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Game Theoretic Pragmatics Michael Franke Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT Game Theoretic Pragmatics . . . aims for mathematically precise models of language use and interpretation . . .

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Game Theoretic Pragmatics

Michael Franke

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Game Theoretic Pragmatics . . .

  • aims for mathematically precise models of language use and

interpretation . . .

  • by formally representing interlocutors’:

(i) action alternatives (ii) preferences (iii) beliefs about (i)–(iii).

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Today’s Agenda

  • introduce game theory
  • charter how to apply gt to linguistic pragmatics

ibr-model

  • compare biotwith ibr-model and reinforcement learning

find an interpretation of strong/weak optimality

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Running Example: Division of Pragmatic Labor

  • Horn (1984)
  • a.k.a. M-Implicature (Levinson, 2000)
  • unmarked form pairs with unmarked meaning

m ↔ t

  • marked form pairs with marked meaning

m∗ ↔ t∗ Example 1 (Black Bart) (1)

  • a. Black Bart killed the sheriff.

m

  • b. Black Bart killed the sheriff in a stereotypical way.

t (2)

  • a. Black Bart caused the sheriff to die.

m∗

  • b. BB killed the sheriff in a non-stereotypical way.

t∗

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Running Example: Division of Pragmatic Labor

  • Horn (1984)
  • a.k.a. M-Implicature (Levinson, 2000)
  • unmarked form pairs with unmarked meaning

m ↔ t

  • marked form pairs with marked meaning

m∗ ↔ t∗ Example 1 (Black Bart) (3)

  • a. Black Bart killed the sheriff.

m

  • b. Black Bart killed the sheriff in a stereotypical way.

t (4)

  • a. Black Bart caused the sheriff to die.

m∗

  • b. BB killed the sheriff in a non-stereotypical way.

t∗

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Example 2 (Sue’s Smile) (5)

  • a. Sue smiled.

m

  • b. Sue smiled genuinely.

t (6)

  • a. The corners of Sue’s lips turned slightly upwards.

m∗

  • b. Sue faked a smile.

t∗ Example 3 (Mrs T’s Song) (7)

  • a. Mrs T sang ‘Home Sweet Home.’

m

  • b. Mrs T sang a lovely song.

t (8)

  • a. Mrs T produced a series of sounds roughly

corresponding to the score of ‘Home Sweet Home.’ m∗

  • b. Mrs T sang very badly.

t∗

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Example 2 (Sue’s Smile) (9)

  • a. Sue smiled.

m

  • b. Sue smiled genuinely.

t (10)

  • a. The corners of Sue’s lips turned slightly upwards.

m∗

  • b. Sue faked a smile.

t∗ Example 3 (Mrs T’s Song) (11)

  • a. Mrs T sang ‘Home Sweet Home.’

m

  • b. Mrs T sang a lovely song.

t (12)

  • a. Mrs T produced a series of sounds roughly

corresponding to the score of ‘Home Sweet Home.’ m∗

  • b. Mrs T sang very badly.

t∗

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

explanation:

  • :

possible

  • :
  • ptimal

× : blocked

biot’s Explanation t t∗ m

  • m∗
  • generator

& preferences

t t∗ m

  • ×

m∗ ×

  • strong optimality

& blocking

t t∗ m

  • ×

m∗ ×

  • strong optimality

& weak optimality

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

explanation:

  • :

possible

  • :
  • ptimal

× : blocked

biot’s Explanation t t∗ m

  • m∗
  • generator

& preferences

t t∗ m

  • ×

m∗ ×

  • strong optimality

& blocking

t t∗ m

  • ×

m∗ ×

  • strong optimality

& weak optimality

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

explanation:

  • :

possible

  • :
  • ptimal

× : blocked

biot’s Explanation t t∗ m

  • m∗
  • generator

& preferences

t t∗ m

  • ×

m∗ ×

  • strong optimality

& blocking

t t∗ m

  • ×

m∗ ×

  • strong optimality

& weak optimality

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Controversial Issue how to interpret strong/weak optimality?

1 online reasoning?

(e.g. Hendriks et al., 2010)

2 diachronic optimization?

(e.g. Blutner and Zeevat, 2008)

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Branches of Game Theory

  • classical game theory

(since 1940)

  • (mostly) assumes perfectly rational agents
  • central notion: Nash equilibrium
  • evolutionary game theory

(since 1970)

  • long-term optimization of boundedly-rational agents
  • central notions: evolutionary stability & replicator dynamics
  • behavioral game theory

(since 1990)

  • studies interactive decision making in the lab
  • seeks regularities in choices beyond perfect rationality

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Let’s Play: p-Beauty Contest (with p = 2 /

3 )

Everybody choose and write down a number from 0 to 100 (including each). We will sum and average all choices. The person(s) closest to 2 /

3 of the average will win.

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Kinds of Games

uncertainty choice points simultaneous in sequence no strategic/static dynamic/sequential with complete info yes Bayesian dynamic/sequential with incomplete info

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Games vs. Solutions

  • Game Models:

representations of a choice situation

  • Solutions Concepts:

capture particular behavior:

good, optimal, rational, stable (. . . )

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Static Games

  • players choose simultaneously
  • players have complete information

Examples ac ad ac 2,2 0,3 ad 3,0 1,1

Prisoner’s Dilemma

astay ago astay 2,2 0,0 ago 0,0 1,1

Coordination Problem

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Static Games

  • players choose simultaneously
  • players have complete information

Examples ac ad ac 2,2 0,3 ad 3,0 1,1

Prisoner’s Dilemma

astay ago astay 2,2 0,0 ago 0,0 1,1

Coordination Problem

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Static Games

  • players choose simultaneously
  • players have complete information

Examples ac ad ac 2,2 0,3 ad 3,0 1,1

Prisoner’s Dilemma

astay ago astay 2,2 0,0 ago 0,0 1,1

Coordination Problem

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Nash Equilibrium (Intuition) Arrangement of strategies, one for each player, such that no player would benefit from unilateral deviation (i.e., no player would be better off doing something else if everybody else keeps doing the same thing).

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Example (Prisoner’s Dilemma) ac ad ac 2,2 0,3 ad 3,0 1,1

  • single pure ne: ad, ad

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Example (Coordination) astay ago astay 2,2 0,0 ago 0,0 1,1

  • two pure nes:
  • astay, astay
  • , and
  • ago, ago
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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Interpretation Game for DoPL (Dekker and van Rooij, 2000) t t∗ m

  • m∗
  • BiOT-System

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Interpretation Game for DoPL (Dekker and van Rooij, 2000) t t∗ m

  • m∗
  • BiOT-System

t t∗ m 2,2 1,1 m∗ 1,1 0,0

Static Game

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Interpretation Game for DoPL (Dekker and van Rooij, 2000) t t′ m

  • ×

m′ ×

  • Strong Optimality

t t∗ m 2,2 1,1 m∗ 1,1 0,0

Static Game

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Interpretation Game for DoPL (Dekker and van Rooij, 2000) t t′ m

  • ×

m′ ×

  • Strong Optimality

t t∗ m 2,2 1,1 m∗ 1,1 0,0

Nash Equilibrium

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Interpretation Game for DoPL (Dekker and van Rooij, 2000) t t′ m

  • ×

m′ ×

  • Weak Optimality

t t∗ m 2,2 1,1 m∗ 1,1 0,0

Nash Equilibrium

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Interpretation Game for DoPL (Dekker and van Rooij, 2000) t t′ m

  • ×

m′ ×

  • Weak Optimality

t t∗ m 2,2 1,1 m∗ 1,1 0,0

???

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Static Games & BiOT (Dekker and van Rooij, 2000)

  • BiOT-Systems ↔ Static Games
  • strong optimality ↔ Nash equilibrium
  • weak optimality ↔ iterated Nash equilibrium

(???)

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Criticism

  • static interpretation game means that:

1 speaker and hearer choose simultaneously 2 speaker chooses utterance independently of meaning that she

wants to express

3 hearer chooses interpretation independently of any utterance that

needs to be interpreted

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Kinds of Games

uncertainty choice points simultaneous in sequence no strategic/static dynamic/sequential with complete info yes Bayesian dynamic/sequential with incomplete info

  • signaling games
  • interpretation games

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Signaling Games

  • models simplest case of information flow between two agents
  • sender has info, sends message, receiver reacts
  • originally to account for evolution of linguistic conventions

(Lewis, 1969)

  • but also many others:
  • economics: Spence (1973)
  • biology: Grafen (1990)
  • pragmatics: Parikh (1991)
  • overview on signaling games:
  • Sobel (2008)
  • Skyrms (2010)

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Signaling Games ts ∈ T m ∈ M tr ∈ T ts = tr

  • success

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Signaling Games ts ∈ T m ∈ M tr ∈ T ts = tr

  • failure

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Signaling Games — Informal Characterization

  • fix set of states T
  • each t ∈ T occurs with some prior probability Pr(t)
  • sender S knows actual state t ∈ T
  • receiver R doesn’t but knows prior distribution Pr ∈ ∆(T)
  • S chooses a message m ∈ M
  • R observes m and chooses an action a ∈ A
  • both S and R receive payoffs depending on t, m and a

N S R

1

a a∗ m R

1

a a∗ m∗ t p S R a

1

a∗ m R a

1

a∗ m∗ t∗ 1 − p

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Interpretation Game for DoPL N S R

1

a a∗ m R

.8

a

−.2

a∗ m∗ t .7 S R a

1

a∗ m R

−.2

a

.8

a∗ m∗ t∗ .3

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Strategy Profiles for DoPL-Game

m m∗ t t∗ t t∗

Horn Convention

m m∗ t t∗ t t∗

Unstable Pattern

m m∗ t t∗ t t∗

Anti-Horn Convention

m m∗ t t∗ t t∗

Smolensky Convention

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Strategy Profiles for DoPL-Game

m m∗ t t∗ t t∗

Horn Convention

m m∗ t t∗ t t∗

Unstable Pattern

m m∗ t t∗ t t∗

Anti-Horn Convention

m m∗ t t∗ t t∗

Smolensky Convention

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Strategy Profiles for DoPL-Game

m m∗ t t∗ t t∗

Horn Convention

m m∗ t t∗ t t∗

Unstable Pattern

m m∗ t t∗ t t∗

Anti-Horn Convention

m m∗ t t∗ t t∗

Smolensky Convention

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Strategy Profiles for DoPL-Game

m m∗ t t∗ t t∗

Horn Convention

m m∗ t t∗ t t∗

Unstable Pattern

m m∗ t t∗ t t∗

Anti-Horn Convention

m m∗ t t∗ t t∗

Smolensky Convention

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Pure Strategy Profiles of the DoPL-Game 13 9 5 1 14 10 6 2 15 11 7 3 16 12 8 4

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Nash Equilibria of the DoPL-Game 13 9 5 1 14 10 6 2 15 11 7 3 16 12 8 4

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Challenge for Game Theoretic Pragmatics

1 construct/motivate interpretation games 2 single out desired solution with adequate solution concept

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Game Model

represents context of utterance

1

IBR Reasoning

captures reasoning about language use

2 Input:

(i) to-be-interpreted utterance & (ii) its alternative expressions

Output / Prediction:

  • ptimal speaker-hearer behavior

Empirical Data:

(i) trained intuitions & (ii) psycholinguistic data

IBR Model

uniquely determines uniquely determines uniquely determines ?matches?

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Iterated Best Response Reasoning a model of stepwise, hypothetical reasoning: (i) player i assumes that player j does X (focal starting point) (ii) then player i considers best response Y to X (iii) then player i considers player j’s best response X′ to Y (iv) . . . (v) terminate when looping Idea: Focality of Semantic Meaning pragmatic reasoning starts from the semantics

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

IBR Reasoning for Signaling Games 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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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Na¨ ıve Receiver m m∗ t t∗ Level-1 Sender m m∗ t t∗ Level-2 Receiver m m∗ t t∗ (forward induction) Level-3 Sender m m∗ t t∗ fixed point

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Na¨ ıve Receiver m m∗ t t∗ Level-1 Sender m m∗ t t∗ Level-2 Receiver m m∗ t t∗ (forward induction) Level-3 Sender m m∗ t t∗ fixed point

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Na¨ ıve Receiver m m∗ t t∗ Level-1 Sender m m∗ t t∗ Level-2 Receiver m m∗ t t∗ (forward induction) Level-3 Sender m m∗ t t∗ fixed point

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Na¨ ıve Receiver m m∗ t t∗ Level-1 Sender m m∗ t t∗ Level-2 Receiver m m∗ t t∗ (forward induction) Level-3 Sender m m∗ t t∗ fixed point

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Na¨ ıve Receiver m m∗ t t∗ Level-1 Sender m m∗ t t∗ Level-2 Receiver m m∗ t t∗ (forward induction) Level-3 Sender m m∗ t t∗ fixed point

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Na¨ ıve Receiver m m∗ t t∗ Level-1 Sender m m∗ t t∗ Level-2 Receiver m m∗ t t∗ (forward induction) Level-3 Sender m m∗ t t∗ fixed point

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Relation BiOT & IBR (Franke and J¨ ager, 2011)

  • not equivalent, but almost
  • main difference: IBR captures quantity reasoning (scalar

implicatures), BiOT only does when proper constraints are given

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Relation BiOT & Reinforcement Learning (Franke and J¨ ager, 2011) t t∗ m

  • m∗
  • BiOT-System

S’s choice m m∗ t .7 .3 t∗ .7 .3 R’s choice t t∗ m .7 .3 m∗ .7 .3

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Relation BiOT & Reinforcement Learning (Franke and J¨ ager, 2011) t t∗ m

  • m∗
  • BiOT-System

S’s choice m m∗ t .8 .2 t∗ .6 .4 R’s choice t t∗ m .8 .2 m∗ .6 .4

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Relation BiOT & Reinforcement Learning (Franke and J¨ ager, 2011) t t∗ m

  • m∗
  • BiOT-System

S’s choice m m∗ t .9 .1 t∗ .5 .5 R’s choice t t∗ m .9 .1 m∗ .5 .5

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Relation BiOT & Reinforcement Learning (Franke and J¨ ager, 2011) t t∗ m

  • m∗
  • BiOT-System

S’s choice m m∗ t 1 t∗ .4 .6 R’s choice t t∗ m 1 m∗ .4 .6

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Relation BiOT & Reinforcement Learning (Franke and J¨ ager, 2011)

  • weak optimality ≈ most likely path of reinforcement learning

NB: tight connection rl-learning & replicator dynamics (B¨

  • rgers and Sarin, 1997)
  • parallel is close but there are divergences:
  • quantity reasoning (as before)
  • biot makes no arbitrary meaning enrichment, rl might

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Preliminaries Game Theory Fundamentals Interpretation Games IBR Interpreting BiOT

Conclusions

  • exact interpretation of biot still open
  • ibr and rl come close
  • upshot of comparison
  • biot is very top-level
  • more concrete reasoning/evolution schemes show limitations

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References

Blutner, Reinhard and Henk Zeevat (2008). “Optimality-Theoretic Pragmatics”. To appear in: Claudia Maienborn, Klaus von Heusinger and Paul Portner (eds.) Semantics: An International Handbook of Natural Language Meaning. B¨

  • rgers, Tilman and Rajiv Sarin (1997). “Learning Through

Reinforcement and Replicator Dynamics”. In: Journal of Economic Theory 77.1, pp. 1–14. Dekker, Paul and Robert van Rooij (2000). “Bi-Directional Optimality Theory: An Application of Game Theory”. In: Journal of Semantics 17, pp. 217–242. Franke, Michael and Gerhard J¨ ager (2011). “Bidirectional Optimization from Reasoning and Learning in Games”. to appear in Journal of Logic, Language and Information. Grafen, Alan (1990). “Biological Signals as Handicaps”. In: Journal of Theoretical Biology 144, pp. 517–546. Hendriks, Petra et al. (2010). Conflicts in Interpretation. London: Equinox Publishing.

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Horn, Laurence R. (1984). “Towards a New Taxonomy for Pragmatic Inference: Q-based and R-based Implicature”. In: Meaning, Form, and Use in Context. Ed. by Deborah Shiffrin. Washington: Georgetown University Press, pp. 11–42. Levinson, Stephen C. (2000). Presumptive Meanings. The Theory of Generalized Conversational Implicature. Cambridge, Massachusetts: MIT Press. Lewis, David (1969). Convention. A Philosophical Study. Harvard University Press. Parikh, Prashant (1991). “Communication and Strategic Inference”. In: Linguistics and Philosophy 473–514.14, p. 3. Skyrms, Brian (2010). Signals. Oxford University Press. Sobel, Joel (2008). “Signaling Games”. In: Encyclopedia of Complexity and Systems Science. Ed. by M. Sotomayor. Springer Verlag. Spence, Andrew Michael (1973). “Job market signaling”. In: Quarterly Journal of Economics 87, pp. 355–374.