Optimal Reasoning About Referential Expressions Judith Degen 1 - - PowerPoint PPT Presentation

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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

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Optimal Reasoning About Referential Expressions

Judith Degen1 Michael Franke2 Gerhard J¨ ager3

1Department of Brain and Cognitive Sciences

University of Rochester

2Institute for Logic, Language and Computation

Universiteit van Amsterdam

3Seminar f¨

ur Sprachwissenschaft Universit¨ at T¨ ubingen

June 8, 2012

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Reference to objects

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Reference to objects

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Reference to objects

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Reference to objects

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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.

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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.

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Outline

1

Game-theoretic pragmatics & IBR

2

Experiment 1 - comprehension

3

Experiment 2 - production

4

Discussion

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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

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The Beauty Contest

(data from Camerer 2003, Behavioral Game Theory)

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Signaling games

sequential game:

1

nature chooses a type t

  • ut of a pool of possible types T

according to a certain probability distribution p∗

2

nature shows t to sender S

3

S chooses a message m out of a set of possible signals M

4

S transmits m to the receiver R

5

R guesses a type t′, based on the sent message.

if t = t′, both players score a point

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An example

Types Messages

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Exogeneous meaning

Messages may have conventional or iconic meaning (which is common knowledge among the players) in our example:

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The Iterated Best Response sequence

S0 R0 S1 R1 S2 R2 . . . . . .

sends any true message interprets mes- sages literally best response to S0 best response to R0 best response to R1 . . . best response to S1 . . .

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Sender

Sender strategy Sk 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

  • f S, given Rk−1

S0

1/2 1/2

1

1/2 1/2

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Receiver

Receiver strategy Rk 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

  • f R, given Sk−1

R0 1 1

1/2 1/2

1

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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

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Iterated Best Response

S0

1/2 1/2

1

1/2 1/2

R1 1 1 1 1 R0 1 1

1/2 1/2

1

S1

1/2 1/2

1 1

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Iterated Best Response (cont.)

S2

1/2 1/2

1 1

R3 1 1 1 1 R2 1 1 1 1

S3

1/2 1/2

1 1

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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)

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Simple implicature trial

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Simple implicature trial - predictions

IBR predictions for distribution

  • f responses over target and

competitor:

20 40 60 80 100 k = 0 k > 0 Proportion of choices Response

target competitor

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Complex implicature trial

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Complex implicature trial - predictions

IBR predictions for distribution

  • f responses over target and

competitor:

20 40 60 80 100 k <= 1 k > 1 Proportion of choices Response

target competitor

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Unambiguous filler

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Ambiguous filler

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Results - proportion of responses by condition

0.0 0.2 0.4 0.6 0.8 1.0 ambiguous filler complex implicature simple implicature unambiguous filler Proportion of choices Response

target distractor competitor

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Results - proportion of responses by condition

0.0 0.2 0.4 0.6 0.8 1.0 ambiguous filler complex implicature simple implicature unambiguous filler Proportion of choices Response

target distractor competitor

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Results - proportion of responses by condition

0.0 0.2 0.4 0.6 0.8 1.0 ambiguous filler complex implicature simple implicature unambiguous filler Proportion of choices Response

target distractor competitor

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Results - distribution of subjects over target choices

5 10 15 20 1 2 3 4 5 6 Number of target choices (out of 6 possible) Number of subjects (out of 28) Implicature complex simple

→ not predicted by standard IBR

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Results - learning effects

simple implicature complex implicature 0.0 0.2 0.4 0.6 0.8 1.0 1 2 3 4 5 6 1 2 3 4 5 6 Relative trial number Proportion of choices Response target distractor competitor

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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)

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Simple implicature trial

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Complex implicature trial

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Results - proportion of responses by condition

0.0 0.2 0.4 0.6 0.8 1.0 ambiguous filler complex implicature simple implicature unambiguous filler

Proportion of choices

Response

target distractors competitor

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Results - proportion of responses by condition

Experiment 1 (comprehension)

0.0 0.2 0.4 0.6 0.8 1.0 a m b i g u

  • u

s f i l l e r c

  • m

p l e x i m p l i c a t u r e s i m p l e i m p l i c a t u r e u n a m b i g u

  • u

s f i l l e r Proportion of choices Response

target distractor competitor

Experiment 2 (production)

0.0 0.2 0.4 0.6 0.8 1.0 a m b i g u

  • u

s f i l l e r c

  • m

p l e x i m p l i c a t u r e s i m p l e i m p l i c a t u r e u n a m b i g u

  • u

s f i l l e r

Proportion of choices

Response

target distractors competitor

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Results - distribution of subjects over target choices

Experiment 1 (comprehension)

5 10 15 20 1 2 3 4 5 6 Number of target choices (out of 6 possible) Number of subjects (out of 28) Implicature complex simple

Experiment 2 (production)

5 10 15 20 1 2 3 4 5 6 Number of target choices (out of 6 possible) Number of subjects (out of 28) Implicature complex simple

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Interim summary

asymmetry in production and comprehension: simple implicatures easier in production than comprehension and vice versa for complex implicatures not predicted by standard IBR

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Predicting behavioral data

Behavioral Game Theory: predict what real people do (in experiments), rather what they ought to do if they were perfectly rational

  • ne 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)

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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

P(ai) =

  • 1

| arg j max ui|

if ui = max juj else

Stochastic choice: (logit) quantal response

P(ai) ∝ exp(λui)

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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

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Level-k thinking

every player:

performs iterated best response a limited number k of times (where k may differ between players), assumes that the other players have a level < k, and assumes that the strategic levels are distributed according to a Poisson distribution P(k) ∝ τ k k!

τ, a free parameter of the model, is the average/expected level of the other players

  • 2

4 6 8 10 0.0 0.1 0.2 0.3

Poisson distribution

k Pr(k)

  • τ = 1.0

τ = 1.5 τ = 2.0 τ = 2.5

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Fitting the data

maximum likelihood estimation of λ and τ on the basis of our experiments: Experiment 1 (comprehension): λ1 = 6.33 τ 1 = 0.87

1 2 3 4 5 6 Number of target responses Number of subjects 5 10 15 20 Implicature simple complex

  • Experiment 2 (production):

λ2 = 6.52 τ 2 = 1.25

1 2 3 4 5 6 Number of target responses Number of subjects 5 10 15 20 Implicature simple complex

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Tentative interpretation

production/comprehension asymmetry: Speakers are more strategic than listeners!

  • 1

2 3 4 5 0.0 0.1 0.2 0.3 0.4

Probability of strategic levels

k Pr(k)

  • τ = 0.87

τ = 1.25

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Alternative hypothesis

This model took it for granted that non-strategic senders simply pick a true message at random. Results of experiment 2 suggest that this is not true; virtually everybody chooses the message that is most informative. Alternative hypothesis: S0 uses the following utility function: uS0 (m|t) =

  • 1

|{t′|t′∈ [ [ m ] ] }|

if t ∈ [ [ m ] ] else

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Fitting the data, # 2

Experiment 2 (production): λ2 ′ = 5.35 τ 2 ′ = 0.23

1 2 3 4 5 6 Number of target responses Number of subjects 5 10 15 20 Implicature simple complex

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Tentative interpretation # 2

production/comprehension asymmetry: Speakers barely reason at all, they just have a useful heuristics!

  • 1

2 3 4 5 0.0 0.2 0.4 0.6 0.8

Probability of strategic levels

k Pr(k)

  • τ = 0.87

τ = 0.23

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Conclusion

interlocutors do take perspective and simulate each others’ beliefs

but not always optimally and less so as the number of required reasoning steps increases

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Future directions

utility manipulation message cost manipulation - moving into the realm of actual language interactive experiments with feedback

?

learning

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Thanks to EURO-XPRAG Tanenhaus lab Mike Tanenhaus & the NIH Florian Jaeger

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References I

Brown-Schmidt, S., Gunlogson, C., & Tanenhaus, M. K. (2008, June). Addressees distinguish shared from private information when interpreting questions during interactive conversation. Cognition, 107(3), 1122–34. Camerer, C., Ho, T., & Chong, K. (2002). Behavioral game theory: Thinking, learning and teaching (Tech. Rep.). Pasadena: CalTech. Camerer, C. F. (2003). Behavioral game theory: Experiments in strategic

  • interaction. Princeton: Princeton University Press.

Clark, H., & Murphy, G. L. (1982). Audience design in meaning and

  • reference. In J. LeNy & W. Kintsch (Eds.), Language and
  • comprehension. Amsterdam: North-Holland.

Hanna, J., Tanenhaus, M. K., & Trueswell, J. C. (2003). The effects of common ground and perspective on domains of referential

  • interpretation. Journal of Memory and Language, 49, 43-61.

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References II

Heller, D., Grodner, D., & Tanenhaus, M. K. (2008). The role of perspective in identifying domains of reference. Cognition, 108, 831-836. Horton, W., & Keysar, B. (1996). When do speakers take into account common ground? Cognition, 59, 91–117. Keysar, B., Barr, D. J., & Brauner, J. S. (2000). Taking perspective in conversation: The role of mutual knowledge in comprehension. Psychological Science, 11, 32-37.

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