Optimal answers and their implicatures A game-theoretic approach

Anton BenzApril 18th, 2006

Overview

1. Conversational Implicatures in the Standard Theory

2. Conventions and Meaning

3. Game Theoretic Pragmatics

4. Implicatures of Answers

Conversational Implicatures

The Standard Theory

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.

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 B’s response: Solve A’s problem of finding petrol for his car.

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.”

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.

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.

The Conversational Maxims

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

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 didn’t say A(all), hence it cannot be the case that all came.

6. Therefore some but not all came to the party.

Conventions and Meaning

The Lewisean Perspective

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

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 M=M’.

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

1. a speaker’s signalling strategy (S: MF)

2. a hearer’s interpretation strategy (H: FM)

such that communication is always successful.

The agenda

Putting Gricean pragmatics on Lewisean feet:

1. Start assumption: semantic meaning is defined by a signalling convention (Semantic Interpretation Game, SIG).

2. Gricean maxims (and other pragmatic conditions) translate into constraints on the SIG.

3. The explanation of a pragmatic phenomenon proceeds by a game theoretic analysis of the constrained SIG.

Game Theoretic Pragmatics

Scalar Implicatures

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.

The Game Theoretic Analysis of Scalar Implicatures(For a scale with three elements: <all, most, some>)

“all”

“some”

“most”

“most”

“some”

“some”

100%

50% >

50% <

50% >

50% >

0; 0

1; 1

0; 0

0; 0

1; 1

1; 1

The Game Theoretic Analysis of Scalar Implicatures(Taking into account the speaker’s preferences)

100%

50% >

50% <

“all”

“some”

“most”

50% >

1; 1

1; 1

1; 1

In all branches that contain “some” the initial situation is “50% < ”

Hence: “some” implicates “50% < ”

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.

Implicatures of Answers

Implicatures and Decision Problems

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.

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.

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.

The general situation

Decision Making

The Model:• Ω: a (countable) set of possible states of the world.

• PI, PE: (discrete) probability measures representing the inquirer’s and the answering expert’s knowledge about the world.

• A : a set of actions.

• UI, UE: Payoff functions that represent the inquirer’s and expert’s preferences over final outcomes of the game.

Decision criterion: an agent chooses an action which maximises his expected utility:

EU(a) = vΩ P(w) U(v,a)

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.

A game tree for the situation where both Salsa and Hip Hop are playing

both play at RS

“Salsa”1

go-to RS

stay home

0

1

go-to RS

stay home

0

1

go-to RS

stay home

0

“both”

“Hip Hop”

RS = Roter Salon

The tree after the first step of backward induction

both

Salsa

Hip Hop

“both”

“Salsa”

“Hip Hop”

“Salsa”

“Hip Hop”

stay home

go-to RS

go-to RS

go-to RS

go-to RS

1

0

0

2

2

The tree after the second step of backward induction

both

Salsa

Hip Hop

“both”

“Salsa”

“Hip Hop”

stay home

go-to RS

go-to RS

1

2

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

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.

The situation where it is possible to book a hotel at the tourist information, a place 2, and a place 3.

“place 2”1

0

1

s. a.

go-to tourist office

0

1/2

0

“tourist office”

“place 3”

go-to pl. 2

go-to pl. 3

s. a.

s. a.

s. a. : search anywhere

The game after the first step of backward induction

booking possible at tour. off.

1

0

1/2

-1

1

1/2

booking not possible

“place 2”

“tourist office”

“place 3”

“place 2”

“tourist office”

“place 3”

go-to t. o.

go-to pl. 2

go-to pl. 3

go-to t. o.

go-to pl. 2

go-to pl. 3

The game after the second step of backward induction

booking possible at tour. off.

1

1booking not possible

“tourist office”

“place 2”

go-to t. o.

go-to pl. 2

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.