Meaning Games

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Meaning GamesLENLS 20072007-06-18

HASIDA KôitiITRI, AIST

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Introduction to Game Theory

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Rational Behavior of Individual action selection as optimization

problem maxx u(x): maximize utility u(x) by action x

optimization process inference perhaps largely precompiled

rather than real-time selection: evolution, imitation, economic

competition, etc. contextual factors: free will, etc.

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Issues of Social Interaction autonomous optimization of entire society?

(invisible hand) Maximization of individual benefits entails

maximization of the social benefit as their total. rational expectation hypothesis Chicago school ... anti-Keynesian

reactionary conservatisms rejecting reformation and supporting imperial colonization, racial discrimination, eugenics, etc. Social Darwinism

In reality, the benefit of the entire society is often impaired!

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Game Theory mathematical theory about interaction

among multiple players (autonomous agents)

John Von Neumann & Oskar Morgenstern (1944) Minimax principle in zero-sum games cooperative games

John F. Nash (1951) equilibria in n-person non-cooperative games

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Prisoner’s Dilemma The prosecutor separately asks two fellow

prisoners (A and B) to confess. Do they cooperate (keep silence) or betray (confess)?

other examples: arms race, global warming, etc.

cooperate betray

cooperate 2＼2 0＼3

betray 3＼0 1＼1

Nash equilibrium

priso

ner B

prisoner A

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Non-Cooperative Gameand Nash equilibrium A (mixed) strategy is a probability

distribution over possible actions. A Nash equilibrium is a state (combination of

players’ strategies) where each player’s strategy is optimal as to the others’ strategies. maxxi ui(x1,…, xi-1, xi, xi+1,…, xn): Each player i

maximizes her payoff ui(x1, …, xn) by action xi. Nobody wants to change her strategy in an

equilibrum. A Nash equilibrium exists in every n-person

non-cooperative game. (Nash, 1951)

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Individual Optimizationvs. Social Optimization Nash equilibria may not be socially

optimal. Prisoners’ Dilemma

Individuals may optimize without optimizing the society, and vice versa. If one person takes a long way round,

then the others’ travel times may become much shorter, reducing the total travel time.

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Communication and Game

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Signaling GameSender S of type t T sends message m M to

receiver R, who does action a A.

TM

A

job market(Spence, 1973)

mate selection(Zahavi, 1975)

S job seeker male deer

R employer female deer

T competence vitality

M education antler size

A salary mating

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Handicap Principle Individual animals communicate their

greater quality through costly morphology or behavior (Zahavi, 1977).

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Communication and Cooperation Even enemies cooperate in

communication. nonnatural meaning (Grice, 1957) i ≡ S’s intention to convey c to R

S’s intention for R to recognize c and i R wants to recognize c and i even if she

doubts S’s honesty.Both S and R want i to hold.

... cooperation!(S: sender, R: receiver, c: content)

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Efficiency of Language One linguistic expression may have

multiple meanings in multiple contexts. (Barwise & Perry, 1983)

Efficiency presupposes disambiguation. Collaboration for disambiguation?

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Collaborative Disambiguationcf. Parikh (1987, 1990, 1992)

Possible correspondences between pieces of content and messages

c1 c2

m1 m2

Optimal correspondencec1 c2

m1 m2

… content

… messages

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Natural Language Anaphorau1: Max scolded Fred.u2: He (Max) was angry with the man (Fred).u2': ??He (Fred) was angry with the man (Max).

Max Fred

he the manu2 u2'

Max Fred

he the man

Max Fred

he the man

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Centering Theory (1/3)Centering (Joshi & Weinstein, 1981; Brennan, Friedman,

& Pollard 1987; Walker, Iida, Cote, 1994; Joshi & Weinstein, 1995)

ui: i-th utterance unit (clause) Cf(ui): list of forward-looking centers (entities

referenced in ui) ordered in saliencesalience in English：

subject＞direct object ＞indirect object ＞other complements ＞adjuncts

preferred center Cp(ui): first entity of Cf(ui) backward-looking center Cb(ui): most salient entity

in Cf(ui-1) referenced in ui

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Centering Theory (2/3)Rule I: If an entity E in Cf(ui) is referenced by a

pronoun in ui+1, then Cb(ui+1) must also be referenced by a pronoun in ui+1 (E may be Cb(ui+1)).

(Generalization: More salient entities are referenced by simpler expressions.)

ui+1: He was angry with the man.(He = Max, the man = Fred)Cb(ui+1): MaxCf(ui+1): [Max, Fred]


ui+1: He was angry with the man.*(He = Fred, the man = Max)Cb(ui+1): Max (violation of Rule I)Cf(ui+1): [Fred, Max]


ui: Max scolded Fred.Cb(ui): ?Cf(ui): [Max, Fred]


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Centering Theory (3/3)Rule II: CONTINUE > RETAIN >

SMOOTH-SHIFT > ROUGH-SHIFT (Walker, Iida & Cote, 1994).

Cb(ui)＝Cb(ui-1) Cb(ui)≠Cb(ui-1)

Cb(ui)＝Cp(ui) CONTINUE SMOOTH-SHIFT

Cb(ui)≠Cp(ui) RETAIN ROUGH-SHIFT

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ui+2: He invited him to dinner.(He = Max, him = Fred)Cb(ui+2): MaxCf(ui+2): [Max, Fred]

ui+2: He invited him to dinner.(He = Max, him = Fred)Cb(ui+2): MaxCf(ui+2): [Max, Fred]

ui+2: He invited him to dinner.*(He = Fred, him = Max)Cb(ui+2): MaxCf(ui+2): [Fred, Max]

ui+2: He invited him to dinner.*(He = Fred, him = Max)Cb(ui+2): MaxCf(ui+2): [Fred, Max]

ui+1: He is waiting for Fred.(He = Max)Cb(ui+1): MaxCf(ui+1): [Max, Fred]

ui+1: He is waiting for Fred.(He = Max)Cb(ui+1): MaxCf(ui+1): [Max, Fred]

CONTINUE or SMOOTH SHIFT

CONTINUE RETAIN

ui: Who is Max waiting for?Cb(ui): ?Cf(ui): [Max]

ui: Who is Max waiting for?Cb(ui): ?Cf(ui): [Max]

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Drawback of Centering Theory no general principles many different versions no quantitative predictions

Let’s reduce CT to decision-theoretic principles while generalizing it by quantitative (statistical) terms.

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Meaning Game (1/2)(Hasida, et al., 1995; Hasida, 1996)a turn: [cs, m, cr] T Sender S sends message m M to

convey content cs C. Receiver R interprets m as meaning

cr C.

Unlike Prisoner’s Dilemma, meaning games entail cooperation due to nonnatural meaning.

c1 c2

m1 m2

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Meaning Game (2/2) An effective solution of a meaning

game should be a Pareto-optimal ESS(evolutionarily stable strategy).

An equilibrium is Pareto-optimal when no other equilibrium is better for all players.

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Evolutionary Game Theorynarrow sense: game-theoretic analysis of

biological evolution behavioral ecology John Maynard Smith

wide sense: game-theoretic analysis of evolution of social interaction in general

evolutionarily stable strategy (ESS)

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Evolutionarily Stable Strategy (ESS) a strategy x preventing others When most players' strategy is x, any

other strategy y is disadvantageous.u(x, (1 –ε) x +εy) > u(y, (1 –ε) x +εy)

When ε= 0, u(x, x) u(y, x).→ An ESS is a Nash equilibrium.

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Stability of Equilibria (Example) Agreement is more beneficial when the

agreement ratio is high, and opposition is more beneficial when the agreement ratio is low.

Nash equilibria are agreement ratio 0, 1, and a.

0 and 1 are ESS, but a is not.

0 1a

benefit of oppositionbenefit of opposition

benefit of agreementbenefit of agreement

agreement ratio

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Rule I of Centering Theory If an entity E in Cf(ui) is referenced by

a pronoun in ui+1, then Cb(ui+1) must also be referenced by a pronoun in ui+1 (E may be Cb(ui+1)).







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Derivation of Centering Theory (Rule I)(Hasida, et al., 1995)

Max Fred

he the man

probability： P1 > P2

utility： U1 > U2

comparison of expected utilityMax Fred

he the manE1=P1U1+P2U2

Max Fred

he the manE2=P1U2+P2U1

E1-E2=(P1-P2)(U1-U2) > 0Pareto Pareto optimal

common belief

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Generalization of Rule I(Shiramatsu, et al., 2007)

More salient entities are referenced by simpler (higher-utility) expressions.

corpus-based verification estimation of salience (reference

probability) of entity (potential referent) 1/(1 + exp(-(b0+b1dist+b2gram+b3chain)))

estimation of utility of referring expression Const. + log(occurrence probability)

distance from last reference

salience of grammatical

function of last occurrence

length of successive

coreference chain

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Annotated Corpora Mainichi Newspaper GDA Corpus Japanese newspaper 1,356 (out of 3,000) articles, 63,562 clauses,

16,728 anaphoric NPs, etc. GDA (Global Document Annotation) tags for

syntax, semantic role, andcoreference/anaphora.

Wall Street Journal GDA Corpus English newspaper 2,412 articles, 135,278 clauses, 95,677

anaphoric NPs, etc. GDA tags translated from Penn TreeBank, plus

coreference/anaphora.

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Estimated Salience (Reference Probability) of Grammatical Function

(Shiramatsu, et al., 2007)

gram. func. #samples #successivereferences prob. gram. func. #samples #successive

references prob.topicsubj.

objectindirect obj.

subj.

indirect obj.

object

comp.

other postpositionno gram. func.

other prepositionno gram. func.

Mainichi Newspaper Wall Street Journal

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Whether Pairs of Referential NPs Respect the Generalized Rule I (Mainichi Newspaper) (Shiramatsu, et al., 2007)

E1-E2

#pa

irsnegative pairspositive pairs

ratio

of p

ositi

ve p

airs

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Whether Pairs of Referential NPs Respect the Generalized Rule I(Wall Street Journal) (Shiramatsu, et al., 2007)

E1-E2

#pa

irs

ratio

of p

ositi

ve p

airs

negative pairspositive pairs

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Rule II of Centering Theory CONTINUE > RETAIN > SMOOTH-

SHIFT > ROUGH-SHIFT Cb(ui)＝Cb(ui-1) > Cb(ui)≠Cb(ui-1)

V Cb(ui)＝Cp(ui) > Cb(ui)≠Cp(ui)

Cb(ui)＝Cb(ui-1) Cb(ui)≠Cb(ui-1)

Cb(ui)＝Cp(ui) CONTINUE SMOOTH-SHIFT

Cb(ui)≠Cp(ui) RETAIN ROUGH-SHIFT

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Derivation of Rule II (Hasida, et al., 1995) Cb(ui) = Cb(ui-1) raises the reference

probability of Cb(ui) in ui, and hence the utility of ui.

assumption: Successive references raise salience (reference probability or attention).

Cb(ui) = Cp(ui) raises the reference probability of Cb(ui) in ui+1, and hence the expected utility of ui+1.

assumption: Factors contributing to salience are somewhat additive.

Preference for Cb(ui) = Cp(ui) is based on prediction of next utterance, and hence weaker than that for Cb(ui) = Cb(ui-1).

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Statistical Verification of Rule II(Shiramatsu, et al., 2007)

transition type

transition type

#samples

#samples

avg. expected utilities of transition types Mainichi Newspaper

Wall Street Journal

avg. exp. utility

avg. exp. utility

95% confidence intvl.

95% confidence intvl.

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More Complex Issues

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Compound Meaning Game(Hasida, 1996)

sentence-level meaning game, besides NP-level one

Fred insulted Max.He (Max) was angry with the man (Fred).

Angry(Fred,Max) Angry(Max,Fred)probability： P1 < P2

utility： U1 U2

He was angry with the man.

P1U1 < P2U2

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Partial Sharing of Context (1/2)(Hasida, et al., 1996)

probability： P1 P2

John1 John2

`John Black' `John' `John White'utility： U1 < U0 > U2

two Pareto-optimal equilibria:John1 John2

`John Black' `John'

John1 John2

`John' `John White'

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Partial Sharing of Context (2/2) actual solution

John1 John2

`John Black' `John White'

The solution of a natural-language meaning game should better be strictly Pareto-optimal in the maximum set of equilibria strictly Pareto-comparable to each other??

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A Similar GameTwo people are asked to write a number from 1 to 10 without communication. They know that they'll get some money if they happen to write the same number, and that the amount of money is $10 each if the number is from 1 to 9, and $9 each if the number is 10. The Pareto-optimal solutions are those in which they write the same number from 1 to 9, but in reality they'll write 10.

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Concluding Remarks There are no language-specific rules on

reference resolution. Centering Theory is reduced to meaning game

and general cognitive principles of attention. Language use consists of Pareto-optimal

ESSs of potentially very complex meaning games.

Much to do to address this complexity. Game theory introduces statistical/analytic

methods into semantics/pragmatics.

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Selected ReferencesKôiti Hasida, Katashi Nagao, and Takashi Miyata (1995) A Game-

Theoretic Account of Collaboration in Communication. Proceedings of the First International Conference on Multi-Agent Systems, 140-147, San Francisco.

Kôiti Hasida (1996) Issues in Communication Game. Proceedings of the 16th International Conference on Computational Linguistics, 531-536, Copenhagen.

Kôiti Hasida, Jerry R. Hobbs, and Megumi Kameyama (1996) Optimality in Communication Games. Proceedings of the Second International Conference on Multi-Agent Systems, Kyoto.

白松俊・駒谷和範・橋田浩一・尾形哲也・奥乃博 (2007予定) ゲーム理論に基づく参照結束性のモデル化と日本語・英語の大規模コーパスを用いた統計的検証. 自然言語処理. (Shun Shiramatsu, Kazunori Komatani, Kôiti Hasida, Tetsuya Ogata, and Hiroshi Okuno (2007, to appear) A Game-Theoretic Model of Referential Coherence and Its Statistical Verification based on Large Japanese and English Corpora. Natural Language Processing.)

Meaning Games

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