Game Theory and Grice

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Game Theory and Grice Theory of Implicatures

• Anton Benz

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Anton Benz Game Theory and Grice Theory of
Implicatures

• Grice approach to pragmatics
• Assumptions about communication
• The Cooperative Principle and the Maxims
• Scalar Implicatures
• The standard explanation
• A game theoretic reconstruction
• Where can game theory improve pragmatic theory?
• A problem for the standard theory predictive
  power
• An example of contradicting inferences
• The game theoretic approach at work
• Implicatures of answers

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A simple picture of communication

• The speaker encodes some proposition p
• He sends it to an addressee
• The addressee decodes it again and writes p in
  his knowledgebase.
• Problem We communicate often much more than we
  literally say!
• Some students failed the exam.
• gt Most of the students passed the exam.

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Gricean Pragmatics

• 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
• 1. Do not say what you believe to be false.
  2. Do not say that for which you lack adequate
  evidence.
• Maxim of Quantity
• 1. Make your contribution to the conversation
  as informative as is required for he current talk
  exchange. 2. 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
• 1. Avoid obscurity. 2. Avoid ambiguity. 3.
  Be brief (avoid unnecessary wordiness). 4. 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

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

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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 Version(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 Version(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 Schema for explaining implicatures

• Start out with a game defined by pure semantics.
• Pragmatic principles define restrictions on this
  game.
• Semantics Pragmatic Principles explain an
  implicature of an utterance if the implicated
  proposition is true in all branches of the
  restricted game in which the utterance occurs.

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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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General method for calculating implicatures
(informal)

• Describe the utterance situation by a game (in
  extensive form, i.e. tree form).
• The game tree shows
• 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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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.

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Scalar implicatures The standard explanation

• The Standard Explanation for a scale with two
  elements
• It holds p1 ? p2 but not p2 ? p1.
• There are two expression e1, e2 of comparable
  complexity.
• e1 means p1 and e2 means p2.
• The speaker said e2.
• If p1 is the case, then the use of e1 is
  preferred (by 1. and Quantity).
• The speaker didnt say e1, hence p1 is not the
  case.
• Therefore p2 ? p1 is the case.

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A Schema for Inferring Implicatures

1. S has said that p
2. it is mutual knowledge that S and H play a
   certain (signalling) game G
3. in order for S to say that p and be indeed
   playing G, it must be the case that q
4. (hence) it is mutual knowledge that q must be the
   case if S utters p
5. S has done nothing to stop H, the addressee,
   thinking that they play G
6. therefore in saying that p S has implicated that
   q.