game theoretic pragmatics session 7: introduction ibr-model
Post on 18-Feb-2022
5 Views
Preview:
TRANSCRIPT
Focal Point and Iterated Best Response Vanilla Model Examples
Game Theoretic PragmaticsSession 7: Introduction IBR-Model
<1>
Roland Muhlenbernd, Michael Franke, Jason Quinley
Game Theoretic Pragmatics Session 7: Introduction IBR-Model <1>
Focal Point and Iterated Best Response Vanilla Model Examples
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 ←
Game Theoretic Pragmatics Session 7: Introduction IBR-Model <2>
Focal Point and Iterated Best Response Vanilla Model Examples
Table of Contents
Focal Point and Iterated Best Response
Vanilla Model
Examples
Game Theoretic Pragmatics Session 7: Introduction IBR-Model <3>
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
Game Theoretic Pragmatics Session 7: Introduction IBR-Model <4>
Focal Point and Iterated Best Response Vanilla Model Examples
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
Game Theoretic Pragmatics Session 7: Introduction IBR-Model <5>
Focal Point and Iterated Best Response Vanilla Model Examples
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)
Game Theoretic Pragmatics Session 7: Introduction IBR-Model <6>
Focal Point and Iterated Best Response Vanilla Model Examples
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
Game Theoretic Pragmatics Session 7: Introduction IBR-Model <7>
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
Game Theoretic Pragmatics Session 7: Introduction IBR-Model <8>
Focal Point and Iterated Best Response Vanilla Model Examples
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
Game Theoretic Pragmatics Session 7: Introduction IBR-Model <9>
Focal Point and Iterated Best Response Vanilla Model Examples
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
Game Theoretic Pragmatics Session 7: Introduction IBR-Model <10>
Focal Point and Iterated Best Response Vanilla Model Examples
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
Game Theoretic Pragmatics Session 7: Introduction IBR-Model <11>
Focal Point and Iterated Best Response Vanilla Model Examples
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
. . .
Game Theoretic Pragmatics Session 7: Introduction IBR-Model <12>
Focal Point and Iterated Best Response Vanilla Model Examples
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〉
Game Theoretic Pragmatics Session 7: Introduction IBR-Model <13>
Focal Point and Iterated Best Response Vanilla Model Examples
Excursion: IBR-lite
IBR-process with starter S0
S0 =
{t∃¬∀ → msome
t∀ → mall ,msome
}R∗1 =
{msome → t∃¬∀mall → t∀
}S∗2 =
{t∃¬∀ → msome
t∀ → mall
}R3 = R1
S4 = S2
IBR-process with starter R0
R0 =
{msome → t∃¬∀, t∀mall → t∀
}S∗1 =
{t∃¬∀ → msome
t∀ → mall
}R∗2 =
{msome → t∃¬∀mall → t∀
}S3 = S1
R4 = R2
Game Theoretic Pragmatics Session 7: Introduction IBR-Model <14>
Focal Point and Iterated Best Response Vanilla Model Examples
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
Game Theoretic Pragmatics Session 7: Introduction IBR-Model <15>
Focal Point and Iterated Best Response Vanilla Model Examples
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
Game Theoretic Pragmatics Session 7: Introduction IBR-Model <16>
Focal Point and Iterated Best Response Vanilla Model Examples
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)
Game Theoretic Pragmatics Session 7: Introduction IBR-Model <17>
Focal Point and Iterated Best Response Vanilla Model Examples
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
Game Theoretic Pragmatics Session 7: Introduction IBR-Model <18>
Focal Point and Iterated Best Response Vanilla Model Examples
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
}
Game Theoretic Pragmatics Session 7: Introduction IBR-Model <19>
Focal Point and Iterated Best Response Vanilla Model Examples
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
}
Game Theoretic Pragmatics Session 7: Introduction IBR-Model <20>
Focal Point and Iterated Best Response Vanilla Model Examples
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
}
Game Theoretic Pragmatics Session 7: Introduction IBR-Model <21>
Focal Point and Iterated Best Response Vanilla Model Examples
IBR model for pragmatics
1. There are focal points in the presentation of the game thatattract the attention of reasoners before they engage infurther strategic deliberation
I In signaling games these focal points are Semanticmeaning/Literal interpretation
2. Starting from this initial focal prejudice of attention, playersuse iterated best response reasoning at different levels ofsophistication
I IBR reaches a fixed point for models with aligned preferencesI Strategies at a fixed point represent pragmatic
meaning/usage/interpretation
Game Theoretic Pragmatics Session 7: Introduction IBR-Model <22>
Focal Point and Iterated Best Response Vanilla Model Examples
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 <23>
top related