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) 4 / 38
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¨ of, 1984; Nordstr¨ om 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 5 / 38
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 6 / 38
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) 7 / 38
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 one 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 10 / 38
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 11 / 38
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 ? 12 / 38
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 13 / 38
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 T 1 , T 2 ∈ Type , then T 1 ⌢ T 2 ∈ Type a : T 1 ⌢ T 2 iff a = x ⌢ y , x : T 1 and y : T 2 2. if T ∈ Type then T + ∈ Type . a : T + iff a = x ⌢ 1 . . . ⌢ x n , n > 0 and for i , 1 ≤ i ≤ n , x i : T . . . 14 / 38
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 1 ¡ 2 ¡ 0 ¡ 3 ¡ 6 ¡ 4 ¡ 5 ¡ 15 / 38
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 1 ¡ 2 ¡ 0 ¡ 3 ¡ 6 ¡ 4 ¡ 5 ¡ (pick up( a , c ) ⌢ attract attention( a , b ) ⌢ throw( a , c ) ⌢ run after( b , c ) ⌢ pick up( b , c ) ⌢ return( b , c , a )) + 15 / 38
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 16 / 38
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 ] 17 / 38
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 � � λ r : agenda=[]:[ RecType ] . � � � � agenda=[ e:pick up( a , c ) ]:[ RecType ] ◮ a non-initial, non-final update function � � � � λ r : agenda=[ e:pick up( a , c ) ]:[ RecType ] � � λ e : e:pick up( a , c ) . � � � � agenda=[ e:attract attention( a , b ) ]:[ RecType ] ◮ a final update function � � � � λ r : agenda=[ e:return( b , c , a ) ]:[ RecType ] � � λ e : e:return( b , c , a ) . � � agenda=[]:[ RecType ] 18 / 38
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) h : Ind c human : human(h) d : Ind λ r ∗ : . c dog : dog(d) s : Ind c stick : stick(s) � � { λ r : agenda=[]:[ RecType ] . � � e:pick up( r ∗ .h, r ∗ .s) � � agenda=[ ]:[ RecType ] , � � � � λ r : agenda=[ e:pick up( r ∗ .h, r ∗ .s) ]:[ RecType ] � � λ e : e:pick up( r ∗ .h, r ∗ .s) . � � � � agenda=[ e:attract attention( r ∗ .h, r ∗ .d) ]:[ RecType ] , . . . , � � e:return( r ∗ .d, r ∗ .s, r ∗ .h) λ e : . � � agenda=[]:[ RecType ] } 19 / 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 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’ 21 / 38
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’ 22 / 38
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