Models of Language Evolution

Models of Language EvolutionSession 07: Evolution of Semantic Meaning 1

Michael Franke

Seminar fur SprachwissenschaftEberhard Karls Universitat Tubingen

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

2 / 37


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

3 / 37


How Do Linguistic Items Get Their Meaning?

• magic

• causation

• baptism

• convention

• . . .

4 / 37


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.

5 / 37


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

6 / 37


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 .

7 / 37


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

8 / 37


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)

9 / 37


Lewis Games

• sgs for type-matching

• uniform priors and cheap talk

Horn Games

• sgs for type-matching

• non-uniform priors and costly signaling

10 / 37


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

11 / 37



• 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

12 / 37



• receiver actions:

r1 :a1

a2

ma

mb

r2 :a1

a2

ma

mb

r3 :a1

a2

ma

mb

r4 :a1

a2

ma

mb

13 / 37



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

14 / 37


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

15 / 37


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

16 / 37



• 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

17 / 37



• receiver actions:

r1 :a1

a2

ma

mb

r2 :a1

a2

ma

mb

r3 :a1

a2

ma

mb

r4 :a1

a2

ma

mb

18 / 37



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

19 / 37


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

20 / 37


Dynamic Solutions: Replicator Dynamics

• recall that for asymmetric games we had:

attractors = strict nes = esss

• what about the basin of attraction?

21 / 37


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%

22 / 37


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


Basin of Attraction: 2-2-2 Horn Game (Conditional Imitation,Completely Connected Network)

Horn

Anti-Horn

Smolensky

Micro 100

.459

.365

.160

24 / 37


Basin of Attraction: 2-2-2 Horn Game (Conditional Imitation,Small World Network)

Horn

Anti-Horn

Smolensky

Micro 100 Small World

.501

.410

.065

25 / 37


Basin of Attraction: 2-2-2 Horn Game (Conditional Imitation,Grid Network)

Horn

Anti-Horn

Smolensky

Micro 100 Grid

.501

.408

.067

26 / 37


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

27 / 37


Evolution of Meaning through Signaling

How plausible is the approach to meaning evolution usingtype-matching signaling games?

28 / 37


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)

29 / 37


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

)

30 / 37


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

31 / 37


Example: Similarity Maximizing Game (Jager and van Rooij, 2007)

32 / 37


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

]

33 / 37


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.

34 / 37


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)

35 / 37


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

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.

Models of Language Evolution

Documents

Transcript of Models of Language Evolution