Modelling Semantic Change as Equilibrium Shiftcebert/LGE/schroeder.pdfGoal modelling semantic change...

Modelling Semantic Change as EquilibriumShift

in a Signalling Game

Bernhard Schröder and Philipp C. Wichard

Bonn

ESSLLI 2007, Dublin

Bernhard Schröder and Philipp C. Wichard Modelling Semantic Change as Equilibrium Shift

Goal

modelling semantic change in a signalling gameidentify the condition for the three major classes ofsemantic change

generalisationspecialisationsemantic shift

characterization by initial and resulting equilibriatransition triggered by environmental or contextualchangesstress the importance of interpretation errors


The three main classes of semantic change

generalisation, e.g. Sache: issue of dispute! thingspecialisation, e.g. Mut: frame of mind! audacitysemantic shift, e.g. Hammer: rock, stone! hammer

special cases: pejorization (Weib), metaphorisation (schwer)


Local game of partial information

HHHHH

��

HHHHH

��

r

r

r

rr

r

r

r

rr

r

�1(�1)

�1(�2)

�2(�1)

�2(�2)

�1(a1)

�2(a2)

�2

�1

e

e

a2

a1

�1

�2

�1

�2

p1

p2 = 1� p1

Nature


Nash equilibria

pure strategiesN1 Separating e1: type �1 plays e, e is correctly interpreted as �1,

and type �2 plays a2 because �2(a2) > �2(�1).N2 Separating e2: type �2 plays e, e is correctly interpreted as �2,

and type �1 plays a1 because �1(a1) > �1(�2).N3 Pooling on e: both sender types play e, and e is interpreted as

�1 if

p1�1(�1) + (1� p1)�2(�1) > p1�1(�2) + (1� p1)�2(�2);

and as �2 if

p1�1(�1) + (1� p1)�2(�1) < p1�1(�2) + (1� p1)�2(�2):

Furthermore �i(��i) > �i(ai).


N1: Separating e1

HHHHH

��

HHHHH

��

r

r

r

rr

r

r

r

rr

r

�1(�1)

�1(�2)

�2(�1)

�2(�2)

�1(a1)

�2(a2)

�2

�1

e

e

a2

a1

�1

�2

�1

�2

p1

p2 = 1� p1

Nature

�2(a2) > �2(�1)


N2: Separating e2

HHHHH

��

HHHHH

��

r

r

r

rr

r

r

r

rr

r

�1(�1)

�1(�2)

�2(�1)

�2(�2)

�1(a1)

�2(a2)

�2

�1

e

e

a2

a1

�1

�2

�1

�2

p1

p2 = 1� p1

Nature

�1(a1) > �1(�2)


N3: Pooling on e

HHHHH

��

HHHHH

��

r

r

r

rr

r

r

r

rr

r

�1(�1)

�1(�2)

�2(�1)

�2(�2)

�1(a1)

�2(a2)

�2

�1

e

e

a2

a1

�1

�2

�1

�2

p1

p2 = 1� p1

Nature

�i(��i) > �i(ai)

p1�1(�1) + (1� p1)�2(�1) > p1�1(�2) + (1� p1)�2(�2)


Type frequencies

reflectenvironmental conditionsrelevance, importance


Payo�s

reflectcost of expressionspenalty of wrong interpretation (depend on the context)


Upward monotonic contexts

[[a]] � [[b]] ) C(a) j= C(b): (1)

Example

John saw a dog.

follows from

Example

John saw a yorkie.

In an upward monotonic context a true sentence remains true ifan expression is interpreted as the superconcept of the intendedconcept.


N3

p1 �1(�1) + (1� p1) �2(�1) > p1 �1(�2) + (1� p1) �2(�2); (2)

�2(�1) > �2(a2); (3)

HHHHH

��

HHHHH

��

r

r

r

rr

r

r

r

rr

r

�1(�1)

�1(�2)

�2(�1)

�2(�2)

�1(a1)

�2(a2)

�2

�1

e

e

a2

a1

�1

�2

�1

�2

p1

p2 = 1� p1

Nature


Variation of frquencies

p1 �1(�1) + (1� p1) �2(�1) > p1 �1(�2) + (1� p1) �2(�2) (4)

> may be reversed by decreasing p1 if

p1 (�1(�1)� �1(�2))︸︷︷︸>0 by ass.

< (1� p1) (�2(�2)� �2(�1))︸︷︷︸>0 by ass.

; (5)

With �1(�2) < �1(a1) we arrive at N2.


N2

HHHHH

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HHHHH

��

r

r

r

rr

r

r

r

rr

r

�1(�1)

�1(�2)

�2(�1)

�2(�2)

�1(a1)

�2(a2)

�2

�1

e

e

a2

a1

�1

�2

�1

�2

p1

p2 = 1� p1

Nature


Generalisation

Basically the reverse of specialization:seperating equilibrium! pooling equilibriumN1:

HHH

��

HHH

��

qq

qqq

qq

qqqq

�1(�1)

�1(�2)

�2(�1)

�2(�2)

�1(a1)

�2(a2)�2

�1

e

e

a2

a1

�1

�2

�1

�2

p1

p2 = 1� p1

Nature

�2(�2) < �2(a2)

Problem: There is no incentive for the �2-type sender or thereceiver to deviate from the current strategy.


The dynamic problem

There is no incentive for the �2-type sender or the receiver todeviate from the current strategy.The is no path in the replicator dynamics which leads from N1to the pooling equlibrium N3.


Semantic Shift

N1! N2 (may go through N3)N1:

HHH

��

HHH

��

qq

qqq

qq

qqqq

�1(�1)

�1(�2)

�2(�1)

�2(�2)

�1(a1)

�2(a2)�2

�1

e

e

a2

a1

�1

�2

�1

�2

p1

p2 = 1� p1

Nature


Semantic Shift

N2:

HHH

��

HHH

��

qq

qqq

qq

qqqq

�1(�1)

�1(�2)

�2(�1)

�2(�2)

�1(a1)

�2(a2)�2

�1

e

e

a2

a1

�1

�2

�1

�2

p1

p2 = 1� p1

Nature

Same problem: There is no incentive for the �2-type sender orthe receiver to deviate from the current strategy.


A Solution to the dynamic problem: trembles

Players deviate from their equilibrium strategies withprobability 0 < " << 1 (“trembles”, Selten 1975).

errorstrialsimperfectness of learning (mutations)


Senders' errors

New precondition for N1:

p1(1�")�1(�1)+(1�p1)"�2(�1) > p1(1�")�1(�2)+(1�p1)"�2(�2):

(6)

HHH

��

HHH

��

qq

qqq

qq

qqqq

�1(�1)

�1(�2)

�2(�1)

�2(�2)

�1(a1)

�2(a2)�2

�1

e

e

a2

a1

�1

�2

�1

�2

p1

p2 = 1� p1

Nature

Decrease of p1 will make �2 more attractive, due to the errors of�2.


Generalisation or shift

Decrease of p1 will make �2 more attractive, due to the errors of�2.This may lead to a generalisation (N3) or a semantic shift (N2).The result depends on the payoffs of �1:Generalisation: �1(�1) > �1(�2) > �1(a1)

Shift: �1(�1) > �1(a1) > �1(�2)


The asymmetry of semantic shift

Let us assume that it is worse to interpret e wrong when sent by�2 than by �1, i.e.

�1(�1)� �1(�2) < �2(�2)� �2(�1)

Then semantic shift N1! N2 occurs if p1 > ".For

�1(�1)� �1(�2) < �2(�2)� �2(�1)

p1 < " is necessary.


Conclusions

Semantic change can be triggered by sender typeprobabilities (environmental, contextual changes,relevance changes).Costs of misinterpretation in upward monotonic contexts.Generalisation and semantic shift need the assumption ofsender or receiver deviations from equilibria (“trembles”).Generalisation and semantic shift are differentiatedaccording to the cost of sending a1 instead of e.Asymmetry of semantic shift


Modelling Semantic Change as Equilibrium Shiftcebert/LGE/schroeder.pdfGoal modelling semantic change...

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