Optimal answers and their implicatures A game-theoretic approach

1
Optimal answers and their implicatures A
game-theoretic approach

• Anton Benz
• April 18th, 2006

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Overview

1. Conversational Implicatures in the Standard
   Theory
2. Conventions and Meaning
3. Game Theoretic Pragmatics
4. Implicatures of Answers

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Conversational Implicatures

• The Standard Theory

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Two components of communicated meaning

• Grice distinguishes between
• What is said.
• What is implicated.
• Some of the boys came to the party.
• said At least two of the boys came to the party.
• implicated Not all of the boys came to the
  party.
• Both part of what is communicated.

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Assumptions about Conversation

• Conversation is a cooperative effort. Each
  participant recognises in their talk exchanges a
  common purpose.
• Example A stands in front of his obviously
  immobilised car.
• A I am out of petrol.
• B There is a garage around the corner.
• Joint purpose of Bs response
• Solve As problem of finding petrol for his car.

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The Cooperative Principle

• Conversation is governed by a set of principles
  which spell out how rational agents behave in
  order to make language use efficient.
• The most important is the so-called cooperative
  principle
• Make your conversational contribution such as is
  required, at the stage at which it occurs, by the
  accepted purpose or direction of the talk
  exchange in which you are engaged.

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The Conversational Maxims

• Maxim of Quality
• Do not say what you believe to be false.
• Do not say that for which you lack adequate
  evidence.
• Maxim of Quantity
• Make your contribution to the conversation as
  informative as is required for he current talk
  exchange.
• Do not make your contribution to the conversation
  more informative than necessary.
• Maxim of Relevance make your contributions
  relevant.
• Maxim of Manner be perspicuous, and
  specifically
• Avoid obscurity.
• Avoid ambiguity.
• Be brief (avoid unnecessary wordiness).
• Be orderly.

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The Conversational Maxims

• Maxim of Quality Be truthful.
• Maxim of Quantity
• Say as much as you can.
• Say no more than you must.
• Maxim of Relevance Be relevant.

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The Conversational Maxims

• Be truthful (Quality) and say as much as you can
  (Quantity) as long as it is relevant (Relevance).

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An example Scalar Implicatures

1. Let A(x) ? x of the boys came to the party
2. It holds A(all) ? A(some).
3. The speaker said A(some).
4. If all of the boys came, then A(all) would have
   been preferred (Maxim of Quantity Relevance).
5. The speaker didnt say A(all), hence it cannot be
   the case that all came.
6. Therefore some but not all came to the party.

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Conventions and Meaning

• The Lewisean Perspective

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Conventions

• A convention is
• a regularity r in behaviour
• partially arbitrary
• that is common ground in a community C
• as a coordination device
• for a recurrent coordination problem
• Clark, 1996, p. 71

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Coordination and Language

• Speaker wants to communicate some meaning M. He
  has to choose a form F for M.
• The hearer has to interpret form F. He has to
  assign a meaning M to it.
• Communication is successful if MM.

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Signalling Conventions

• The meaning of signals
• is arbitrary
• answers a recurrent coordination problem
• is common ground in a language community
• A signalling convention (Lewis 1969) is a pair of
  
• a speakers signalling strategy (S M?F)
• a hearers interpretation strategy (H F?M)
• such that communication is always successful.

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The agenda

• Putting Gricean pragmatics on Lewisean feet
• Start assumption semantic meaning is defined by
  a signalling convention (Semantic Interpretation
  Game, SIG).
• Gricean maxims (and other pragmatic conditions)
  translate into constraints on the SIG.
• The explanation of a pragmatic phenomenon
  proceeds by a game theoretic analysis of the
  constrained SIG.

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Game Theoretic Pragmatics

• Scalar Implicatures

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Game Theory

• In a very general sense we can say that we play a
  game together with other people whenever we have
  to decide between several actions such that the
  decision depends on
• the choice of actions by others
• our preferences over the ultimate results.
• Whether or not an utterance is successful depends
  on
• how it is taken up by its addressee
• the overall purpose of the current conversation.

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The Game Theoretic Analysis of Scalar
Implicatures(For a scale with three elements
ltall, most, somegt)
?
1 1
all
50 gt
most
100
0 0
some
?
0 0
most
50 gt
50 gt
1 1
some
?
0 0
some
?
50 lt
1 1
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The Game Theoretic Analysis of Scalar
Implicatures(Taking into account the speakers
preferences)
all
?
100
1 1
most
50 gt
50 gt
1 1
some
?
1 1
50 lt
In all branches that contain some the initial
situation is 50 lt Hence some implicates
50 lt
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General method for calculating implicatures
(informal)

• Describe the utterance situation by a game (in
  extensive form, i.e. tree form).
• Possible states of the world
• Utterances the speaker can choose
• Their interpretations as defined by semantics.
• Preferences over outcomes (given by context)
• Simplify tree by backward induction.
• Read off the implicature from the game tree
  that cannot be simplified anymore.

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Implicatures of Answers

• Implicatures and Decision Problems

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An example of contradicting inferences I

• Situation A stands in front of his obviously
  immobilised car.
• A I am out of petrol.
• B There is a garage around the corner. (G)
• Implicated The garage is open. (H)
• How should one formally account for the
  implicature?
• Set H The negation of H
• B said that G but not that H.
• H is relevant and G ? H ? G.
• Hence if G ? H, then B should have said G ? H
  (Quantity).
• Hence H cannot be true, and therefore H.

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An example of contradicting inferences II

• Problem We can exchange H and H and still get a
  valid inference
• B said that G but not that H.
• H is relevant and G ? H ? G.
• Hence if G ? H, then B should have said G ? H
  (Quantity).
• Hence H cannot be true, and therefore H.
• Missing Precise definitions of basic concepts
  like relevance.

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The Utility of Answers

• Questions and answers are often subordinated to a
  decision problem of the inquirer.
• Example Somewhere in Amsterdam
• I Where can I buy an Italian newspaper?
• E At the station and at the palace.
• Decision problem of A Where should I go to in
  order to buy an Italian newspaper.

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The general situation
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Decision Making

• The Model
• O a (countable) set of possible states of the
  world.
• PI, PE (discrete) probability measures
  representing the inquirers and the answering
  experts knowledge about the world.
• A a set of actions.
• UI, UE Payoff functions that represent the
  inquirers and experts preferences over final
  outcomes of the game.
• Decision criterion an agent chooses an action
  which maximises his expected utility
• EU(a) ?v?O P(w) ? U(v,a)

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An Example

• 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. It is common knowledge
  that E knows always which kind of music plays at
  which place.
• J I want to dance tonight. Where can I go to?
• E Oh, tonight they play Hip Hop at the Roter
  Salon.
• implicated No Salsa at the Roter Salon.

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A game tree for the situation where both Salsa
and Hip Hop are playing
RS Roter Salon
1
stay home
0
go-to RS
both
1
stay home
both play at RS
Salsa
0
go-to RS
1
stay home
Hip Hop
0
go-to RS
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The tree after the first step of backward
induction
stay home
1
both
both
Salsa
go-to RS
0
Hip Hop
go-to RS
0
Salsa
Salsa
go-to RS
2
Hip Hop
Hip Hop
go-to RS
2
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The tree after the second step of backward
induction
both
stay home
both
1
Salsa
go-to RS
Salsa
2
Hip Hop
go-to RS
Hip Hop
2
In all branches that contain Salsa the initial
situation is such that only Salsa is playing at
the Roter Salon. Hence Salsa implicates that
only Salsa is playing at Roter Salon
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Another Example

• J approaches the information desk at the city
  railway station.
• J I need a hotel. Where can I book one?
• E There is a tourist office in front of the
  building.
• (E There is a hairdresser in front of the
  building.)
• implicated It is possible to book hotels at the
  tourist office.

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The situation where it is possible to book a
hotel at the tourist information, a place 2, and
a place 3.
1
go-to tourist office
s. a. search anywhere
0
s. a.
tourist office
1
go-to pl. 2
place 2
s. a.
0
1/2
place 3
go-to pl. 3
s. a.
0
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The game after the first step of backward
induction
go-to t. o.
1
tourist office
booking possible at tour. off.
place 2
go-to pl. 2
0
place 3
go-to pl. 3
1/2
go-to t. o.
-1
tourist office
booking not possible
place 2
go-to pl. 2
1
place 3
go-to pl. 3
1/2
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The game after the second step of backward
induction
tourist office
booking possible at tour. off.
go-to t. o.
1
booking not possible
place 2
go-to pl. 2
1
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Conclusions

• Advantages of using Game Theory
• provides an established framework for studying
  cooperative agents
• basic concepts of linguistic pragmatics can be
  defined precisely
• extra-linguistic context can easily be
  represented
• allows fine-grained predictions depending on
  context parameters.