Optimal Reasoning About Referential Expressions Judith Degen 1 Michael Franke 2 ager 3 Gerhard J¨ 1 Department of Brain and Cognitive Sciences University of Rochester 2 Institute for Logic, Language and Computation Universiteit van Amsterdam 3 Seminar f¨ ur Sprachwissenschaft Universit¨ at T¨ ubingen June 8, 2012 Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 1 / 47
Reference to objects Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 2 / 47
Reference to objects Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 2 / 47
Reference to objects Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 2 / 47
Reference to objects Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 2 / 47
A hard problem Production (audience design) Clark & Murphy, 1982; Horton & Keysar, 1996; Brown-Schmidt et al., 2008 Choose a message to convey a given intended meaning with sufficiently high probability. Comprehension (perspective-taking) Keysar et al., 2000; Hanna et al., 2003; Heller et al., 2008 Infer the most likely intended interpretation upon observing an utterance. Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 3 / 47
Today Provide a game-theoretic model of the inferences involved in production and comprehension of referential expression that provides a benchmark model of rationality. Provide experimental evidence from two experiments that language users’ choices are boundedly rational. Provide a sketch of how to update the standard model that better captures participants’ probabilistic choices. Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 4 / 47
Outline Game-theoretic pragmatics & IBR 1 Experiment 1 - comprehension 2 Experiment 2 - production 3 Discussion 4 Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 5 / 47
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 Reasoning About Referential Expressions June 8, 2012 6 / 47
The Beauty Contest (data from Camerer 2003, Behavioral Game Theory ) Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 7 / 47
Signaling games sequential game: nature chooses a type t 1 out of a pool of possible types T according to a certain probability distribution p ∗ nature shows t 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 guesses a type t ′ , based on the sent message. 5 if t = t ′ , both players score a point Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 8 / 47
An example Types Messages Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 9 / 47
Exogeneous meaning Messages may have conventional or iconic meaning (which is common knowledge among the players) in our example: • ◦ ◦ • ◦ ◦ • ◦ ◦ • • ◦ Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 10 / 47
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 Reasoning About Referential Expressions June 8, 2012 11 / 47
Sender Sender strategy S k gives probabilistic function from types to messages if several options are equally good, they are chosen with the same probability if k > 0, only messages are chosen that maximize the expected utility of S , given R k − 1 S 0 1 / 2 0 0 1 / 2 0 0 1 0 0 1 / 2 1 / 2 0 Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 12 / 47
Receiver Receiver strategy R k gives stochastic function from messages to types if several options are equally good, they are chosen with the same probability if k > 0, only messages are chosen that maximize the expected utility of R , given S k − 1 R 0 1 0 0 0 0 1 0 1 / 2 1 / 2 1 0 0 Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 13 / 47
Computing best responses to compute the best response to a matrix A : transpose A put a 1 in each cell that is maximal within its row, and a 0 everywhere else normalize row-wise Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 14 / 47
Iterated Best Response S 0 R 0 1 0 0 1 / 2 0 0 1 / 2 0 0 1 0 0 1 0 0 1 / 2 1 / 2 1 0 0 0 1 / 2 1 / 2 0 S 1 R 1 1 / 2 0 0 1 / 2 1 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 15 / 47
Iterated Best Response (cont.) S 2 R 2 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 S 3 R 3 1 / 2 0 0 1 / 2 1 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 16 / 47
Experiment 1 - comprehension test participants’ behavior in a comprehension task implementing previously described signaling games 30 participants on Amazon’s Mechanical Turk initially 4 trials as senders 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 Reasoning About Referential Expressions June 8, 2012 17 / 47
Simple implicature trial Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 18 / 47
Simple implicature trial - predictions IBR predictions for distribution of responses over target and competitor: 100 80 Proportion of choices 60 Response target competitor 40 20 0 k = 0 k > 0 Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 19 / 47
Complex implicature trial Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 20 / 47
Complex implicature trial - predictions IBR predictions for distribution of responses over target and competitor: 100 80 Proportion of choices 60 Response target competitor 40 20 0 k <= 1 k > 1 Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 21 / 47
Unambiguous filler Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 22 / 47
Ambiguous filler Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 23 / 47
Results - proportion of responses by condition 1.0 0.8 Proportion of choices Response 0.6 target distractor 0.4 competitor 0.2 0.0 ambiguous filler complex implicature simple implicature unambiguous filler Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 24 / 47
Results - proportion of responses by condition 1.0 0.8 Proportion of choices Response 0.6 target distractor 0.4 competitor 0.2 0.0 ambiguous filler complex implicature simple implicature unambiguous filler Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 24 / 47
Results - proportion of responses by condition 1.0 0.8 Proportion of choices Response 0.6 target distractor 0.4 competitor 0.2 0.0 ambiguous filler complex implicature simple implicature unambiguous filler Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 24 / 47
Results - distribution of subjects over target choices 20 Number of subjects (out of 28) 15 Implicature complex 10 simple 5 0 0 1 2 3 4 5 6 Number of target choices (out of 6 possible) → not predicted by standard IBR Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 25 / 47
Results - learning effects simple implicature complex implicature 1.0 0.8 Proportion of choices Response 0.6 target distractor 0.4 competitor 0.2 0.0 1 2 3 4 5 6 1 2 3 4 5 6 Relative trial number Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 26 / 47
Experiment 2 - production test participants’ behavior in the analogous production task 30 participants on Amazon’s Mechanical Turk 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 Reasoning About Referential Expressions June 8, 2012 27 / 47
Simple implicature trial Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 28 / 47
Complex implicature trial Degen, Franke & J¨ ager Reasoning About Referential Expressions June 8, 2012 29 / 47
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