Cost-based pragmatic implicatures in an artificial language experiment Judith Degen, Michael Franke & Gerhard J¨ ager Rochester/Stanford Amsterdam T¨ ubingen July 27, 2013 Workshop on Artificial Grammar Learning T¨ ubingen Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 1 / 42
The Beauty Contest each participant has to write down a number between 0 and 100 all numbers are collected the person whose guess is closest to 2/3 of the arithmetic mean of all numbers submitted is the winner Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 2 / 42
The Beauty Contest (data from Camerer 2003, Behavioral Game Theory ) Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 3 / 42
Signaling games sequential game: nature chooses a world w 1 out of a pool of possible worlds W according to a certain probability distribution p ∗ nature shows w to sender S 2 S chooses a message m out of a set of possible signals M 3 S transmits m to the receiver R 4 R chooses an action a , based on the sent message. 5 Both S and R have preferences regarding R’s action, depending on w . S might also have preferences regarding the choice of m (to minimize signaling costs). Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 4 / 42
The Iterated Best Response sequence sends any interprets mes- S 0 R 0 sages literally true message best response best response R 1 S 1 to S 0 to R 0 best response best response to R 1 to S 1 S 2 R 2 . . . . . . . . . . . . Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 5 / 42
Quantity implicatures (1) a. Who came to the party? b. some : Some boys came to interpretation function: the party. c. all : All boys came to the � some � = { w ∃¬∀ , w ∀ } party. � all � = { w ∀ } Game construction ct = ∅ utilities: W = { w ∃¬∀ , w ∀ } a ∃¬∀ a ∀ w ∃¬∀ = { some } , w ∀ = w ∃¬∀ 1 , 1 0 , 0 { some , all } w ∀ 0 , 0 1 , 1 p ∗ = ( 1 / 2 , 1 / 2 ) Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 6 / 42
Truth conditions some all w ∃¬∀ 1 0 w ∀ 1 1 Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 7 / 42
Example: Quantity implicatures S 0 R 0 w ∃¬∀ w ∀ some all w ∃¬∀ 1 0 1 / 2 1 / 2 some w ∀ 1 / 2 1 / 2 0 1 all R 1 w ∃¬∀ w ∀ S 1 some all 1 0 w ∃¬∀ 1 0 some 0 1 w ∀ 0 1 all F = ( R 1 , S 1 ) In the fixed point, some is interpreted as entailing ¬ all , i.e. exhaustively. Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 8 / 42
Lifted games Truth conditions a. Ann or Bert showed up. (= 1 or ) b. Ann showed up. (= a ) or a b and c. Bert showed up. (= b ) { w a } 1 1 0 0 d. Ann and Bert showed up. (= { w b } 1 0 1 0 and ) { w ab } 1 1 1 1 { w a , w b } 1 0 0 0 w a : Only Ann showed up. { w a , w ab } 1 1 0 0 w b : Only Bert showed up. { w b , w ab } 1 0 1 0 w ab : Both showed up. { w a , w b , w ab } 1 0 0 0 Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 9 / 42
Lifted games IBR sequence: 1 S 0 or a b and { w a } 1 / 2 1 / 2 0 0 { w b } 1 / 2 0 1 / 2 0 { w ab } 1 / 4 1 / 4 1 / 4 1 / 4 { w a , w b } 1 0 0 0 { w a , w ab } 1 / 2 1 / 2 0 0 { w b , w ab } 1 / 2 0 1 / 2 0 { w a , w b , w ab } 1 0 0 0 Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 10 / 42
Lifted games IBR sequence: 2 R 1 { w a } { w b } { w ab } { w a , w b } { w a , w ab } { w b , w ab } { w a , w b , w ab } 0 0 0 1 0 0 0 or 1 0 0 0 0 0 0 a 0 1 0 0 0 0 0 b 0 0 1 0 0 0 0 and Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 11 / 42
Lifted games IBR sequence: 3 S 2 or a b and { w a } 0 1 0 0 { w b } 0 0 1 0 { w ab } 0 0 0 1 { w a , w b } 1 0 0 0 { w a , w ab } 1 / 2 1 / 2 0 0 { w b , w ab } 1 / 2 0 1 / 2 0 { w a , w b , w ab } 1 0 0 0 Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 12 / 42
Lifted games or is only used in { w a , w b } in the fixed point this means that it carries two implicatures: exhaustivity: Ann and Bert did not both show up ignorance: Sally does not know which one of the two disjuncts is true Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 13 / 42
Predicting behavioral data Behavioral Game Theory : predict what real people do (in experiments), rather what they ought to do if they were perfectly rational one implementation (Camerer, Ho & Chong, TechReport CalTech): stochastic choice: people try to maximize their utility, but they make errors level- k thinking: every agent performs a fixed number of best response iterations, and they assume that everybody else is less smart (i.e., has a lower strategic level) Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 14 / 42
Stochastic choice real people are not perfect utility maximizers they make mistakes ❀ sub-optimal choices still, high utility choices are more likely than low-utility ones Rational choice: best response � 1 if u i = max j u j | arg j max u i | P ( a i ) = 0 else Stochastic choice: (logit) quantal response P ( a i ) ∝ e λu i Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 15 / 42
Stochastic choice λ measures degree of rationality λ = 0 : completely irrational behavior all actions are equally likely, regardless of expected utility λ → ∞ convergence towards behavior of rational choice probability mass of sub-optimal actions converges to 0 Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 16 / 42
Iterated Quantal Response (IQR) variant of IBR model best response ist replaced by quantal response predictions now depend on value for λ no 0-probabilities IQR converges gradually Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 17 / 42
Level- k thinking every player: performs iterated quantal response a Poisson distribution limited number k of times (where k ● ● may differ between players), ● assumes that the other players have a 0.3 τ = 1.0 ● τ = 1.5 ● level < k , and ● ● τ = 2.0 ● ● τ = 2.5 ● ● assumes that the strategic levels are ● ● 0.2 ● Pr(k) ● ● distributed according to a Poisson ● ● distribution ● 0.1 ● ● ● ● P ( k ) ∝ τ k / k ! ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 2 4 6 8 10 τ , a free parameter of the model, is the k average/expected level of the other players Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 18 / 42
The experimental setup Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 19 / 42
The experimental setup Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 19 / 42
The experimental setup Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 20 / 42
The experimental setup Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 21 / 42
The experimental setup Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 22 / 42
Simple condition: Literal meanings R 0 S 0 1 0 0 1 / 2 0 0 1 / 2 0 0 1 0 0 1 0 0 1 / 2 1 / 2 0 1 / 2 1 / 2 0 1 0 0 Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 23 / 42
Simple condition: Iterated Best Response R 1 S 1 1 0 0 1 / 2 0 0 1 / 2 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0 Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 24 / 42
Complex condition: Literal meanings R 0 S 0 1 / 3 1 / 3 1 / 3 0 1 / 2 0 1 / 2 1 / 2 1 / 2 0 0 1 / 2 1 / 2 0 0 1 0 0 0 0 1 1 / 2 0 1 / 2 Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 25 / 42
Complex condition: Iterated Best response R 1 S 1 1 / 3 1 / 3 1 / 3 0 1 / 2 0 1 / 2 1 / 2 1 / 2 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 26 / 42
Complex condition: Iterated Best response R 2 S 2 1 / 3 1 / 3 1 / 3 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 27 / 42
Experiment 1 - comprehension test participants’ behavior in a comprehension task implementing previously described signaling games 48 participants on Amazon’s Mechanical Turk two stages: language learning inference 36 experimental trials 6 simple (one-step) implicature trials 6 complex (two-step) implicature trials 24 filler trials (entirely unambiguous/ entirely ambiguous target) Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 28 / 42
Artificial language Zorx XEK RAV ZUB KOR ∅ ∅ Three stages of language learning: 1 2 3 Degen, Franke & J¨ ager (AGL-Workshop) Cost-based implicatures 7/27/2013 29 / 42
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