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
Bernhard Schröder and Philipp C. Wichard Modelling Semantic Change as Equilibrium Shift
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)
Bernhard Schröder and Philipp C. Wichard Modelling Semantic Change as Equilibrium Shift
Local game of partial information
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Nature
Bernhard Schröder and Philipp C. Wichard Modelling Semantic Change as Equilibrium Shift
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).
Bernhard Schröder and Philipp C. Wichard Modelling Semantic Change as Equilibrium Shift
N1: Separating e1
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Nature
�2(a2) > �2(�1)
Bernhard Schröder and Philipp C. Wichard Modelling Semantic Change as Equilibrium Shift
N2: Separating e2
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p1
p2 = 1� p1
Nature
�1(a1) > �1(�2)
Bernhard Schröder and Philipp C. Wichard Modelling Semantic Change as Equilibrium Shift
N3: Pooling on e
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Nature
�i(��i) > �i(ai)
p1�1(�1) + (1� p1)�2(�1) > p1�1(�2) + (1� p1)�2(�2)
Bernhard Schröder and Philipp C. Wichard Modelling Semantic Change as Equilibrium Shift
Type frequencies
reflectenvironmental conditionsrelevance, importance
Bernhard Schröder and Philipp C. Wichard Modelling Semantic Change as Equilibrium Shift
Payo�s
reflectcost of expressionspenalty of wrong interpretation (depend on the context)
Bernhard Schröder and Philipp C. Wichard Modelling Semantic Change as Equilibrium Shift
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.
Bernhard Schröder and Philipp C. Wichard Modelling Semantic Change as Equilibrium Shift
N3
p1 �1(�1) + (1� p1) �2(�1) > p1 �1(�2) + (1� p1) �2(�2); (2)
�2(�1) > �2(a2); (3)
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Nature
Bernhard Schröder and Philipp C. Wichard Modelling Semantic Change as Equilibrium Shift
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.
Bernhard Schröder and Philipp C. Wichard Modelling Semantic Change as Equilibrium Shift
N2
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Nature
Bernhard Schröder and Philipp C. Wichard Modelling Semantic Change as Equilibrium Shift
Generalisation
Basically the reverse of specialization:seperating equilibrium! pooling equilibriumN1:
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�2(�2) < �2(a2)
Problem: There is no incentive for the �2-type sender or thereceiver to deviate from the current strategy.
Bernhard Schröder and Philipp C. Wichard Modelling Semantic Change as Equilibrium Shift
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.
Bernhard Schröder and Philipp C. Wichard Modelling Semantic Change as Equilibrium Shift
Semantic Shift
N1! N2 (may go through N3)N1:
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a2
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Nature
Bernhard Schröder and Philipp C. Wichard Modelling Semantic Change as Equilibrium Shift
Semantic Shift
N2:
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Nature
Same problem: There is no incentive for the �2-type sender orthe receiver to deviate from the current strategy.
Bernhard Schröder and Philipp C. Wichard Modelling Semantic Change as Equilibrium Shift
A Solution to the dynamic problem: trembles
Players deviate from their equilibrium strategies withprobability 0 < " << 1 (“trembles”, Selten 1975).
errorstrialsimperfectness of learning (mutations)
Bernhard Schröder and Philipp C. Wichard Modelling Semantic Change as Equilibrium Shift
Senders' errors
New precondition for N1:
p1(1�")�1(�1)+(1�p1)"�2(�1) > p1(1�")�1(�2)+(1�p1)"�2(�2):
(6)
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Nature
Decrease of p1 will make �2 more attractive, due to the errors of�2.
Bernhard Schröder and Philipp C. Wichard Modelling Semantic Change as Equilibrium Shift
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)
Bernhard Schröder and Philipp C. Wichard Modelling Semantic Change as Equilibrium Shift
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.
Bernhard Schröder and Philipp C. Wichard Modelling Semantic Change as Equilibrium Shift
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
Bernhard Schröder and Philipp C. Wichard Modelling Semantic Change as Equilibrium Shift