Game Theoretic Pragmatics

Transcription

1 Game Theoretic Pragmatics Session 8: The IBR Model Revisited Michael Franke, Roland Mühlenbernd & Jason Quinley Seminar für Sprachwissenschaft Eberhard Karls Universität Tübingen

2 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)

3 Today s Session 1 IBR Reasoning 2 Interpretation Games 3 IBR Interpretation 3 / 14

4 Basic Scaffolding of IBR Reasoning (in a Signaling Game) Base: S 0 S R 0 R Step: S k+1 = {s S ρ Π S : (s1) ρ is a belief based in some fashion on R k (s2) s BR(ρ) } R k+1 = {r R π R = Pr, σ, µ Π R : (r1) σ is a belief based in some fashion on S k (r2) π R is consistent Limit: (r3) r } BR(µ) } S = {s S i j > i : s S j } R = {r R i j > i : r R j IBR = S, R 4 / 14

5 Vanilla IBR Model semantics as focal starting point: S 0 = {s S t : t [[s(t)]]} R 0 = BR(µ 0 ) µ 0 (m) = Pr( [[m]]) step-assumptions: (i) myopia (belief in exactly level-k behavior) (ii) unbiased beliefs (all level-k strategies equiprobable) (iii) truth ceteris paribus (tcp) (all-else-equal stick to semantic meaning) S k+1 = {s S ρ Π S : (v-s1) ρ = R k (s2) s BR(ρ) (tcp) t ( s BR(R k ) t [[s (t)]]) t [[s(t)]] } R k+1 = {r R π R = Pr, σ, µ Π R : (v-r1) σ = S k (r2) π R is consistent (r3) r BR(µ) } 5 / 14

6 Example (Credibility) a 1 a 2 a 3 a 4 m 12 m 23 m 13 t 1 4,5 5,4 0,0 1,4 t 2 0,0 4,5 5,4 1,4 t 3 5,4 0,0 4,5 1,4 6 / 14

7 Definition (Interpretation Game) An interpretation game is a signaling games with meaningful messages: {S, R}, T, Pr, M, [[ ]], A, U S, U R such that: talk is cheap (untrue signaling possible) actions are interpretations T = A priors are flat Pr(t) = Pr(t ) S and R want to coordinate interpretation cooperatively: { 1 if t = a U R (t, m, a) = U S (t, m, a) = 0 otherwise. 7 / 14

8 Constructing an Interpretation Game take to-be-interpreted message m Neo-Gricean alternatives Alt(m ) = M all with standard logical semantics state distinctions: T = {X M m X and X M \ X is consistent} Motivation model of a default context of utterance (think: out-of-the-blue utterance) only distinctions expressible with alternatives to be included 8 / 14

9 Example (Some-All Game) Pr(t) a a m some m all t 1 / 2 1,1 0,0 t 1 / 2 0,0 1,1 (1) a. I ate some of the cookies. m some b. I ate all of the cookies. m all c. The Speaker did not eat all of the cookies. 9 / 14

10 Example (Free Choice Implicature) (2) a. You may take an apple or a pear. (A B) b. You may take an apple and you may take a pear. A B (3) a. You may take an apple. A b. You may take a pear. B target utterance: m (A B) } M = {m A, m B, m (A B) we can distinguish three states within [[ (A B)]] : } t A = {m A, m (A B) t B = } {m B, m (A B) t AB = } {m A, m B, m (A B) 10 / 14

11 Example (Free Choice Implicature) Pr(t) a A a B a AB m A m B m (A B) t 1 A / 3 1,1 0,0 0,0 t 1 B / 3 0,0 1,1 0,0 t 1 AB / 3 0,0 0,0 1,1 11 / 14

12 Theorem (IBR Lite) In interpretation games, the vanilla ibr model (with tcp assumption for both sender and receiver) is equivalent to the following set-theoretic formulation: S 0 (t) = [[t]] 1 R 0 (t) = [[m]] arg max S k+1 (t) = m R 1 k (t) R k(m) if R 1 k (t) = S 0 (t) otherwise arg max R k+1 (m) = t S 1 k (m) S k(t) if S 1 k (m) = R 0 (m) otherwise. 12 / 14

13 Theorem (Equilibrium Selection) For finite interpretation games, each ibr sequence terminates in a fixed point S, R that gives rise to a perfect Bayesian equilibrium. Sketch of Proof. (i) expected gain EG( ) is monotone increasing along ibr sequence: EG(σ, ρ) = Pr(t) σ(t, m) ρ(m, a) U(t, m, a) t m a (ii) for finite games, EG( ) must reach an upper bound (iii) EG(S i, R i+1 ) = EG(S i+2, R i+1 ), implies S i (t) S i+2 (t) for all t (iv) for finite games, this implies a fixed point S, R (v) take triple S, R, µ with µ consistent with S, such that for all surprise messages m we have µ (t m) = [[m]] 1 (vi) S, R, µ is a pbe 13 / 14

14 Homework read: Michael Franke (2010). Free Choice from Iterated Best Response. In: Amsterdam Colloquium Ed. by Maria Aloni and Katrin Schulz. LNAI Heidelberg: Springer. Pp Next Sessions clausal/epistemic implicatures reasoning about unawareness Learning from Neighbors??? 14 / 14