game theoretic pragmatics session 7: introduction ibr-model

Focal Point and Iterated Best Response Vanilla Model Examples

Game Theoretic PragmaticsSession 7: Introduction IBR-Model

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Roland Muhlenbernd, Michael Franke, Jason Quinley

Game Theoretic Pragmatics Session 7: Introduction IBR-Model <1>


Introduction

1. Status Quo: Basic knowledge of game theoretic toolsI to model language use (signaling games)I to analyse emerging phenomena (Solution concepts: Nash

Equilibrium, Iterated Strict Dominance, Rationalizability)

2. But now?I How do I know, how a model for a particular Implicature

should look like? (what kind of parameters?)I Which solution concept is appropriate for Implicatures?

3. Outstanding Work:I More formal concepts of Implic.: Neo-Gricean Pragmatics

√

I Solution concept with a adequate epistemic interpretation,which at best gives the right results: IBR-Model ←



Table of Contents

Focal Point and Iterated Best Response

Vanilla Model

Examples



IBR model with focal starting points

1. There are focal points in the presentation of the game thatattract the attention of reasoners before they engage infurther strategic deliberation

2. Starting from this initial focal prejudice of attention, playersuse iterated best response reasoning at different levels ofsophistication



Hide-and-Seek Game

A B A ADoor 1 Door 2 Door 3 Door 4

Door 1 Door 2 Door 3 Door 4

Door 1 0,1 1,0 1,0 1,0Door 2 1,0 0,1 1,0 1,0Door 3 1,0 1,0 0,1 1,0Door 4 1,0 1,0 1,0 0,1

Tabelle: Parameters of the Hide-and-Seek game



Hide-and-Seek Game

A B A ADoor 1 Door 2 Door 3 Door 4

A B A A

Hider 9% 36% 40% 15%Seeker 13% 31% 45% 11%

Tabelle: Experimental results (Rubinstein et al. 1996)

I Unique mixed Nash Equilibrium: Random choose with 1/4

I Results depict non-neutral psychological framing effectsI Results can be best explained by a Iterated best response

model with focal starting points (Crawford and Iriberri 2007)



Focal meaning assumption

I Semantic meaning is a psychological attraction point ofparticipants’ attention

I Semantic meaning is not binding, but it’s fairly intuitive tostart pondering how to use or interpret an expression byassessing it

I Semantic meaning is focal and therefore a plausible startingpoint of IBR reasoning to find a rational pragmatic usage orinterpretation of an expression



IBR model with focal starting points





Iterated best response reasoningTheory of Mind reasoning:

I A level-0 player is a naive rational player

I A level-k player is a rational player and believes that heropponent is a level-(k − 1) player

Player 1

〈3, 2〉

Player 2

〈1, 4〉

Player 1

〈2, 6〉

〈4, 3〉c c c

q q q

Results:

I Player 1 as level-1 player would play c

I Player 1 as level-2 player would play q



ConclusionPlayers behaviour in a signaling game:

I a level-0 player does not engage in strategic reasoning

I she only takes into account the semantic meaning of messages

I a level-k player believes that his opponent is a level-(k − 1)player and will play a best response to his belief

Benchmarks of an IBR-process:

I Semantic meaning as focal point is realized by starting withsender or receiver as level-0 player

I Strategic rational behaviour is realized by a step-by-step ToMreasoning

I Pragmatic meaning should be depicted by a resulting strategyof level-n? player



Definition of a signaling game

I 〈{S ,R},T ,Pr ,M, J·K,A,US ,UR〉I sets of states T , of messages M, of actions A

I probability function Pr and semantic meaning J·KI utility functions US,R : T ×M × A→ RI Sk is used ambiguously as

1. a sender of strategic level k as an abstract entity2. the set of pure strategies representing the belief of Rk+1

I Rk is used ambiguously as

1. a receiver of strategic level k as an abstract entity2. the set of pure strategies representing the belief of Sk+1



IBR-SequenceLevel-0 playersI S0 sends any true message: S0 = {s ∈ S |∀t ∈ T : t ∈ Js(t)K}I R0 interprets literally: R0 = BR(Pr(·|JmK))

Level-k + 1 playersI Sk+1 = BR(Rk)I Rk+1 = BR(ΠRk+1

) with ΠRk+1= 〈Pr ,Sk , µ〉

S0

send any true message

R1

best response to S0

S2

best response to R1

. . .

R0

interprets messages literally

S1

best response to R0

R2

best response to S1

. . .



Excursion: IBR-liteAn IBR-lite system is an equivalent reformulation of the previousIBR system, if the game model satisfied the following conditions:

1. T = A2. US,R(t,m, a) = 1 if t = a; 0 else3. Pr(t) = Pr(t ′) for all t, t ′

4. JmK 6= � for all m5. JtK−1 6= � for all t

Example for IBR-lite: Scalar Implicature

Pr(t) t∀ t∃¬∀ mall msome

t∀ 1/2 1,1 0,0√ √

t∃∀ 1/2 0,0 1,1 −√

Tabelle: Parameters of a signaling game for 〈all , some〉



Excursion: IBR-lite

IBR-process with starter S0

S0 =

{t∃¬∀ → msome

t∀ → mall ,msome

}R∗1 =

{msome → t∃¬∀mall → t∀

}S∗2 =


t∀ → mall

}R3 = R1

S4 = S2

IBR-process with starter R0

R0 =

{msome → t∃¬∀, t∀mall → t∀

}S∗1 =


t∀ → mall

}R∗2 =

{msome → t∃¬∀mall → t∀

}S3 = S1

R4 = R2



Excursion: IBR-litePragmatic phenomenon: Free choice Disjunction

Example:

You may take an apple or a pear. ♦(A ∨ B)# You may take an apple and you may take a pear. ♦A ∧ ♦B

Expression alternatives

You may take an apple. ♦AYou may take a pear. ♦B

Pr(t) tA tB tAB m♦A m♦B m♦(A∨B)

tA 1/3 1,1 0,0 0,0√

−√

tB 1/3 0,0 1,1 0,0 −√ √

tAB 1/3 0,0 0,0 1,1√ √ √

Tabelle: Parameters of a signaling game for free choice disjunction



Excursion: IBR-lite

IBR-process with starter S0

S0 =

tA → m♦A, m♦(A∨B)tB → m♦B , m♦(A∨B)tAB → m♦A, m♦B , m♦(A∨B)

R1 =

m♦A → tAm♦B → tBm♦(A∨B) → tA, tB

S2 =

tA → m♦AtB → m♦BtAB → m♦A, m♦B , m♦(A∨B)

R∗3 =

m♦A → tAm♦B → tBm♦(A∨B) → tAB

S∗4 =

tA → m♦AtB → m♦BtAB → m♦(A∨B)

IBR-process with starter R0

R0 =

m♦A → tA, tABm♦B → tB , tABm♦(A∨B) → tA, tB , tAB

S1 =

tA → m♦AtB → m♦BtAB → m♦A, m♦B

R2 =

m♦A → tAm♦B → tBm♦(A∨B) → tA, tB , tAB

S∗3 =

tA → m♦AtB → m♦BtAB → m♦(A∨B)

R∗4 =

m♦A → tAm♦B → tBm♦(A∨B) → tAB



Unexpected messages

Example:

. . .

S1 =

tA → m♦AtB → m♦BtAB → m♦A, m♦B

R2 =

m♦A → tAm♦B → tBm♦(A∨B) → tA, tB , tAB

. . .

m♦A m♦B m♦(A∨B)

tA√

−√

tB −√ √

tAB√ √ √

Zero-Order rationalizable actionsA∗(m) = {a ∈ A|∃µ ∈ (∆(T ))Ma ∈ arg maxa′∈A EUR(a′,m, µ)}

Handling surprise messages

Rk+1(m) = A∗(m)



Stable strategy

I for finite sets T and M the IBR sequence will cycle

I any strategy that occurs in a cycle is repeated infinitely manytimes

Infinitely repeated strategies

S∗ = {s ∈ S |∀i∃j > i : s ∈ Sj}R∗ = {r ∈ R|∀i∃j > i : r ∈ Rj}

I The tuple 〈S∗,R∗〉 is the IBR model’s idealized solution

I 〈S∗,R∗〉 is a fixed point, if the cycle has length 1



Example: Division of pragmatic labour

Pr(t) ap ar mu mm

tp 3/4 1,1 0,0√ √

tr 1/4 0,0 1,1√ √

.1 .2

S0 =

{tp → mu,mm

tr → mu,mm

}R1 =

{mu → ap

mm → ap

}S2 =

{tp → mu

tr → mu

}R3 =

{mu → ap

mm → ap, ar

}S4∗ =

{tp → mu

tr → mm

}R5∗ =

{mu → ap

mm → ar

}



Example: Division of pragmatic labour

Pr(t) ap ar mu mm

tp 3/4 1,1 0,0√ √

tr 1/4 0,0 1,1√ √

.1 .2

R0 =

{mu → ap

mm → ap

}S1 =

{tp → mu

tr → mu

}R2 =

{mu → ap

mm → ap, ar

}S3∗ =

{tp → mu

tr → mm

}R4∗ =

{mu → ap

mm → ar

}



Example: I-Heuristic

Pr(t) ac ag mc mg mm

tc 3/4 1,1 0,0√

−√

tg 1/4 0,0 1,1 −√ √

.2 .2 .1

S0 =

{tc → mc ,mm

tg → mg ,mm

}

R1∗ =

mc → ac

mg → ag

mm → ac

S2∗ =

{tc → mm

tg → mg

}

R0∗ =

mc → ac

mg → ag

mm → ac

S1∗ =

{tc → mm

tg → mg

}



IBR model for pragmatics


I In signaling games these focal points are Semanticmeaning/Literal interpretation


I IBR reaches a fixed point for models with aligned preferencesI Strategies at a fixed point represent pragmatic

meaning/usage/interpretation



Homework:I read

I ’script’: PhD of M. Franke Chapter 2.1 & 2.2I Michael Franke (2009). ”Free Choice from Iterated Best

Response”. In: M. Aloni and K. Schulz (Eds.): AmsterdamColloquium 2009, LNAI 6042, pp 295-304.

Next Session:

I IBR-Model Part 2


game theoretic pragmatics session 7: introduction ibr-model

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