Game Theory And Grice’ Theory Of Implicatures

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Game Theory And Grice’ Theory Of Implicatures

  1. 1. Game Theory and Grice’ Theory of Implicatures Anton Benz
  2. 2. Anton Benz: Game Theory and Grice’ Theory of Implicatures <ul><li>Grice’ approach to pragmatics </li></ul><ul><ul><ul><li>Assumptions about communication </li></ul></ul></ul><ul><ul><ul><li>The Cooperative Principle and the Maxims </li></ul></ul></ul><ul><li>Scalar Implicatures </li></ul><ul><ul><ul><li>The `standard explanation’ </li></ul></ul></ul><ul><ul><ul><li>A game theoretic reconstruction </li></ul></ul></ul><ul><li>Where can game theory improve pragmatic theory? </li></ul><ul><ul><ul><li>A problem for the standard theory: predictive power </li></ul></ul></ul><ul><ul><ul><li>An example of contradicting inferences </li></ul></ul></ul><ul><li>The game theoretic approach at work </li></ul><ul><ul><ul><li>Implicatures of answers </li></ul></ul></ul>
  3. 3. A simple picture of communication <ul><li>The speaker encodes some proposition p </li></ul><ul><li>He sends it to an addressee </li></ul><ul><li>The addressee decodes it again and writes p in his knowledgebase. </li></ul><ul><li>Problem: We communicate often much more than we literally say! </li></ul><ul><ul><li>Some students failed the exam. </li></ul></ul><ul><ul><li>+> Most of the students passed the exam. </li></ul></ul>
  4. 4. Gricean Pragmatics <ul><li>Grice distinguishes between: </li></ul><ul><li>What is said . </li></ul><ul><li>What is implicated . </li></ul><ul><ul><li>“Some of the boys came to the party.” </li></ul></ul><ul><ul><ul><li>said: At least two of the boys came to the party. </li></ul></ul></ul><ul><ul><ul><li>implicated: Not all of the boys came to the party. </li></ul></ul></ul><ul><li>Both part of what is communicated . </li></ul>
  5. 5. Assumptions about Conversation <ul><li>Conversation is a cooperative effort . Each participant recognises in their talk exchanges a common purpose . </li></ul><ul><li>Example: A stands in front of his obviously immobilised car. </li></ul><ul><ul><li>A: I am out of petrol. </li></ul></ul><ul><ul><li>B: There is a garage around the corner. </li></ul></ul><ul><ul><li>Joint purpose of B’s response : Solve A’s problem of finding petrol for his car. </li></ul></ul>
  6. 6. The Cooperative Principle <ul><li>Conversation is governed by a set of principles which spell out how rational agents behave in order to make language use efficient. </li></ul><ul><li>The most important is the so-called cooperative principle : </li></ul><ul><li>“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.” </li></ul>
  7. 7. The Conversational Maxims <ul><li>Maxim of Quality : </li></ul><ul><li>1. Do not say what you believe to be false. 2. Do not say that for which you lack adequate evidence. </li></ul><ul><li>Maxim of Quantity : </li></ul><ul><li>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. </li></ul><ul><li>Maxim of Relevance : make your contributions relevant. </li></ul><ul><li>Maxim of Manner : be perspicuous, and specifically: </li></ul><ul><li>1. Avoid obscurity. 2. Avoid ambiguity. 3. Be brief (avoid unnecessary wordiness). 4. Be orderly. </li></ul>
  8. 8. The Conversational Maxims <ul><li>Maxim of Quality : Be truthful. </li></ul><ul><li>Maxim of Quantity : </li></ul><ul><li>Say as much as you can. </li></ul><ul><li>Say no more than you must. </li></ul><ul><li>Maxim of Relevance : Be relevant. </li></ul>
  9. 9. The Conversational Maxims <ul><li>Be truthful (Quality) and say as much as you can (Quantity) as long as it is relevant (Relevance). </li></ul>
  10. 10. An example: Scalar Implicatures <ul><li>Let A(x)  “x of the boys came to the party” </li></ul><ul><li>It holds A(all)  A(some). </li></ul><ul><li>The speaker said A(some). </li></ul><ul><li>If all of the boys came, then A(all) would have been preferred (Maxim of Quantity). </li></ul><ul><li>The speaker didn’t say A(all), hence it cannot be the case that all came. </li></ul><ul><li>Therefore some but not all came to the party. </li></ul>
  11. 11. Game Theory <ul><li>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: </li></ul><ul><ul><li>the choice of actions by others </li></ul></ul><ul><ul><li>our preferences over the ultimate results. </li></ul></ul><ul><li>Whether or not an utterance is successful depends on </li></ul><ul><ul><li>how it is taken up by its addressee </li></ul></ul><ul><ul><li>the overall purpose of the current conversation. </li></ul></ul>
  12. 12. 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
  13. 13. 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% < ”
  14. 14. General Schema for explaining implicatures <ul><li>Start out with a game defined by pure semantics. </li></ul><ul><li>Pragmatic principles define restrictions on this game. </li></ul><ul><li>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. </li></ul>
  15. 15. An example of contradicting inferences I <ul><li>Situation: A stands in front of his obviously immobilised car. </li></ul><ul><ul><li>A: I am out of petrol. </li></ul></ul><ul><ul><li>B: There is a garage around the corner. (G) </li></ul></ul><ul><ul><li>Implicated: The garage is open. (H) </li></ul></ul><ul><li>How should one formally account for the implicature? </li></ul><ul><li>Set H*:= The negation of H </li></ul><ul><li>B said that G but not that H*. </li></ul><ul><li>H* is relevant and G  H*  G. </li></ul><ul><li>Hence if G  H*, then B should have said G  H* (Quantity). </li></ul><ul><li>Hence H* cannot be true, and therefore H. </li></ul>
  16. 16. An example of contradicting inferences II <ul><li>Problem: We can exchange H and H* and still get a valid inference: </li></ul><ul><li>B said that G but not that H. </li></ul><ul><li>H is relevant and G  H  G. </li></ul><ul><li>Hence if G  H, then B should have said G  H (Quantity). </li></ul><ul><li>Hence H cannot be true, and therefore H*. </li></ul><ul><li>Missing: Precise definitions of basic concepts like relevance . </li></ul>
  17. 17. The Utility of Answers <ul><li>Questions and answers are often subordinated to a decision problem of the inquirer. </li></ul><ul><li>Example: Somewhere in Amsterdam </li></ul><ul><ul><li>I: Where can I buy an Italian newspaper? </li></ul></ul><ul><ul><li>E: At the station and at the palace. </li></ul></ul><ul><ul><li>Decision problem of A: Where should I go to in order to buy an Italian newspaper. </li></ul></ul>
  18. 18. The general situation
  19. 19. Decision Making <ul><li>The Model: </li></ul><ul><li>Ω: a (countable) set of possible states of the world. </li></ul><ul><li>P I , P E : (discrete) probability measures representing the inquirer’s and the answering expert’s knowledge about the world. </li></ul><ul><li>A : a set of actions. </li></ul><ul><li>U I , U E : Payoff functions that represent the inquirer’s and expert’s preferences over final outcomes of the game. </li></ul><ul><li>Decision criterion: an agent chooses an action which maximises his expected utility : </li></ul><ul><li>EU(a) =  v  Ω P(w)  U(v,a) </li></ul>
  20. 20. An Example <ul><li>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 . </li></ul><ul><ul><li>J: I want to dance tonight. Where can I go to? </li></ul></ul><ul><ul><li>E: Oh, tonight they play Hip Hop at the Roter Salon. </li></ul></ul><ul><ul><li>implicated: No Salsa at the Roter Salon. </li></ul></ul>
  21. 21. 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
  22. 22. 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
  23. 23. 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
  24. 24. General method for calculating implicatures (informal) <ul><li>Describe the utterance situation by a game (in extensive form, i.e. tree form). </li></ul><ul><ul><ul><li>The game tree shows: </li></ul></ul></ul><ul><ul><li>Possible states of the world </li></ul></ul><ul><ul><li>Utterances the speaker can choose </li></ul></ul><ul><ul><li>Their interpretations as defined by semantics. </li></ul></ul><ul><ul><li>Preferences over outcomes (given by context) </li></ul></ul><ul><li>Simplify tree by backward induction. </li></ul><ul><li>‘Read off’ the implicature from the game tree that cannot be simplified anymore. </li></ul>
  25. 25. Another Example <ul><li>J approaches the information desk at the city railway station. </li></ul><ul><ul><li>J: I need a hotel. Where can I book one? </li></ul></ul><ul><ul><li>E: There is a tourist office in front of the building. </li></ul></ul><ul><ul><li>(E: *There is a hairdresser in front of the building.) </li></ul></ul><ul><ul><li>implicated: It is possible to book hotels at the tourist office. </li></ul></ul>
  26. 26. 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
  27. 27. 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
  28. 28. The game after the second step of backward induction booking possible at tour. off. 1 1 booking not possible “ tourist office” “ place 2” go-to t. o. go-to pl. 2
  29. 29. Conclusions <ul><li>Advantages of using Game Theory: </li></ul><ul><li>provides an established framework for studying cooperative agents; </li></ul><ul><li>basic concepts of linguistic pragmatics can be defined precisely; </li></ul><ul><li>extra-linguistic context can easily be represented; </li></ul><ul><li>allows fine-grained predictions depending on context parameters. </li></ul>
  30. 30. Scalar implicatures: The standard explanation <ul><li>The ‘Standard Explanation’ for a scale with two elements: </li></ul><ul><li>It holds p 1  p 2 but not p 2  p 1 . </li></ul><ul><li>There are two expression e 1 , e 2 of comparable complexity. </li></ul><ul><li>e 1 means p 1 and e 2 means p 2 . </li></ul><ul><li>The speaker said e 2 . </li></ul><ul><li>If p 1 is the case, then the use of e 1 is preferred (by 1. and Quantity). </li></ul><ul><li>The speaker didn’t say e 1 , hence p 1 is not the case. </li></ul><ul><li>Therefore p 2  ¬ p 1 is the case. </li></ul>
  31. 31. A Schema for Inferring Implicatures <ul><li>S has said that p; </li></ul><ul><li>it is mutual knowledge that S and H play a certain (signalling) game G; </li></ul><ul><li>in order for S to say that p and be indeed playing G, it must be the case that q; </li></ul><ul><li>(hence) it is mutual knowledge that q must be the case if S utters p; </li></ul><ul><li>S has done nothing to stop H, the addressee, thinking that they play G; </li></ul><ul><li>therefore in saying that p S has implicated that q. </li></ul>

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