How to play games with types Robin Cooper Centre for Linguistic - - PowerPoint PPT Presentation

How to play games with types

Robin Cooper Centre for Linguistic Theory and Studies in Probability (CLASP) Department of Philosophy, Linguistics and Theory of Science (FLoV) Supported in part by VR project 2016-01162, Incremental Reasoning in Dialogue (IncReD).

Outline

Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Outline

Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Social meaning games

◮ Social Meaning, Sociolinguistic Variation and Game-Theoretic

Pragmatics — Heather Burnett’s and E. Allyn Smith’s course at ESSLLI 2017 https://www.irit.fr/esslli2017/courses/6

◮ Heather Burnett — Signalling Games, Sociolinguistic

Variation and the Construction of Style (circulated, forthcoming in L&P)

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

TTR — a type theory with records

◮ originally inspired by constructive type theory (Martin-L¨

• f,

1984; Nordstr¨

• m et al., 1990)

◮ a rich type theory, as opposed to simple type theory used by

Montague

◮ judgement that an object a is of a type T, a : T ◮ judgements as type acts ◮ among our types we include types of events – more generally,

situations (Ranta, 1994)

◮ Some references: Cooper (2005a,b, 2012); Cooper and

Ginzburg (2015), an partial book draft on https://sites. google.com/site/typetheorywithrecords/drafts, general TTR references on https://sites.google.com/ site/typetheorywithrecords/publications

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

KoS

◮ theory of dialogue in terms of information state update

(Ginzburg, 2012 and much else)

◮ uses TTR

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Why might it be useful to put GT together with KoS-TTR?

◮ Both talk about types of action ◮ In GT “possible worlds” refers to what is called “type of

situation” in TTR as opposed to the total logically possible worlds used in traditional linguistic semantics deriving from modal and intensional logic.

◮ cf. “possible worlds” in probability theory (Lappin, 2012,

2015; Cooper et al., 2015)

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Possible advantages

◮ KoS-TTR might provide:

◮ a framework for choosing which games to play ◮ an account of misunderstandings about which game is being

played

◮ accommodation of games on the basis of interlocutor’s

behaviour

◮ explain how a single action can represent a move in more than

• ne game — What’s cookin’?

◮ GT might provide:

◮ a theory of strategy in non-deterministic games ◮ an account of variation in probabilistic terms ◮ a variety of overall interactive strategies: ◮ male rationalism – maximize own utility ◮ collaborative – maximize utility (regardless of whose) ◮ altruistic – maximize other’s utility 8 / 38

Outline

Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Games in TTR

◮ Cooper (in prep), Ch. 1 (discussed here) ◮ Breitholtz (2014) in relation to enthymematic reasoning ◮ related to Ginzburg on genre and conversation types

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Fetch – a game of interaction and coordination

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Query – is this the beginning of an event of type FetchGame?

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Creation – the dog must predict and carry out its contribution to an event of type FetchGame

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

String types

• cf. work by Tim Fernando, e.g. Fernando (2015)
• 1. if T1, T2 ∈ Type, then T1⌢T2 ∈ Type

a : T1⌢T2 iff a = x⌢y, x : T1 and y : T2

• 2. if T ∈ Type then T + ∈ Type.

a : T + iff a = x⌢

1 . . .⌢xn, n > 0 and for i, 1 ≤ i ≤ n, xi : T

. . .

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

A game of fetch

0 ¡ 1 ¡ 2 ¡ 3 ¡ 4 ¡ 5 ¡ 6 ¡

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

A game of fetch

0 ¡ 1 ¡ 2 ¡ 3 ¡ 4 ¡ 5 ¡ 6 ¡

(pick up(a,c)⌢attract attention(a,b)⌢throw(a,c)⌢run after(b,c)⌢ pick up(b,c)⌢return(b,c,a))+

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Information states and gameboards

◮ Information states (gameboards) are used by agents to keep

track of where they are in the creation of an event belonging to a certain type

◮ each agent has their own view of the state of the game ◮ plays an essential role in coordination ◮ information state (Larsson, 2002) and gameboard (Ginzburg,

1994, 2012, originally Lewis, 1979) are adopted from the literature on dialogue

◮ we shall model information states as records and use

‘gameboard’ to refer to types of information states

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

The types InfoState and InitInfoState

InfoState

• agenda

: [RecType]

• InitInfoState
• agenda=[]

: [RecType]

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Game of fetch (human, a, dog, b, and stick, c)

◮ game as a set of update functions corresponding to transitions

in a finite state automaton

◮ an initial update function

λr:

• agenda=[]:[RecType]
• .
• agenda=[
• e:pick up(a,c)
• ]:[RecType]
• ◮ a non-initial, non-final update function

λr:

• agenda=[
• e:pick up(a,c)
• ]:[RecType]
• λe:
• e:pick up(a,c)
• .
• agenda=[
• e:attract attention(a,b)
• ]:[RecType]
• ◮ a final update function

λr:

• agenda=[
• e:return(b,c,a)
• ]:[RecType]
• λe:
• e:return(b,c,a)
• .
• agenda=[]:[RecType]
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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Game of fetch (with roles abstracted)

λr∗:         h : Ind chuman : human(h) d : Ind cdog : dog(d) s : Ind cstick : stick(s)         . { λr:

• agenda=[]:[RecType]
• .
• agenda=[
• e:pick up(r∗.h,r∗.s)
• ]:[RecType]
• ,

λr:

• agenda=[
• e:pick up(r∗.h,r∗.s)
• ]:[RecType]
• λe:
• e:pick up(r∗.h,r∗.s)
• .
• agenda=[
• e:attract attention(r∗.h,r∗.d)
• ]:[RecType]
• ,

. . . , λe:

• e:return(r∗.d,r∗.s,r∗.h)
• .
• agenda=[]:[RecType]
• }

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Outline

Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Obama

◮ Use of -ing/-in’ verbal morphology (Labov, 2012, p. 22, cited

by Burnett and Smith)

◮ at a barbeque — 72% -in’ ◮ meeting press after barbecue — 33% -in’ ◮ acceptance speech at Democratic National Convention — 3%

• in’

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Social meaning

◮ -in’ — less educated, lower class ◮ -ing — more educated, higher class ◮ -in’ indicates ‘friendly’, but also possibly ‘incompetent’ ◮ -ing indicates ‘competent’, but also possibly ‘aloof’

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Social meaning games

forthcoming work by Burnett

Definition 4.1. A Social Meaning Game is a tuple {S, L}, P, >, M, C, [·], µ, Pr where:

• 1. S and L are the players.
• 2. P, > is the universe (a relational structure), where
• P = {p1, . . . , pn} is a finite set of properties.
• > is a relation on P that is irreflexive.
• 3. M is a finite set of messages.
• 4. C is a measure function on M describing the cost of each message.
• 5. [·] is the indexation relation (to be described below).
• 6. µ is a measure function on sets of properties describing S’s values in the context.
• 7. Pr is a probability distribution over sets of properties describing L’s prior beliefs

about S.

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Social meaning games

forthcoming work by Burnett

Definition 4.1. A Social Meaning Game is a tuple {S, L}, P, >, M, C, [·], µ, Pr where:

• 1. S and L are the players.
• 2. P, > is the universe (a relational structure), where
• P = {p1, . . . , pn} is a finite set of properties.
• > is a relation on P that is irreflexive.
• 3. M is a finite set of messages.
• 4. C is a measure function on M describing the cost of each message.
• 5. [·] is the indexation relation (to be described below).
• 6. µ is a measure function on sets of properties describing S’s values in the context.
• 7. Pr is a probability distribution over sets of properties describing L’s prior beliefs

about S.

TTR properties (dependent types) friendly, aloof, competent, incompetent 23 / 38

Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Social meaning games

forthcoming work by Burnett

Definition 4.1. A Social Meaning Game is a tuple {S, L}, P, >, M, C, [·], µ, Pr where:

• 1. S and L are the players.
• 2. P, > is the universe (a relational structure), where
• P = {p1, . . . , pn} is a finite set of properties.
• > is a relation on P that is irreflexive.
• 3. M is a finite set of messages.
• 4. C is a measure function on M describing the cost of each message.
• 5. [·] is the indexation relation (to be described below).
• 6. µ is a measure function on sets of properties describing S’s values in the context.
• 7. Pr is a probability distribution over sets of properties describing L’s prior beliefs

about S.

TTR properties (dependent types) friendly, aloof, competent, incompetent preclude relation on types friendly | aloof, competent | incompetent 23 / 38

Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Social meaning games

forthcoming work by Burnett

Definition 4.1. A Social Meaning Game is a tuple {S, L}, P, >, M, C, [·], µ, Pr where:

• 1. S and L are the players.
• 2. P, > is the universe (a relational structure), where
• P = {p1, . . . , pn} is a finite set of properties.
• > is a relation on P that is irreflexive.
• 3. M is a finite set of messages.
• 4. C is a measure function on M describing the cost of each message.
• 5. [·] is the indexation relation (to be described below).
• 6. µ is a measure function on sets of properties describing S’s values in the context.
• 7. Pr is a probability distribution over sets of properties describing L’s prior beliefs

about S.

TTR properties (dependent types) friendly, aloof, competent, incompetent preclude relation on types friendly | aloof, competent | incompetent utterance types Ing, In’ 23 / 38

Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Integrating with an information state update game

Domain for ING-game, INGdom:                    sp:Ind au:Ind props=     comp =competent(⇑sp) incomp=incompetent(⇑sp) fr =friendly(⇑sp) aloof =aloof(⇑sp)    :Rec msgs= ing=Ing in’ =In’

• :Rec

index=   types=⇑msgs mngs= ing=(⇑2props.comp∨⇑2props.aloof) in’ =(⇑2props.fr∨⇑2props.incomp)

• 

:Rec                   

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Integrating with an information state update game

Towards update rules: λr∗:INGdom . { λr:       private:   agenda:list(RecType) relevant beliefs: max:MaxCons(r∗.props) beliefs:ProbDis(max)

• 

 shared:

• latest-utterance:

sp=r∗.sp:Ind u:In’

• 

     .   relevant beliefs=conditionalize(r.relevant beliefs.bels, r∗.index.mngs.in’): ProbDis(r.private.relevant beliefs.max)  , . . . }

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Integrating with an information state update game

Towards update rules: λr∗:INGdom . { λr:       private:   agenda:list(RecType) relevant beliefs: max:MaxCons(r∗.props) beliefs:ProbDis(max)

• 

 shared:

• latest-utterance:

sp=r∗.sp:Ind u:In’

• 

     .   relevant beliefs=conditionalize(r.relevant beliefs.bels, r∗.index.mngs.in’): ProbDis(r.private.relevant beliefs.max)  , . . . }

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Beliefs as probability distributions

◮ Burnett’s idea: your beliefs are a probability distribution over

the set of propositions which are maximal and consistent with respect to the relevant propositions for the game

◮ Current work by Shalom Lappin suggests that beliefs are

probability distributions

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Maximal consistent sets

Let r∗ be     comp =competent(a) incomp=incompetent(a) fr =friendly(a) aloof =aloof(a)     Since competent(a)⊥incompetent(a) and friendly(a)⊥aloof(a), r : MaxCons(r∗) iff

• 1. r is simple (flat)
• 2. the multiset extension of r is

{ r∗.competent∧r∗.fr, r∗.competent∧r∗.aloof, r∗.incomp∧r∗.fr, r∗.incomp∧r∗.aloof }

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Sample witness for MaxCons(r ∗)

    p0 = r∗.competent∧r∗.fr p1 = r∗.competent∧r∗.aloof p2 = r∗.incomp∧r∗.fr p3 = r∗.incomp∧r∗.aloof    

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Probability distributions

Let r∗ be   ℓ0 = a0 . . . ℓn = an   r : ProbDis(r∗) iff

• 1. r :

   

• bjs=r∗

: Rec probs :   ℓ0 : Real . . . ℓn : Real       2.

• ℓ∈labels(r∗)

r.probs.ℓ = 1

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Outline

Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Two possible roles for probability

◮ beliefs as probability distributions over types (as we have seen) ◮ non-deterministic update functions as returning probability

distributions over potential updates

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Deterministic update functions

tacit λr :“current info state” . “type to update with” event driven λr :“current info state” . λe:“event” . “type to update with”

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Nondeterministic update functions

tacit λr :“current info state” . “probability distribution over types to update with” event driven λr :“current info state” . λe:“event” . “probability distribution over types to update with”

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Playing several games simultaneously

◮ Obama says, “What’s cookin’? ” ◮ Playing two games simultaneously: Question-Answer game

and ING game

◮ His utterance is of several types, including Question and In’ ◮ Our game involves message types rather than messages ◮ The update rule we gave was tacit, i.e. it did not require new

event input

◮ As a consequence it is possible to play both game

simultaneously

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Games on the gameboard

◮ Games should be on the gameboard (as suggested by

Breitholtz, 2014 )

◮ Facilitates choice of games, mismatch of games between

dialogue participants, accommodation

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Allowing things to happen – licensing conditions

◮ If A is an agent, si is A’s current information state,

si :A

• agenda=T|R

: [RecType]

• , then :A T! is

licensed.

◮ If f : (T1 → (T2 → Type)) is an update function, A is an

agent, si is A’s current information state, si :A Ti, Ti ⊑ T1 (and si : T1), then an event e :A T2 (and e : T2) licenses si+1 :A f (si)(e).

◮ If f : (T1 → Type) is an update function, A is an agent, si is

A’s current information state, si :A Ti, Ti ⊑ T1 (and si : T1), si+1 :A f (si) is licensed.

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Adding GT to licensing conditions

◮ We currently talk of “licensing” ◮ This could be refined and strategy for choice could be made

explicit by the addition of GT

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Adding game theory to KoS-TTR Games in TTR Obama, social meaning and information state update Towards a theory of action

Conclusions

◮ we have suggested a project and some preliminary ideas ◮ combine GT and KoS-TTR ◮ one way of putting probability and strategy into our work on

dialogue

◮ a way of relating GT to work on information state update in

dialogue

◮ potential advantages:

◮ dialogue strategies like accommodation and repair may involve

choice of games

◮ strategies for playing non-deterministic games 38 / 38

References

Bibliography I

Breitholtz, Ellen (2014) Enthymemes in Dialogue: A mico-rhetorical approach, PhD dissertation, University of Gothenburg. Cooper, Robin (2005a) Austinian truth, attitudes and type theory, Research on Language and Computation, Vol. 3, pp. 333–362. Cooper, Robin (2005b) Records and Record Types in Semantic Theory, Journal of Logic and Computation, Vol. 15, No. 2, pp. 99–112. Cooper, Robin (2012) Type Theory and Semantics in Flux, in R. Kempson, N. Asher and T. Fernando (eds.), Handbook of the Philosophy of Science, Vol. 14: Philosophy of Linguistics, pp. 271–323, Elsevier BV. General editors: Dov M. Gabbay, Paul Thagard and John Woods.

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References

Bibliography II

Cooper, Robin (in prep) Type theory and language: from perception to linguistic communication. Draft of book chapters available from https://sites.google.com/site/ typetheorywithrecords/drafts. Cooper, Robin, Simon Dobnik, Shalom Lappin and Staffan Larsson (2015) Probabilistic Type Theory and Natural Language Semantics, Linguistic Issues in Language Technology, Vol. 10,

• No. 4, pp. 1–45.

Cooper, Robin and Jonathan Ginzburg (2015) Type Theory with Records for Natural Language Semantics, in Lappin and Fox (2015), pp. 375–407. Fernando, Tim (2015) The Semantics of Tense and Aspect: A Finite-State Perspective, in Lappin and Fox (2015).

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References

Bibliography III

Ginzburg, Jonathan (1994) An update semantics for dialogue, in

• H. Bunt (ed.), Proceedings of the 1st International Workshop on

Computational Semantics, Tilburg University. Ginzburg, Jonathan (2012) The Interactive Stance: Meaning for Conversation, Oxford University Press, Oxford. Labov, William (2012) Dialect diversity in America: The politics of language change, University of Virginia Press. Lappin, Shalom (2012) An operational approach to fine-grained intensionality, in T. Graf, D. Paperno, A. Szabolcsi and J. Tellings (eds.), Theories of Everything: In Honor of Ed Keenan, UCLA Working Papers in Linguistics 17, Department of Linguistics, UCLA. Lappin, Shalom (2015) Curry Typing, Polymorphism, and Fine-Grained Intensionality, in Lappin and Fox (2015).

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References

Bibliography IV

Lappin, Shalom and Chris Fox, eds. (2015) The Handbook of Contemporary Semantic Theory, second edition, Wiley-Blackwell. Larsson, Staffan (2002) Issue-based Dialogue Management, PhD dissertation, University of Gothenburg. Lewis, David (1979) Scorekeeping in a Language Game, Journal of Philosophical Logic, Vol. 8, pp. 339–359. Martin-L¨

• f, Per (1984) Intuitionistic Type Theory, Bibliopolis,

Naples. Nordstr¨

• m, Bengt, Kent Petersson and Jan M. Smith (1990)

Programming in Martin-L¨

• f’s Type Theory ( International Series
• f Monographs on Computer Science 7), Clarendon Press,

Oxford. Ranta, Aarne (1994) Type-Theoretical Grammar, Clarendon Press, Oxford.

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