Game Theory and Grice’ Theory of Implicatures

Anton Benz

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 workImplicatures of answers

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.+> Most of the students passed the

exam.

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.

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 MaximsMaxim 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 MaximsMaxim of Quality: Be

truthful.

Maxim of Quantity: 1. Say as much as you can. 2. Say no more than you must.

Maxim of Relevance: Be relevant.

The Conversational MaximsBe truthful (Quality) and

say as much as you can (Quantity) as long as it is relevant (Relevance).

An example: Scalar ImplicaturesLet 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 didn’t say A(all), hence it cannot be the case that all came.

Therefore some but not all came to the party.

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 Version(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 Version(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 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.

An example of contradicting inferences ISituation: 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

1. B said that G but not that H*. 2. H* is relevant and G H* G. 3. Hence if G H*, then B should have said G

H* (Quantity). 4. Hence H* cannot be true, and therefore H.

An example of contradicting inferences IIProblem: We can exchange H and H* and still

get a valid inference:

1. B said that G but not that H. 2. H is relevant and G H G. 3. Hence if G H, then B should have

said G H (Quantity). 4. 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 AmsterdamI: 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

General method for calculating implicatures (informal)1. 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)

2. Simplify tree by backward induction.

3. ‘Read off’ the implicature from the game tree that cannot be simplified anymore.

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.

Scalar implicatures: The standard explanationThe ‘Standard Explanation’ for a scale with two

elements:

1. It holds p1 p2 but not p2 p1.

2. There are two expression e1, e2 of comparable complexity.

3. e1 means p1 and e2 means p2.

4. The speaker said e2.

5. If p1 is the case, then the use of e1 is preferred (by 1. and Quantity).

6. The speaker didn’t say e1, hence p1 is not the case.

7. Therefore p2 ¬ p1 is the case.

A Schema for Inferring Implicatures1. 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.