Game Theoretic Pragmatics

Transcription

1 Game Theoretic Pragmatics Session 2: Relevance of Information & Relevance Implicatures Michael Franke, Roland Mühlenbernd & Jason Quinley Seminar für Sprachwissenschaft Eberhard Karls Universität Tübingen

2 Today s Session 1 Information Dynamics 2 Relevance of Information 3 Relevance Implicatures 2 / 24

3 Definition (Updating a Belief Set) Let X T be a belief set. Learning of information Y T is modelled as updating the belief set X with Y: X[Y] = X Y. Example (Deutscher Fußballmeister 09/10) T = { } t Bayern, t Schalke,..., t Hertha as before belief state: X = { } t Bayern, t Schalke, t Hertha proposition neither Hertha nor Eintracht : Y = T \ {t Hertha, t Eintracht } update: X[Y] = { } t Bayern, t Schalke 3 / 24

4 Definition (Bayesian Update) Let Pr (T). The conditional probability of event X T given event Y T is calculated by Bayesian update as follows: Pr(X Y) = Pr(X Y) Pr(Y). Remark only defined if Pr(Y) = 0 if Pr(Y) = 0, we need belief revision (complicated) 4 / 24

5 Example (Deutscher Fußballmeister 09/10) T = { } t Bayern, t Schalke,..., t Hertha as before probabilistic belief:.6 if y = Bayern.3 if y = Schalke Pr(t y ) =.1 if y = Hertha 0 otherwise proposition neither Hertha nor Eintracht : Y = T \ {t Hertha, t Eintracht } update:.6 if y = Bayern.3 if y = Schalke Pr(t y Y) = 0 if y = Hertha 0 otherwise 5 / 24

6 Definition (Updated Decision Problem) If D = T, A, Pr, U and P T such that Pr(P) > 0, then D[P] = T, A, Pr( P), U is the decision problem D updated with information P. 6 / 24

7 Example (Beer, Wine or Vodka?) original problem D is: Pr(t) a beer a wine a vodka t cool t classy t punk EU D (a) } information showers daily : P = {t cool, t classy updated problem D[P] is: Pr(t) a beer a wine a vodka t cool t classy t punk EU D[P] (a) / 24

8 Relevance of Information When is information P relevant for D? compare D with D[P] two notions: 1 Utility Value 2 Value of Sample Information 8 / 24

9 Definition (Utility Value) The utility value UV D (P) of a proposition P in decision problem D is: UV D (P) = Value(D[P]) Value(D). P is uv-relevant iff UV(P) = 0. Intuition Compare whether D[P] is better or worse (in expectation) than D. Properties 1 can be negative, zero or positive 2 it is possible that UV D (P) < UV D (Q), for P Q 9 / 24

10 Example (Beer, Wine or Vodka?) D Pr(t) a beer a wine a vodka t cool t classy t punk EU D (a) D[P] Pr(t) a beer a wine a vodka t cool t classy t punk EU D[P] (a) P is uv-relevant: UV D (P) = Value(D[P]) Value(D) = = / 24

11 Definition ((Conditional) Value of Sample Information) Let a BR(D) be the unique action the agent chooses in D, and define the (conditional) value of sample information P as: VSI D (P) = Value(D[P]) EU D[P] (a ). P is vsi-relevant iff VSI(P) = 0. Intuition vsi compares the uninformed choice a with some informed choice a BR(D[P]) in the light of D[P]. Properties 1 VSI D (P) 0 for all D and P. 2 If information P does not change the agent s choice of action, it is irrelevant according to vsi: i.e., if a = a, then VSI D (P) = 0. 3 Information can also be irrelevant if it does change the agent s choice of action, i.e., it is not the case that if VSI D (P) = 0, then a = a. 11 / 24

12 Example (Beer, Wine or Vodka?) D Pr(t) a beer a wine a vodka t cool t classy t punk EU D (a) D[P] Pr(t) a beer a wine a vodka t cool t classy t punk EU D[P] (a) P is not vsi-relevant: VSI D (P) = Value(D[P]) EU D[P] (a wine ) = EU D[P] (a wine ) EU D[P] (a wine ) = 0 12 / 24

13 Relevance Implicatures Enrich the semantic meaning of P by maximization of relevance : Prag D (P) = P + X where X is an inference derived from the assumption that P is (maximally/optimally) relevant in D. Relevance Orders (4 possibilities) P more relevant than P (or P is relevant beyond P) iff: measure method uv-based vsi-based naïve UV D (P ) > UV D (P) UV D (P ) > UV D (P) ex post UV D[P] (P ) = 0 VSI D[P] (P ) = 0 13 / 24

14 Naïve Approach Interpret P as at least as relevant as all stronger P P: Prag D (P) = P } {P P P Rel D (P ) > Rel D (P) where Rel D ( ) is either uv or vsi. Intuition Assume that: proposition P is true, and any more relevant stronger proposition P must be false, because else the speaker would have used it. 14 / 24

15 Example (Grice s Garage) (1) a. A: I am out of petrol. b. B: There is a garage round the corner. c. The garage is open. decision problem D: Pr(t) a stroll a go t open t closed t (1b) is P = { } t open, t closed compare this to P open = { t open } and Pclosed = {t closed } 15 / 24

16 Example (Grice s Garage): Naïve Approach with uv D Pr(t) a stroll a go t open t closed t calculate uv: uv D (P ) = Value(D[P ]) Value(D) = EU D[P ] (a go) EU D (a stroll ) =.5.3 =.2 uv D (P open ) = EU D[Popen ] (a go) EU D (a stroll ) =.7 uv D (P closed ) = EU D[Pclosed ] (a stroll) EU D (a stroll ) = 0 ordering: UV D (P open ) > UV D (P ) > UV D (P closed ) pragmatic interpretation: Prag D (P ) = P { } P open = {tclosed } (wrong prediction) 16 / 24

17 Example (Grice s Garage): Naïve Approach with vsi D Pr(t) a stroll a go t open t closed t calculate vsi: vsi D (P ) = EU D[P ] (a go) EU D[P ] (a stroll) =.5.3 =.2 vsi D (P open ) =.7 vsi D (P closed ) = 0 ordering: UV D (P open ) > UV D (P ) > UV D (P closed ) pragmatic interpretation: Prag D (P ) = P { } P open = {tclosed } (wrong prediction) 17 / 24

18 Conclusion Naïve Approach does not work. Relevance Ex Post Interpret P as relevant enough, so that after knowing P no P P is relevant: Prag D (P) = P } {P P P Rel D[P] (P ) = 0 where Rel D[P] ( ) is either uv or vsi. Intuition Assume that: P is true, and all stronger P that would have been relevant beyond P, i.e., relevant information when we know P, are false, because else the speaker would have said it. 18 / 24

19 Example (Grice s Garage): ex post with uv D Pr(t) a stroll a go t open t closed t calculate UV D[P ] ( ): uv D[P ] (P open) = Value(D[P open ]) Value(D[P ]) = EU D[Popen ] (a go) EU D[P ] (a go) =.5 uv D[P ] (P closed) =.2 both propositions are relevant beyond P pragmatic interpretation: Prag D (P ) = P P open P closed =. 19 / 24

20 Example (Grice s Garage): ex post with vsi D Pr(t) a stroll a go t open t closed t calculate VSI D[P ] ( ): vsi D[P ] (P open) = EU D[Popen ] (a go) EU D[Popen ] (a go) = 1 1 = 0 vsi D[P ] (P closed) = EU D[Pclosed ] (a stroll) EU D[Pclosed ] (a go) =.3 0 =.3 only P closed is relevant beyond P pragmatic interpretation: Prag D (P ) = P P close = { } t open. 20 / 24

21 Reflection what counts as intuitively relevant information? how does this property influence language interpretation? why did only the ex post + vsi approach work? what s wrong with naïve? satisficing vs. maximizing? what s wrong with uv? relevance = painting a pink future what s good about ex post + vsi? Observation Suppose the speaker knows which action a is best to perform. Then, if she is cooperative, she will try to induce this a in the hearer. ex post + vsi assimilates to reasoning that: the induced action a BR(D[P]) is the/a best action a = a, more specific propositions that would have triggered another action = a are therefore false 21 / 24

22 Reflection does ex post + vsi always work? do we need relevance for all this? expression alternatives? no! game theory Further Reading Anton Benz (2006). Utility and Relevance of Answers. In: Game Theory and Pragmatics. Ed. by Anton Benz et al. Palgrave. Pp Anton Benz (2007). On Relevance Scale Approaches. In: Proceedings of Sinn und Bedeutung 11. Ed. by Estela Puig-Waldmüller. Pp / 24

23 Course Overview (partly tentative) date content 21-4 Gricean Pragmatics & Decision Theory rm, mf 28-4 Relevance & Implicatures rm, mf 05-5 Questions and Decision Problems mf 12-5 Introduction to Game Theory mf 19-5 Game Theory in Pragmatics rm, mf 26-5 Pentecost no class 02-6 Neo-Gricean Pragmatics rm 09-6 ibr model 1 rm, mf 16-6 ibr model 2 rm, mf 23-6 Pragmatic Reasoning about Unawareness mf 30-6 Language Learning in Network Games rm 07-7 Politeness & the Handicap Principle jq 14-7 exam homework dates (due 1 week after) 23 / 24

24 Homework 1 st homework sheet (due May 5) read: script section 2 sections 1 3 of: Dan Sperber and Deirde Wilson (2004). Relevance Theory. In: Handbook of Pragmatics. Ed. by Laurence R. Horn and Gregory Ward. Oxford: Blackwell. Pp Next Session script section 3 Robert van Rooij (1999). Questioning to Resolve Decision Problems. In: Proceedings of the 12 th Amsterdam Colloquium. Ed. by Paul Dekker 24 / 24