Meaning, Evolution and the Structure of Society

Meaning, Evolution and the Structure of Society

Roland M ¨uhlenbernd

November 7, 2014

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Game Theory and Linguistics Pragm. Reasoning Language Evolution

Signaling Games

Replicator Dyn. Imitation Dyn. Learning Dyn.

Reinforcement BL & BR GT in Lang. Use

I Accounts differ inpsychologyof language users

I What role does thesociologyof language users play?

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”[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

”Two savages, who had never been taught to speak, but had been bred up remote from the societies of men, would naturally begin to form that language by which they would endeavor to make their mutual wants intelligible to each other, by uttering certain sounds, whenever they meant to denote certain objects.”

Smith, Considerations Concerning the First Formation of Languages

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(Lewis game)

Signaling GameSG=h{S,R},T,M,A,Pr,C,Ui

I set of players{S,R}

I set of statesT(T={t₁,t₂})

I state probability functionPr∈∆(T)(Pr(t₁) =Pr(t₂) =.5)

I set of messagesM(M={m₁,m₂})

I cost functionC:M→R(C(m₁) =C(m₂) =0)

I set of actionsA(A={a₁,a2})

I utility functionU(t_i,m_j,a_k) =

1−C(m_j) ifi=k 0−C(m_j) else a₁ a₂

t₁ 1,1 0,0 t2 0,0 1,1

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I the players’ ”actions” can be represented as pure strategies

I set of pure sender strategiesS={s|s∈[T→M]}

I set of pure receiver strategiesR={r|r∈[M→A]}

I for the Lewis game there are 4 strategies for each player

I S={s₁,s₂,s₃,s₄}

I R={r₁,r₂,r₃,r₄}

s₁: ^{t1 m1}

t2 m2

s₂: ^t1

m2 t2

m1

s₃: ^{t1 m1}

t2 m2

s₄: ^t1

m₂ t2

m1

r₁: ^{m1 a1}

m2 a2

r₂: ^m1

a2 m2

a1

r₃: ^{m1 a1}

m2 a2

r₄: ^m1

a₂ m2

a1

S

S

...

I are combinations of pure strategies. The Lewis game has two:L₁ =hs₁,r₁iandL2=hs2,r2i

L₁: tL

t_S

m₁ m₂

a_L a_S L₂:

tL

t_S

m₁ m₂

a_L a_S

I are strictNash equilibriaover expected utilities

I associate messages to states in an unique way

I areevolutionary stable states

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The expected utility for a combination of strategies is given as:

EU(s,r) =X

t∈T

Pr(t)×U(t,s(t),r(s(t))) (1)

r₁ r2 r3 r₄ s₁ 1 0 .5 .5 s₂ 0 1 .5 .5 s₃ .5 .5 .5 .5 s₄ .5 .5 .5 .5

Table: The static Lewis game: Expected utilities for all strategy combinations of the dynamic Lewis game

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

I T={t_f,t_r},M={m_u,m_m},A={a_f,a_r}

I Pr(t_f)>Pr(t_r)

I C(m_u)<C(m_m)

I S={s_h,sa,ss,sy}

I R={r_h,r_a,r_s,r_y}

s_h:

t_f mu

tr mm

s_a:

t_f mm tr

mu

s_s:

t_f mu

tr mm

s_y: ^tf

mm tr

mu

r_h: ^mu

a_f

mm ar

ra: ^mu

ar mm

a_f

rs: ^mu

a_f

mm ar

r_y: ^mu

ar mm

a_f

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Exemplary parameters for the Horn Game:

I Pr(t_f) =.7

I C(m_u) =.1,C(m_m) =.2

r_h r_a r_s r_y s_h .87 -.13 .57 .17 s_a -.17 .83 .53 .13

ss .6 .2 .6 .2

sy .1 .5 .5 .1

Table: Payoff table of the asymmetric static Horn game with Pr(tf) =.75,C(mu) =.1 andC(mm) =.2

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I Which strategy emerges under evolutionary (population) dynamics like the replicator dynamics?

I How might the outcome differ for agent-based dynamics like imitation or learning?

I How does the outcome vary for different typologies?

I What role might structural aspects play for the emergence of specific strategies?

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Given a very large (effectively infinite) population of agents playing a symmetric static gameh{P₁,P₂},S,U:S×S→Ri randomly against each other. Then we can define

I p(si): proportion of agents in the population playing strategysi I U(si) =P

sj∈Sp(sj)U(si,sj): expected utility for agents playingsi I U=P

si∈Sp(si)U(si)the average fitness of the whole population

Definition (Replicator Dynamics)

The RD is defined by the following differential equation:

dp(s_i)

dt =p(s_i)[U(s_i)−U]

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Definition (Evolutionary stable strategy)

For a symmetric static gameh{P₁,P₂},S,U:S×S→Ria strategy s_i ∈Sis said to be a evolutionary stable strategy if and only if one of the following two conditions holds:

1. U(s_i,s_i)>U(s_i,s_j)for all s_j 6=s_i

2. if U(s_i,s_i) =U(s_i,s_j)for some s_j6=s_i, then U(s_i,s_j)>U(s_j,s_j) A ESSs_ihas the following properties:

I s_ihas an invasion barrier: resistant against the invasion of a small proportion of mutants withs_j 6=s_i

I s_iis a attractor under the replicator dynamics

I Strict NE ⊂ESS ⊂NE

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Strategies:•Horn,•anti-Horn,•Smolensky,•anti-Smolensky Start population:

[.25, .25, .25, .25]

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Horn game 1:

I Pr: [.6, .4]

I C: [.05, .1]

Lh: 55%,La: 43%,Ls: 2%

Horn game 2:

I Pr: [.7, .3]

I C: [.1, .2]

Lh: 59%,La: 35%,Ls: 6%

Horn game 3:

I Pr: [.9, .1]

I C: [.1, .3]

Lh: 64%,La: 22%,Ls: 14%

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”...the replicator dynamics is a natural place to begin investigations of dynamical models of cultural evolution, but I do not believe that it is the whole story.”

Skyrms, Evolution of the Social Contract

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The replicator dynamics describes the most likely path of strategy distribution in a virtually infinite and

homogeneous population, when every agent updates her behavior by conditional imitation.

shown by e.g. Helbing 1996, Schlag 1998

Conditional Imitation: the probability of an agent playing strategysto switch to a randomly chosen participant’s strategys⁰is given as:

P(s→s⁰) =

( 0 ifU(s,s⁰)≥U(s⁰,s⁰) α×^U(s_U^{0,s0)−U(s,s0)}

max−Umin else (2)

with:

I α: scaling factor

I Umax: maximal value of the utility table

I Umin: minimal value of the utility table

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Horn

Anti-Horn Smolensky

Macro .5 .374 .122 .003

(a)Replicator Dynamics

Horn

Anti-Horn Smolensky

Micro 100 .459 .365 .160

(b)Conditional Imitation Figure: Basins of attraction for the replicator dynamics and the conditional imitation rule by numerical simulation for a population of 100 randomly interacting agents.

→the Correspondence of basins of attraction is93.2%

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Simulations with 100 agents on different typologies:

I β-graph (Watts-Strogatz algorithm, starting ring) I scale-free networks (Holme-Kim algorithm) I 10×10 toroid lattice

percentage

0 0.2 0.4 0.6

complete (93.2%) scale-free (91%) β-graph (89.1%) grid (88.2 %) Smolensky

anti-Horn Horn

Figure: Comparing distributions of basin of attraction and

RD-correspondence (percentage in brackets) for different network topologies

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Zollman (2005) made experi- ments with agents

I playing the Lewis game

I updating by Imitation

I placed on a 100×100 toroid lattice

Result: the emergence of re- gional meaning

Source: Kevin J. S. Zollman (2005): Talking to Neighbors:

The Evolution of Regional Meaning. Philosophy of Science 72.1, 69–85.

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Agents play the dynamic gameSG=h(S,R),T,M,A,Pr,C,Ui

I as probabilistic strategiess:T→∆(M),r:M→∆(A)

I communicative outcome shifts probabilities in appropriate directions

I successful communication viaht∈T,m∈M,a∈Aimakes the choicess(m|t)andr(a|m)more probable in subsequent rounds

I agents might learn a pure strategy profile that constitutes a signaling system

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Experiments on a 30×30 toroid lattice:

The Lewis Game

L1 L2 other

The Horn Game

Horn Anti-Horn other Regional Convention evolve not always for the Horn Game.

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I the closer the distance on the lattice, the more probable the interaction I relationship controlled bydegree of localityγ

γ prob. for neighborhood

communication

0 1 2 3 4 5 6 7 8

0 0.2 0.4 0.6 0.8 1

percentage

0%

20%

40%

60%

80%

100%

Horn game Lewis game

Local interaction supports the emergence of local conventions.

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β-graph network (Watts & Strogatz 1998)

scale-free network (Holme & Kin 2002)

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Simulation runs for populations of 300 agents

I placed on small world network interaction structures

I ’interact’ via Lewis game and Horn game

I decide and update by learning dynamics: reinforcement learning or belief learning

I interact until 90%of all agents are language learners Analyzis of

I properties of language regions

I interrelationships between agents’ network position and learning behavior

I specific roles of agent in the emerging process of conventions/signaling languages

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density

n

0 50 100 150 200 250 300

0 0.2 0.4 0.6 0.8 1

language region (β−graph) language region (scale-free network)

average clustering

n

0 50 100 150 200 250 300

0.2 0.4 0.6 0.8 1

language region (β−graph) language region (scale-free network)

Cliquishness supports the emergence of language regions

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Social structure matters to the evolution and establishment of conventional meaning!

I different interaction structures lead to shifts in the basin of attraction of communication strategies

I locality of structure supports regional variety

I particular network properties support the emergence of regional conventions