
Lewis Signalling Conventions
Parikh s Radical Underspecification Model
Lesson 3: The Optimal Answer Approach I
Lesson 4: The Optimal Answer Approach II
Comparison with Relevance Scale Approaches
Decision Contexts with Multiple ObjectivesZ0ZZIZQZVZ0H&()Vo
VcOptimal Answer Approach 0Lesson III April, 5th ( > dOverview of Lesson III
&Natural Information and Conversational Implicatures
An Example: Scalar Implicatures
Natural Information and Conversational Implicatures
Calculating Implicatures in Signalling Games
Optimal Answers
Core Examples
Optimal Answers in Support Problems
Examples
Support Problems and Signalling Games 4ZZZ;Z'Z4;%"
'
The AgendaPutting Grice on Lewisean feet! (f4Natural Information and Conversational Implicatures 55.
PJ3Explanation of Implicatures Optimal Answer Approach$4* PStart with a signalling game where the hearer interprets forms by their literal meaning.
Impose pragmatic constraints and calculate equilibria that solve this game.
Implicature F +> j is explained if for all solutions (S,H):
S1(F) = jfnn+O
ContrastIn an information based approach:
Implicatures emerge from indicated meaning (in the sense of Lewis).
Implicatures are not initial candidate interpretations.
Speaker does not maximise relevance.
No diachronic process.X"ZZ;+1,8$hAssumption: speaker and hearer use language according to a semantic convention.
Goal: Explain how implicatures can emerge out of semantic language use.
Nonreductionist perspective..iRepresentation of AssumptionSemantics defines interpretation of forms.
Let [F] denote the semantic meaning.
Hence, assumption: H(F)=[F], i.e.:
H(F) is the semantic meaning of F
Semantic meaning @ Lewis imperative signal.VtZ"ZZZ8K Background (Repetition)
Lewis (IV.4,1996) distinguishes between
indicative signals
imperative signals
Two possible definitions of meaning:
Indicative:
[F] = M :iff S1(F)=M
Imperative:
[F] = M :iff H(F)=M(P&P(PPP
PPN
, Q
An ExamplexWe consider the standard example:
Some of the boys came to the party.
said: at least two came
implicated: not all came\"% 1HSThe GameUThe Solved GameV2The hearer can infer after receiving A(some) that:33(%[3Natural Information and Conversational Implicatures443 VNatural and NonNatural Meaning (mGrice distinguished between
natural meaning
nonnatural meaning
Communicated meaning is nonnatural meaning.
8$nWExample$I show Mr. X a photograph of Mr. Y displaying undue familiarity to Mrs. X.
I draw a picture of Mr. Y behaving in this manner and show it to Mr. X.
The photograph naturally means that Mr. Y was unduly familiar to Mrs. X
The picture nonnaturally means that Mr. Y was unduly familiar to Mrs. XnZnZZ <
/X
eTaking a photo of a scene necessarily entails that the scene is real.
Every branch which contains a showing of a photo must contain a situation which is depicted by it.
The showing of the photo means naturally that there was a situation where Mr. Y was unduly familiar with Mrs. X.
The drawing of a picture does not imply that the depicted scene is real.BGZZIZGIYNatural Information of SignalsLet G be a signalling game.
Let S be a set of strategy pairs (S,H).
We identify the natural information of a form F in G with respect to S with:
The set of all branches of G where the speaker chooses F. ;h<<VZInformation coincides with S1(F) in case of simple Lewisean signalling games.
Generalises to arbitrary games which contain semantic interpretation games in embedded form.
Conversational Implicatures are implied by the natural information of an utterance.<S4!The Standard Example reconsidered
"V
Some of the boys came to the party.
said: at least two came
implicated: not all cameF& 1& UX"The game defined by pure semantics##(YXThe game after optimising speaker s strategy(\The possible worldsw1: 100% of the boys came to the party.
w2: More than 50% of the boys came to the party.
w3: Less than 50% of the boys came to the party.H'0/^&The possible Branches of the Game Tree''(`5The unique signalling strategy that solves this game:66(Z4The Natural Information carried by utterance A(some)55(NThe branches allowed by strategy S:
w1,A(all), {w1}
w2,A(most), {w1,w2}
w3,A(some), {w1,w2,w3}
Natural information carried by A(some):
{w3,A(some), {w1,w2,w3}}$ZAZ(ZPZ2$
(
!"#$%&'()*+,./012 b Implicatures in Signalling Games!!(A special case
c!As Signalling Game (Repetition)&
A signalling game is a tuple:
N,, p, (A1,A2), (u1, u2)
N: Set of two players S,H.
: Set of types representing the speakers private information.
p: A probability measure over representing the hearer s expectations about the speaker s type. "
"d"p(A1,A2): the speaker s and hearer s action sets:
A1 is a set of forms F / meanings M.
A2 is a set of actions.
(u1,u2): the speaker s and hearer s payoff functions with
ui: A1A2 RP1>:+5 e#Strategies in a Signalling GameLet [ ] : F M be a given semantics.
The speaker s strategies are of the form:
S : A1 such that
S() = F [F]
i.e. if the speaker says F, then he knows that F is true (Maxim of Quality).Pu
+
@NfDf$(Definition of Implicature(special case))*Given a signalling game as before, then an implicature
F +> y
is explained iff the following set is a subset of [y] = {w  w = y}:V8 H =><L,h&Preview
Later, we apply this criterion to calculating implicatures of answers.
The definition depends on the method of finding solutions.n[First we need a method for calculating optimal answers.
The resulting signalling and interpretation strategies are then the solutions which we use as imput for calculating implicatures. *'<8Ng%Optimal Answersi'
Core Examplesj(Italian NewspaperSomewhere in the streets of Amsterdam...
J: Where can I buy an Italian newspaper?
E: At the station and at the Palace but nowhere else. (SE)
E: At the station. (A) / At the Palace. (B)
.)n)k)The answer (SE) is called strongly exhaustive.
The answers (A) and (B) are called mention some answers.
A and B are as good as SE or as A B:
E: There are Italian newspapers at the station but none at the Palace./9@)@Gn#$Gl*Partial AnswersIf E knows only that A, then A is an optimal answer:
E: There are no Italian newspapers at the station.
If E only knows that the Palace sells foreign newspapers, then this is an optimal answer:
E: The Palace has foreign newspapers.^74nZ&n74Z&m+Partial answers may also arise in situations where speaker E has full knowledge:
I: I need patrol for my car. Where can I get it?
E: There is a garage round the corner.
J: Where can I buy an Italian newspaper?
E: There is a news shop round the corner.NQ@1'@ )* n,#Optimal Answers in Support Problems$$#
The Framework
oSupport ProblemDefinition: A support problem is a five tuple (,PE,PI,A,u) such that
(, PE) and (, PI) are finite probability spaces,
(,PI,A, u) is a decision problem.
Let K:= {wW PE(w) > 0 } (E s knowledge set).
Then, we assume in addition:
for all A : PE(A) = PI(AK) .G3$@M!n#"
"
"
!"
=
x/&"np.Support Problemq/,I s Decision Situation,I optimises expected utilities of actions:
,r0I will choose an action aA that optimises expected utility, i.e. for all actions b
EU(b,A) EU(aA,A)
Given answer A, H(A) = aA.
For simplicity we assume that I s choice aA is commonly known.HT ?6+t: s1,E s Decision Situation*E optimises expected utilities of answers:++t2
^(Quality): The speaker can only say what he thinks to be true.
(Quality) restricts answers to:$_>$ u3Examples The Italian Newspaper Examplesv5Italian NewspaperSomewhere in the streets of Amsterdam...
J: Where can I buy an Italian newspaper?
E: At the station and at the Palace but nowhere else. (SE)
E: At the station. (A) / At the Palace. (B)
.)n)w4"Possible Worlds (equally probable)##(x6Actions and AnswersI s actions:
a: going to station;
b: going to Palace;
Answers:
A: at the station (A = {w1,w2})
B: at the Palace (B = {w1,w3})
* ?
* _Let utilities be such that they only distinguish between success (value 1) and failure (value 0).
Let s consider answer A = {w1,w2}.
Assume that the speaker knows that A, i.e. there are Italian newspapers at the station.
6~]`The CalculationIf hearing A induces hearer to choose a (i.e. aA=a going to station ):
If hearing A induces hearer to choose b (i.e. aA=b going to Palace ):
If PE(B) = 1, then EUE(A) = EUE(B) = 1.
PE(B) < 1 leads to a contradiction.L# J.H3+bPE(B) < 1 leads to a contradiction:
aA = b implies EUI(bA) EUI(aA) = 1.
Hence, EUI(bA) = vA PI(v) u(v,b) =1.
Therefore PI(BA) =1, hence PI(BA) = PI(A), hence PI(A\B)=0.
PE(A\B)=0, because $K: PE(X)= PI(XK).
PE(BA)=PE(A)=1, hence PE(B) = 1.%ZnZ
%
O"a=Case: Speaker knows that Italian newspaper are at both places>>(fCalculation showed that EUE(A) = 1.
Expected utility cannot be higher than 1 (due to assumptions).
Similar: EUE(B) = 1; EUE(AB) = 1.
Hence, all these answers are equally optimal.O5y7
More Cases,E knows that A and B:
EUE(A) = EUE(B) = EUE(AB)
E knows that A and B:
EUE(A) = EUE(A B)
E knows only that A:
For all admissible C: EUE(C) EUE(A)` &
The following example shows how the method of finding optimal answers in support problems interacts with the general theory of implicatures in signalling games.$
DHip Hop at Roter SalonJohn loves to dance to Salsa music and he loves to dance to Hip Hop but he can t stand it if a club mixes both styles.
J: I want to dance tonight. Is the Music in Roter Salon ok?
E: Tonight they play Hip Hop at the Roter Salon.
+> They play only Hip Hop.0wnw,/#]FA game tree for the situation where both Salsa and Hip Hop are playingGG G+After the first step of backward induction:,,(^,After the second step of backward induction:(If we say that a proposition is the more relevant the higher the expected utility after learning it, then relevance scale approaches predict that Hip Hop implicates that both, Salsa and Hip Hop, are playing.
Worst case compatible with what was said!(*(2$EHip Hop at Roter SalonAbbreviations:
Good(x) := HAssumptions hEqual Probabilities
Independence: X,Y{H,S,Good}T5n&
ILearning H(x) or S(x) raises expected utility of going to salon x:
EUI(goingtox) < EUI(stayhome) < EUI(goingtoxH(x))
EUI(goingtox) < EUI(stayhome) < EUI(goingtoxS(x)) Cnrn ,
/ KViolating Assumptions IIThe Roter Salon and the Grner Salon share two DJs. One of them only plays Salsa, the other one mainly plays Hip Hop but mixes into it some Salsa. There are only these two Djs, and if one of them is at the Roter Salon, then the other one is at the Grner Salon. John loves to dance to Salsa music and he loves to dance to Hip Hop but he can t stand it if a club mixes both styles.
J: I want to dance tonight. Is the Music in Roter Salon ok?
E: Tonight they play Hip Hop at the Roter Salon.^~Zn Z"" % 0%Support Problems and Signalling Games%In our model, the speaker finds an optimal answer by backward induction in support problems.
This is not a standard method for solving coordination problems in signalling games.H$~Signalling Game
A signalling game is a tuple:
N,, p, (A1,A2), (u1, u2)
N: Set of two players S,H.
: Set of types representing the speakers private information
p: A probability measure over representing the hearer s expectations about S type. "
"Solution to a Signalling GameThe standard solution concept for Signalling games is that of a perfect Bayesian equilibrium!
(S,H) strategies:
S : A1
H : A1 A2q @$"$$$
$,$$
$"$$$
$$_
"Perfect Bayesian equilibrium (S,H)6#((((" S() argmaxF u1(F,H(F),)
"F H(F) argmaxM m(F)u2(F,M,)
where m is defined by
m(F) = 0 if S()F
m(F) = p() / p(S1[F]) if S()=F
if p(S1[F]) > 0, else m(F) is arbitrary.rExx:x,x H$
Task
,Given:
a set of support problems S with fixed decision problem (,PI,A,u) for a
Wanted:
Representation as signalling game:
N,, p, (AE,AI), (uE, uI)I#! 3 T>2Construction
(
tLet s=(,PE,PI,A,u) be a given support problem.
Remember: there is a common prior P on such that:
PE(X) = PI(XKs) for Ks:= {wW PE(w) > 0}
Add Ks to (i.e. = {Ks sS})
The speaker s action set AE is identical with a set of forms F / meanings M.
The hearer s action set is identical to the action set of s.dZ,Z!ZZ"
E
"
""""*!"#
bbcbb:"\
=(<
The game is a game of pure coordination with respect to joint payoff functions
ui: F AI R
uI(A,a,K) := EUI(aK)
uE(A,a,K) := EUE(aK) (= EUI(aK))
On; O !
RN>
`p is arbitrary (as long as p(q)>0 for q).
Forms F have to be interpreted by their semantic meaning [F].
The speaker has to conform to the maxim of quality, i.e. S(Ks) Adms
,n
a ^=FResult
pThe strategy pairs defined by:
S(Ks) Ops, H(A) = aA
are Perfect Bayesian Equilibria of the associated signalling game.
they (weakly) Pareto dominate all other strategy pairs (S ,H )."
/p
p F(
x
c$ `
c$d `p
"p`PpH
0h ? }ff___PPT10i.^̽+D=' =
@B +rp
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