models of language evolution
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Models of Language EvolutionSession 07: Evolution of Semantic Meaning 1
Michael Franke
Seminar fur SprachwissenschaftEberhard Karls Universitat Tubingen
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Course Overview (old)
date content
20-4 MoLE: Aims & Challenges
27-4 Evolutionary Game Theory 1: Statics04-5 Evolutionary Game Theory 2: Macro-Dynamics11-5 Guest Lecture by Gerhard Jager18-5 egt 3: Micro-Dynamics & Multi-Agent Systems
25-5 Communication, Cooperation & Relevance01-6 Combinatoriality, Compositionality & Recursion08-6 Evolution of Semantic Meaning & Pragmatic Strategies
15-6 Pentecost — no class
22-6 work on student projects29-6 work on student projects06-7 work on student projects
13-7 presentations
20-7 presentations
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Course Overview (new, still tentative)
date content
20-4 MoLE: Aims & Challenges
27-4 Evolutionary Game Theory 1: Statics04-5 Evolutionary Game Theory 2: Macro-Dynamics11-5 Guest Lecture by Gerhard Jager18-5 egt 3: Micro-Dynamics & Multi-Agent Systems
25-5 Evolution of Semantic Meaning01-6 Semantic Meaning & Conceptual Space08-6 Evolution of Pragmatic Strategies
15-6 Pentecost — no class
22-6 Combinatoriality, Compositionality & Recursion29-6 work on student projects06-7 work on student projects
13-7 presentations
20-7 presentations
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
How Do Linguistic Items Get Their Meaning?
• magic
• causation
• baptism
• convention
• . . .
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Paradox of Conventionalist Accounts of Meaning
“We can hardly suppose a parliament of hitherto speechless eldersmeeting together and agreeing to call a cow a cow and a wolf a wolf.The association of words with their meanings must have grown upby some natural process, though at present the nature of the process isunknown.” (Russell, 1921, p. 190)
Bertrand Russell (1921). The Analysis of Mind. Unwin Brothers Ltd.
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Behavioral Emergence of Meaning (Lewis, 1969)
• signaling games = coordination problems with informationasymmetry
• strict Nash equilibrium = evolutionary stable strategy (ess)
• meaning of signals emerges from the behavior of agents in aness of certain signaling games
(cf. Nowak and Krakauer, 1999; Skyrms, 2010)
David Lewis (1969). Convention. A Philosophical Study. Harvard University PressMartin A. Nowak and David C. Krakauer (1999). “The Evolution of Language”. In:PNAS 96, pp. 8028–8033
Brian Skyrms (2010). Signals. Oxford University Press
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Definition (Signaling Game)
A signaling game is a tuple
sg = 〈{S, R} , T, Pr, M, A, US, UR〉with:
{S, R} set of players
T set of states
Pr prior beliefs: Pr ∈ ∆(T)
M set of messages
A set of receiver actions
US,R utility functions: T×M×A→ R .
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Further Properties
• priors are uniform iff Pr(t) = Pr(t′) for all t, t′ ∈ T
• sg is a cheap talk game iff US,R do not depend on messages;otherwise we speak of costly signaling
• sg is a cooperative game if (but not only if) US = UR
• sg is an interpretation game if T = A
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Types of sgs
• sgs for type-matching:• cooperative interpretation game• U(t, a) = 1 if t = a; 0 otherwise
• sgs for similarity-maximizing:• cooperative interpretation game• U(t, a) = similarity(t, a)
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Lewis Games
• sgs for type-matching
• uniform priors and cheap talk
Horn Games
• sgs for type-matching
• non-uniform priors and costly signaling
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Example (2-2-2 Lewis Game)
N
S
R
〈1, 1〉
a1
〈0, 0〉
a2
ma
R
〈1, 1〉
a1
〈0, 0〉
a2
mb
t1
.5
S
R
〈0, 0〉
a1
〈1, 1〉
a2
ma
R
〈0, 0〉
a1
〈1, 1〉
a2
mb
t2
.5
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Example (2-2-2 Lewis Game)
• we can represent this game also as a static game
• sender actions:
s1 :ma
mb
t1
t2
s2 :ma
mb
t1
t2
s3 :ma
mb
t1
t2
s4 :ma
mb
t1
t2
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Example (2-2-2 Lewis Game)
• receiver actions:
r1 :a1
a2
ma
mb
r2 :a1
a2
ma
mb
r3 :a1
a2
ma
mb
r4 :a1
a2
ma
mb
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Example (2-2-2 Lewis Game)
r1 r2 r3 r4
s1 1, 1 0, 0 .5, .5 .5, .5s2 0, 0 1, 1 .5, .5 .5, .5s3 .5, .5 .5, .5 .5,.5 .5,.5s4 .5, .5 .5, .5 .5,.5 .5,.5
• 2 strict nes
• 4 non-strict nes
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Nash Equilibria of the 2-2-2 Lewis Game
13
9
5
1
14
10
6
2
15
11
7
3
16
12
8
4
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Example (2-2-2 Horn Game)
N
S
R
〈.9, 1〉
a1
〈−.1, 0〉
a2
ma
R
〈.8, 1〉
a1
〈−.2, 0〉
a2
mb
t1
.75
S
R
〈−.1, 0〉
a1
〈.9, 1〉
a2
ma
R
〈−.2, 0〉
a1
〈.8, 1〉
a2
mb
t2
.25
• non-uniform priors: t1 is three times as likely as t2
• costly signaling: sender pays .1 util for ma and .2 for mb
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Example (2-2-2 Horn Game)
• we can represent this game also as a static game
• sender actions:
s1 :ma
mb
t1
t2
s2 :ma
mb
t1
t2
s3 :ma
mb
t1
t2
s4 :ma
mb
t1
t2
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Example (2-2-2 Horn Game)
• receiver actions:
r1 :a1
a2
ma
mb
r2 :a1
a2
ma
mb
r3 :a1
a2
ma
mb
r4 :a1
a2
ma
mb
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Example (2-2-2 Horn Game)
r1 r2 r3 r4
s1 .875, 1 −.125, 0 .625, 75 .125, .25
s2 −.175, 0 .825, 1 .575, .75 .075, .25
s3 .65, .75 .15, .25 .65, .75 .15, .25
s4 .05, .25 .55, .75 .55, .75 .05, .25
• 2 strict nes
• 1 non-strict ne
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Nash Equilibria of the 2-2-2 Horn Game
13
9
5
1
14
10
6
2
15
11
7
3
16
12
8
4
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Dynamic Solutions: Replicator Dynamics
• recall that for asymmetric games we had:
attractors = strict nes = esss
• what about the basin of attraction?
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Replicator Dynamics: Basin of Attraction
• 2-2-2 Lewis game:• half of all interior points converge to one strict ne the other half to
the other
• 2-2-2 Horn game:• interior points converge to: (NB: mistake in Jager (2004)!)
• Horn: ∼ 52%• Anti-Horn: ∼ 35%• Smolensky: ∼ 12%
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Basin of Attraction: 2-2-2 Horn Game (Macro-Level RD)
Horn
Anti-Horn
Smolensky
Macro
.5
.374
.122
.003
graphics thanks to Roland Muhlenbernd23 / 37
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Basin of Attraction: 2-2-2 Horn Game (Conditional Imitation,Completely Connected Network)
Horn
Anti-Horn
Smolensky
Micro 100
.459
.365
.160
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Basin of Attraction: 2-2-2 Horn Game (Conditional Imitation,Small World Network)
Horn
Anti-Horn
Smolensky
Micro 100 Small World
.501
.410
.065
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Basin of Attraction: 2-2-2 Horn Game (Conditional Imitation,Grid Network)
Horn
Anti-Horn
Smolensky
Micro 100 Grid
.501
.408
.067
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Evolution of “Regional Meaning” (cf. Zollman, 2005)
• agents distributed on a grid structure
• “imitate the best” dynamics (asymmetric!)
Kevin J. S. Zollman (2005). “Talking to Neighbors: The Evolution of Regional
Meaning”. In: Philosophy of Science 72, pp. 69–85
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Evolution of Meaning through Signaling
How plausible is the approach to meaning evolution usingtype-matching signaling games?
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Types of sgs (repeated)
• sgs for type-matching:• cooperative interpretation game• U(t, a) = 1 if t = a; 0 otherwise
• sgs for similarity-maximizing:• cooperative interpretation game• U(t, a) = similarity(t, a)
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Meaning Spaces and Perceptual Similarity (naıve approach)
• T ⊆ Rn (n-dimensional Euclidean space)• t = 〈x1, . . . , xn〉 describes a perceptual property t along n
dimensions• e.g.: t =
⟨x1, x2, x3
⟩∈ R3 could be a representation of a color in
terms of its hue (x1), saturation (x2) and lightness (x3)
• for t1 = 〈x1, . . . , xn〉 and t2 = 〈y1, . . . , yn〉 define:• Euclidean distance:
dist(t1, t2) =
√n
∑i=1
(xi − yi)2
• perceived similarity: (Gaussian function of distance)
similarity(t1, t2) = exp(−dist(t1, t2)2
σ2
)
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Example: Similarity Maximizing Game (Jager and van Rooij, 2007)
• T ⊂ N2 (finite approximation of infinite space)
• uniform priors
• small number of messages (e.g. |M| = 3)
• T = A
• US,R(t1, t2) = similarity(t1, t2)
Gerhard Jager and Robert van Rooij (2007). “Language Structure: Psychological and
Social Constraints”. In: Synthese 159.1, pp. 99–130
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Example: Similarity Maximizing Game (Jager and van Rooij, 2007)
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Definition (Behavioral Strategies)
Behavioral strategies are functions that map choice points toprobability distributions over actions available in that choice point.
• behavioral sender strategy:
σ ∈ S = (∆(M))T .
• behavioral receiver strategy:
ρ ∈ R = (∆(A))M .
Example
ρ =
ma 7→
[a1 7→ .35
a2 7→ .65
]
mb 7→[
a1 7→ 0
a2 7→ 1
]
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Evolved Meaning
A pair 〈σ, ρ〉 fixes the way sender and receiver use and interpretsignals. This defines the meaning of signals that arises from signaluse as follows:
• descriptive meaning of m as constituted by σ:
Fσ(m) = {t ∈ T | σ(m|t) 6= 0}• imperative meaning of m as constituted by ρ:
Fρ(m) = {a ∈ A | ρ(a|m) 6= 0}We are particularly interested in the meaning constituted bybehavior that is an ess.
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Results of Simulation (suggestive!)
• descriptive meaning in esss:• convex regions of state space (≈ natural concepts)
• imperative meaning in esss:• center of gravity of region (≈ prototypes)
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Signaling & Meaning Evolution Type-Matching Similarity-Maximizing Homework & Stuff
Homework
• look at and play with scripts:1 imitate best asym.py
2 sim max signaling.py
• try to change the parameter settings for sim max signaling.py
(priors, utilities etc.)
Background Reading Material for this Session
• evolutionary analysis of Lewis games:• Simon M. Huttegger (2007). “Evolution and the Explanation of
Meaning”. In: Philosophy of Science 74, pp. 1–27
• similarity maximizing games:• Gerhard Jager and Robert van Rooij (2007). “Language Structure:
Psychological and Social Constraints”. In: Synthese 159.1,pp. 99–130
(especially: Sec 4.2)36 / 37
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ReferencesHuttegger, Simon M. (2007). “Evolution and the Explanation of
Meaning”. In: Philosophy of Science 74, pp. 1–27.Jager, Gerhard (2004). “Evolutionary Game Theory for Linguists. A
Primer”. Unpublished manuscript, Stanford/Potsdam.Jager, Gerhard and Robert van Rooij (2007). “Language Structure:
Psychological and Social Constraints”. In: Synthese 159.1,pp. 99–130.
Lewis, David (1969). Convention. A Philosophical Study. HarvardUniversity Press.
Nowak, Martin A. and David C. Krakauer (1999). “The Evolution ofLanguage”. In: PNAS 96, pp. 8028–8033.
Russell, Bertrand (1921). The Analysis of Mind. Unwin Brothers Ltd.Skyrms, Brian (2010). Signals. Oxford University Press.Zollman, Kevin J. S. (2005). “Talking to Neighbors: The Evolution of
Regional Meaning”. In: Philosophy of Science 72, pp. 69–85.