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Structure and Variation of Signaling Conventions

Roland M ¨uhlenbernd

November 7, 2014

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T

ABLE

O

F

C

ONTENTS

I Signaling Games

I Reinforcement Learning

I Simulations on Grid Structures

I Simulations on Scale-Free Networks

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T

HE

C

ONVENTION OF

S

EMANTIC

M

EANING

”A name is a spoken sound significant by convention... I say ’by convention’ because no name is a name naturally but only when it has become a symbol.”

Aristotle, De Interpretatione

”[w]e can hardly suppose a parliament of hitherto speechless elders meeting together and agreeing to call a cow a cow and a wolf a wolf.”

Russell, The Analysis of Mind

Paradox: language is needed for language to emerge

Lewis: semantic convention can emerge in ways dif- ferent to verbal agreements

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Signaling Game

:SG=h{S,R},T,M,A,Pr,Ui

N S

R

1 0

R

1 0

S R

0 1

R

0 1

.5 .5

tL tS

m1 m2 m1 m2

aL aS aL aS aL aS aL aS

I T={tL,tS}

I M={m1,m2}

I A={aL,aS}

I Pr(tL) =Pr(tS) =.5

I U(ti,aj) =

1 ifi=j

0 else aL aS

tL 1 0 tS 0 1

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P

URE STRATEGIES

Pure strategies are contingency plans, players act according to.

I sender strategy:σ :T→M

I receiver strategy:ρ:M→A

σ1: tL m1

tS m2

σ2: tL

m2 tS

m1

σ3: tL m1

tS m2

σ4: tL

m2 tS

m1

ρ1: m1 aL

m2 aS

ρ2: m1

aS m2

aL

ρ3: m1 aL

m2 aS

ρ4: m1

aS m2

aL

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S

IGNALING

S

YSTEMS

...

I are combinations of pure strategies. The Lewis game has two:L1 =hσ1, ρ1iandL2=hσ2, ρ2i

L1: tL tS

m1 m2

aL aS L2:

tL tS

m1 m2

aL aS

I are strictNash equilibriaof theEU-table:

ρ1 ρ2 ρ3 ρ4

σ1 1 0 .5 .5

σ2 0 1 .5 .5

σ3 .5 .5 .5 .5 σ4 .5 .5 .5 .5

I associate messages to states in an unique way

I areevolutionary stable states

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C

ONCLUSION

I signaling systems explain stability of semantic meaning...

I but not how it might emergewithout verbal agreement

I idea: participantlearna signaling system

I they start withunbiasedbehavior

I the update their behavior after each encounter

I Huttegger and Zollman (2011) asked: “How little cognitive ability is needed to learn a signaling system?”

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R

EINFORCEMENT

L

EARNING IN

S

IGNALING

G

AMES

S R

tl

ts

m1

m2

al

as

f f

f f

I the sender has an urnftfor each statetT

I each urn contains balls of each messagemM

I the sender decides by drawing from urnft

I the receiver has an urnfm

for each messagemM

I each urn contains balls of each actionaA

I the receiver decides by drawing from urnfm I successful communicationurn update

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R

EINFORCEMENT

L

EARNING

U

PDATE

R

ULES

Communication viaht,m,aiis successful

I Roth-Erev reinforcement:increase successful balls in appropriate urn byα∈R

I lateral inhibition:additionally decrease all other balls by γ ∈R

I negative reinforcement: if communication viaht,m,aiis not successful, decrease appropriate balls byβ ∈R

I Bush-Mosteller reinforcement:reinforce and scale down urn contents toΩ∈R

I RL=hα, β, γ,Ω, φi

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T

HE

T

ROUBLE

W

ITH

R

ICHER

G

AMES What happens in differentn×k-signaling games?

(n=|A|=|T|,k=|M|)

Game RFR

2×2 0%

3×3 9.6%

4×4 21.9%

8×8 59.4%

Table: Barrett’s results for Roth-Erev reinf. learning:

run failure rates (RFR) for diverse games (1000 runs)

RFR

0%

20%

40%

60%

3×3 4×4 8×8

B-M + LI + NRL Bush-Mosteller Roth-Erev

Figure: Run failure rates for different reinforcement learning accounts Source: Barret, J. A. (2009): The Evolution of Coding in Signaling Games.

Theory and Decision67, 223–237

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T

HE

T

ROUBLE

W

ITH

R

ICHER

G

AMES

2×2 3×3

4×4 5×5

6×6 7×7

8×8 0%

20%

40%

60%

80%

100%

01 2 3

01 2 3

0 1 2

3 01 2 3

0 1

2 3

0

1

23 0

1 23 1 no learners

1 1 learner

2 2 learners

3 3 learners

Figure: Experiments with 3 agents:

percentage of particular number of signaling system learners (averaged over 1000 simulation runs)

2×2 3×3

4×4 5×5

6×6 7×7

8×8 0%

20%

40%

60%

80%

100%

5 agents 3 agents

Figure: Experiments with 3 and 5 agents: percentage of signaling system learners (averaged over 1000 simulation runs)

What feature supports the emergence of signaling sys- tems in richer games and larger populations?

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I

NNOVATION

I each sender urn containsblack ballsthat – if drawn – produce a new message

I result: all agents learn a signaling system, even in rich games

I BUT: only in small populations

I solution: black ball induces a random message from the (fixed) message set

I to keep innovative nature, choose signaling games with

|M|>>|T|

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T

ESTRUN WITH

3 A

GENTS AND A

3 × 9-G

AME

I communicative success (CS): utility value (population average)

I force of innovation (FOI): # black balls (population average) FOI CS

100 150 200 250 300 350

0 .01 .02 .03 .04 .05

-.3-.2 -.1.1.2.3.4.5.6.7.8.901

Figure: Simulation run of a 3×9 signaling game with innovation in a 3-agents population:

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I

NNOVATION

VS. C

OMMUNICATIVE

S

UCCESS

FOI

CS

0 .01 .02 .03 .04 .05

-.6 -.4 -.2 0 .2 .4 .6 .8 1

Figure: 40,000 data points of 10 simulation runs: CS and FOI reveal a negativePearsoncorrelation of−.6

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C

ONCLUSION

I agents learn signaling systems by repeated play and simple update mechanismreinforcement learning

I but not necessarily for rich games and/or large populations

I idea: participant can beinnovative

I by sending a random message from setM

I when drawing the specialblack ballfrom a sender urn

I result: agents learn signaling systems for richer games in large populations.

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E

XPERIMENTS ON

G

RID

S

TRUCTURES

Settings:

I network: 10,000 agents on a 100×100 toroid lattice

I game type: 3×30 game

I update: Bush-Mosteller reinforcement learning with negative reinforcement, lateral inhibition and innovation (α=1,β =1,γ =0.5,Ω =20,φ: all balls uniformly distributed over all urns)

I break condition: after 50,000 simulation steps

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R

ESULTS

O

N

A T

OROID

L

ATTICE

Figure: Structure after 2,000 simulation steps

CS:≈.8,#RC:≈500

Figure: Structure after 50,000 simulation steps

CS:≥.9,#RC:≈170

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R

ESULTS

O

N

A T

OROID

L

ATTICE

L55: t1 t2 t3

m8 m21 m22

a1 a2

a3

L72: t1 t2 t3

m23 m21 m22

a1 a2 a3

L139: t1 t2 t3

m7 m8 m15

a1 a2 a3

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S

IMILARITY

M

EASURES AND

R

EGIONAL

D

ISTANCE

Lexical SimilarityLS:

LS(L1,L2) = |{mM|∃tT:m=s1(t)} ∩ {mM|∃tT:m=s2(t)}|

|T|

Mutual IntelligibilityMI:

MI(L1,L2) = P

t(Ux(t,s1,r2) +Ux(t,s2,r1)) 2× |T|

Regional DistanceRD:

RD(G1,G2) = P

ni∈N1

P

nj∈N2SP(ni,nj)

|N1| × |N2|

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R

ESULTS

O

N

A T

OROID

L

ATTICE

Mutual Intelligibility (MI) Lexical Similarity (LS)

regional distance

0 5 10 15 20 25 30 35 40

0 1 2 3 4

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1

Figure: LS and MI between two language regions in dependence of the distance between them, averaged over all pairs of language regions of 10 simulation runs (10×1702= 289,000 data points).

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E

XPERIMENTS ON

S

CALE

-F

REE

N

ETWORKS

Settings:

I network: 500 agents on a scale-free network (Holme-Kim algorithm withm=2,p=.8)

I game type: 3×9 game

I update: Bush-Mosteller reinforcement learning with negative reinforcement, lateral inhibition and innovation (α=1,β =1,γ =0.5,Ω =20,φ: all balls uniformly distributed over all urns)

I break condition: after 100,000 simulation steps

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R

ESULTS ON

S

CALE

-F

REE

N

ETWORKS

Figure: After 100,000 simulation steps 50 regions of different sizes emerged.

Figure: Histogram of region sizes bins for 10 simulation runs (ca. 500 data points).

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Negative Pearson Correlation of−.4 between the size of a language region and its members’ degree centralityDC(n) = N−1d(n)

Figure: Data plot of agent’s degree centrality (y-axis) in comparison to their region’s size (x-axis) for 5,000 data points.

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C

ONCLUSION

I repeated signaling gamesplus update dynamics might simulate the pathes of the emergence of semantic meaning

I reinforcement learningcannot guarantee the emergence of signaling systems for rich games in large populations

I innovationas additional feature overcomes this problematic nature

I onlattice structuresa regional structure similar to a dialect spectrum emerges, whereby regions coalesce with each other bit by bit

I onscale-free networksregions of different sizes emerge that contain agents with anti-proportional degree centralities

I theinnovative natureof the learning dynamics reorganizes structures of communities and realizessemantic change

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