models of language evolution - session 7: game theoretic

RolandMühlenbernd

Homework 2

The Evolution ofLanguageThe Evolution of SymbolicCommunication

The Evolution of WordFormations

The Evolution of Syntax

Case Marking Study

Further Studies

Models of Language EvolutionSession 7: Game Theoretic Applications

Roland Mühlenbernd

2016/12/07

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies

Exercise 1 (Nowak 2011)

Some notable dates (billion years ago):

13.7 big bang4.567 solar system? origin of life3.8 b-life (bacteria)1.8 e-life (eucaria)0.6 m-life (complex multi-cellularity)0.001 i-life (human language)

In comparison to all other evolutionary steps of progress, what is sospecial about the emergence of ’i-life’ (human language)?

▶ evolutionary and cultural progresses speed up exponentially

▶ evolutionary processes are not dependent on genetic changes√

▶ evolutionary development has reached an end point

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies

Exercise 2 (Nowak 2011)What is the crucial difference in Lamarck’s and Darwin’s pointof view concerning evolution?

▶ While both believed in the change of species, only Darwinbelieved in a ’tree of evolution’ (

√)

▶ While both agreed on the fact that all species are related, onlyDarwin believed in ’natural selection’

▶ While both believed in the dynamics of natural selection, onlyDarwin believed in ’adaptation’ of species

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies


What is - according thNowak - the only thingthat really evolves inevolution?

▶ behavioralcharacteristics

▶ language▶ populations (

√)

▶ species▶ genes

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies


What is cooperation in a game-theoretical sense?

▶ An interaction between two individuals: a donor that paysa cost and a recipient that gets a benefit (

√)

▶ An exchange process of values and goods betweendifferent parties

▶ A behavioral pattern in a group of individuals thatcoordinates efficient behavior

b > c > 0

coop. defectcoop. b-c -cdefect b 0

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies


What are the 5 mechanisms for the evolution of cooperation?

▶ threat of punishment▶ direct reciprocity (

√)

▶ indirect reciprocity (√

)▶ selfishness▶ spatial selection (

√)

▶ group selection (√

)▶ rationality▶ kin selection (

√)

▶ compassion▶ sense of justice

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies

The Evolution of Language (1999)

In their study “the Evolution of Language” Nowak & Krakauerused methods of evolutionary game theory to investigate thefollowing research questions:

1. How can a protolanguage emerge from an initialpopulation without language: how do signal-objectassociations evolve?

2. How can a protolanguage produce complex structures:why and how did word formation emerge?

3. How did syntax and grammar evolve??

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies

The Evolution of Language (1999)How do signal-object associations evolve?

▶ Initial situation: a population of early hominids, which canproduce a number of m sounds that can be used to carryinformation about a number of n objects

▶ active matrix P contains values pij ∈ R: probabilities that objecti is associated with signal j by the speaker

▶ passive matrix Q contains values qij ∈ R: probabilities thatsignal j is construed as object i by the listener

s1 s2 s3 s4 s5

o1 0.2 0.1 0.2 0.3 0.2o2 0.1 0.0 0.4 0.1 0.4o3 0.4 0.3 0.1 0.1 0.1o4 0.0 0.1 0.2 0.2 0.5o5 0.3 0.3 0.2 0.2 0.0

Table: Matrix for 5 objects (rows) and 5 signals (columns)

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies


How do signal-object associations evolve?

▶ an individual language L is defined by an active and a passivematrix: L = ⟨P,Q⟩

▶ fitness is communicative success, defined as utility of mutualintelligibility:

F(L,L′) =12

n∑i=1

m∑j=1

(pijq′ij + p′ijqij) (1)

▶ Simulation experiment:

▶ a population of 100 individuals▶ 5 objects, 5 signals▶ everybody communicates with everybody and computes the total

fitness by communicative success▶ replicator dynamics: higher fitness leads to higher reproduction

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies

Exercise 1 (Nowak & Krakauer 1999)

F(L,L′) =12

n∑i=1

m∑j=1

(pijq′ij + p′ijqij) (2)

In the initial simulation experiment each individual can reach a’total payoff’ per round. What does this payoff represent?

▶ fitness: the individual’s survival chance, which leads to ahigher number of offspring

▶ communicative success: the individual’s ability tocommunicate information with other individuals in thecommunity (

√)

▶ multilingualism: the individual’s faculty to communicatesuccessfully with different languages

Exercise: how does the the formula for the total payofflooks like? T(L) =

∑i∈P F(L,Li)

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies


Result of the simulation experiment

▶ signals are successively associated with objects▶ in some runs an optimal communication system emerged:

per object one signal▶ sometimes suboptimal systems emerged▶ exemplary course:

Figure: Emergence of a sup-optimal system: signal 1 isambiguous, signal 5 is not used

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies


Next step: including perception errors and a sound spectrum

▶ Original equation:

F(L, L′) =12

n∑i=1

m∑j=1

(pijq′ij + p′

ijqij) (3)

▶ error by noise: uij = probability of perceiving sound i as sound j

▶ Equation with perception errors:

F(L, L′) =12

n∑i=1

m∑j=1

[pij

(m∑

k=1

u′jkq′

ki

)+ p′

ij

(m∑

k=1

ujkqij

)](4)

▶ uij can defined by similarity of sounds sij (sound spectrum)

uij =sij∑m

k=1 sik(5)

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies


Next step: assuming that objects can have different values ai

▶ to inform about an leopard may be more valuable thanabout a python, and this information more valuable than abanana

▶ objects are ranked according to values: a1 > a2 > ... > an

▶ the utility function is influenced by these values

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies

The Evolution of Language (1999)Result of an simulation experiment with 20 objects and 40signals:

▶ If not all objects are associated, than the highest ranked ones

▶ Only a subset of all signals is used

▶ Used signals are selected to minimize similarity

▶ Exemplary result (9 signals/11 objects-system):

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies

The Evolution of Language (1999)Question: Why weren’t more signals used to communicate allobjects?

▶ a higher number of signals increases the number of objects thatcan be communicated, but it also increases the probability ofmiscommunication

▶ the result is that the overall ability of transfer information doesnot improve

Figure: 100 × 100-Game: communicative payoff is maximal if28 signals are used for the 28 highest ranked objects.

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies





3. How did syntax and grammar evolve?

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies

The Evolution of Language (1999)▶ error limit: the number of used signals that optimizes communicative

payoff▶ to overcome the error limit, sounds can be combined to word

formations:

▶ words as combined signals: m signals and word length l →ml possible words

▶ word similarity is the product of position-wise signalsimilarity

Figure: Communicative payoff over used signals/objects. Left:maximum at 28/100 single signals. Right: maximum at 52/121double signals (11 single signals).

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies


How do Nowak & Krakauer define similarity of words?

▶ As an inversely proportional function of word differences▶ In relation to the number and conformity of sounds of

syllables▶ As the product of the similarities between individual

phonemes in corresponding positions (√

)

Exercise: bill, pill, bull

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies





3. How did syntax and grammar evolve?

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies


How did syntax and grammar evolve?

▶ the combination of signals gives a possible unlimited number ofways to describe objects and actions as words

▶ therefore the combination of words (action+object) allows foran unlimited number of phrases

▶ For e.g. n objects and h actions there are n × h possibleintransitive constructions1

▶ Experiment: compare two strategies or transitive phrases (TP)

▶ Non-grammatical: like for simple words, for each TPcreate a new arbitrary word

▶ Grammatical: for each TP construct formations of words,that encode word information

1But not all make sense: ”tiger runs.” VS ”banana runs.”

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies


When is a grammatical system favored to a non-grammaticalsystem (accoring to N&K’s calculations)?

▶ If the number of relevant objects exceeds the number ofdistinguishable strings of sounds

▶ If the number of relevant sentences exceeds the number ofwords that make up these sentences (

√)

▶ If the grammatical system does not exceed the individuals’cognitive abilities to use it

▶ If the grammatical system is evolutionary stable and thenon-grammatical is not

Exercise: tiger, rabbit, banana, run away from, eat, hunt

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies

How did syntax and grammar evolve?

▶ (a) Grammatical and non-grammatical strategy

▶ (b) Agents communicate grammatical with probability x andnon-grammatical with probability 1 − x (see active matrices)

▶ (c) Both (pure) strategies are ESS, but not all probabilisticstrategies lead to the pure grammatical strategy.

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies


What is - according to Nowak & Krakauer - an importantprerequisite for the evolution of communication and language?

that...▶ communication favors fitness and selection▶ there is a will for cooperation between individuals (

√)

▶ populations are sufficiently large and complex structuredand therefore need more sophisticated forms of interaction

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies


What are - according to Nowak & Krakauer - the ’anatomicunits’ that make up the edifice of human language?

▶ emergence of simple syntax or grammar (√

)▶ association of signals to objects (

√)

▶ usage of language as building and maintaining socialrelationships

▶ realization of simple semantic relationships▶ adaptation of a serial communication channel to produce

complex structures▶ formation of words (

√)

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies


1. How can a protolanguage emerge from an initial population withoutlanguage: how do signal-object associations evolve?Answer: it can emerge by merely assuming the will for cooperation andmutual intelligibility (that’s what the fitness function F(L, L′) is basedon).

2. How can a protolanguage produce complex structures: why and howdid word formation emerge?Answer: communication by simple signals reaches an error limit thatmaximizes communicative fitness/payoff for a particular number ofobjects to describe. It is possible to overcome this limit value bycombining signals to words.

3. How did syntax and grammar evolve??Answer: by assuming non-grammatical and grammatical strategies ofencoding events, it can be shown the grammatical strategies are theevolutionary stable strategies with a maximal basin of attraction.

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies

Case Marking

Example: Case Marking Game (Jäger 2007)

The game G = {(S,H),M,F,P,C,US,UH} is a communication situationbetween speaker and hearer, whereby

▶ M is a set of meanings: pA/nA + pO/nO

▶ F is a set of forms: a, e, z

▶ Pr ∈ ∆(M) is a probability distribution over meanings M (informed bycorpus frequency)

▶ C : F → R is a cost function over forms F: C(a) = C(e) = 1,C(z) = 0

▶ US : S× H → R is the speaker’s utility function

▶ UH : S× H → R is the hearer’s utility function

▶ whereby:

▶ S : m → f is a speaker strategy▶ H : f → m is a hearer strategy

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies

Case Marking

pA nA pO nO systeme e z z full erg.z z a a full acc.e z a zz e z az e a z split erg.e z z z erg. iDSMz e z z erg. DSMz z a z acc. DOMz z z a acc. iDOMz z z z neutral

Table: Speaker strategy set restricted to 10 ‘reasonable’ startegies byruling out double erg./acc. marking and strictly dominated strategies.

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies

Case Marking

Formal Evolutionary Analysis

Figure: The static asymmetric case marking game for k = .1

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies

Asymmetric Game & Case MarkingWhat are the evolutionary stable states?

Figure: The static asymmetric game of case marking for k = .1

▶ the split ergative pattern zeaz/pA

▶ note that e.g. zzaa/SO (accusative marking + SO word order) isa Nash equilibrium, but not evolutionary stable

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies

Asymmetric Game & Case Marking

Let’s analyze the following sub-game:SO pA

zzaa 0.90 0.90zeaz 0.61 0.97

▶ zzaa/SO is an attractor state, but not an ESS

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies

Asymmetric Game & Case Marking


▶ the only ESS is split ergative zeaz/pA, that is very commonamong Australian aborigenes languages

▶ note that the values are influenced by the value k

▶ What will happen if we change this value?

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies

Asymmetric Game & Case MarkingWhat are the evolutionary stable states for k = 0.45?


▶ the differential object marking (DOM) pattern zzaz/pA;e.g. English, Dutch

▶ the inverse differential subject marking (inverse DSM) ezzz/pO;e.g. Wahki (Iranian, Northern Pakistan)

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies

Asymmetric Game & Case MarkingWhat are the evolutionary stable states for k = 0.55?


▶ the differential subject marking (DSM) pattern zezz/pA;e.g. several caucasian languages

▶ the inverse differential object marking (inverse DOM) zzza/pO;e.g. Nganasan (North Siberia)

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies

Asymmetric Game & Case MarkingWhat are the evolutionary stable states for k = 1?

Figure: The static asymmetric game of case marking for k = 1

▶ no case marking zzzz/pA;e.g. Bantu languages (south to middle Africa)

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies

Results

▶ only very few languages are not evolutionarily stable inthis sense:

▶ zzaa: Hungarian▶ ezza: Arrernte (Central Australia)▶ eeaa: Wangkumara (Australia, already extinct)

▶ curious asymmetry: if there are two competing stablestates, one is common and the other one rare

▶ it can be shown: the rare ones are not stochastically stablestates (temporal information)

▶ the only stochastically stable strategies:▶ split ergative zeaz▶ differential object marking zzaz▶ differential subject marking zezz▶ no case marking zzzz

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies

Results▶ the configuration of evolutionary strategies depend on the

case marking costs k and the split point betweenprominent and non-prominent arguments

▶ a low split point makes pp > nn more probable, a highsplit point nn > pp

▶

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies

Conclusion

▶ out of 4 × 16 = 64 possible case marking patterns onlyfour are stochastically stable

▶ vast majority of all languages that fit into thiscategorization are stochastically stable

▶ linguistic universals need not be based on innate“language instinct” but can be result of evolutionarypressure in the sense of cultural evolution

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies

Applications of Signaling Games I

Functional explanations for typological universals:▶ morphology: evolutionary stability of case marking

systems for semantic core roles via (i) replicatordynamics, and (ii) stability analysis (Jäger 2007)

▶ phonology: stability of vowel systems (separating acousticspace) via (i) exemplar learning, and (ii) simulationexperiments (Jäger 2008)

▶ pragmatics: evolutionary stability of the ‘Horn strategy’via (i) reinforcement learning, fictitious play, and (ii)simulation experiments (Mühlenbernd 2011)

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies

Applications of Signaling Games II

Functional explanations for ‘diachronic’ universals:▶ negation: evolutionary trajectories of the ‘Jespersen cycle’

via (i) replicator dynamics, and (ii) stability/trajectoryanalysis (Ahern & Clark 2014, Ahern 2016)

▶ morphosemantics: evolutionary trajectories of the‘progressive cycle’ via (i) replicator dynamics, and (ii)stability/trajectory analysis (Deo 2015, Yanovich 2016)

▶ morphosemantics: evolutionary trajectories of the‘progressive cycle’ via (i) reinforcement learning, and (ii)simulation experiments (Enke, Mühlenbernd, Yanovich2016)

RolandMühlenbernd

Homework 2




Case Marking Study

Further Studies

Timescale of Literature1990 Pinker & Bloom: language evolution theory19911992199319941995 Bickerton: PL-fossils in form of language behavior1996199719981999 Jackendoff: PL-fossils in instances of Human language

Nowak & Krakauer: The Evolution of Language20002001 Simulating the Evolution of Language←←←←←←2002 Hauser, Chomsky & Fitch: FLN = FLB + recursion20032004200520062007 Bickerton: perspective from linguistics

Kirby: perspective from LE-modelers

models of language evolution - session 7: game theoretic

Documents