Models of Language Evolution Evolutionary game theory & the evolution of meaning Michael Franke
Topics for today 1 evolutionary stability 2 meaning of signals 3 replicator dynamic
Population games Evolutionary Stability Meaning Evolution Population games Evolutionary Stability Meaning Evolution 6 / 29
Population games Evolutionary Stability Meaning Evolution (One-Population) Symmetric Game A (one-population) symmetric game is a pair � A , U � , where: • A is a set of acts, and • U : A × A → R is a utility function (matrix). Example (Prisoner’s dilemma) Example (Hawk & Dove) � a c a d � a h a d � � a c 2 0 a h 1 7 U = U = a d a d 3 1 2 3 7 / 29
Population games Evolutionary Stability Meaning Evolution Symmetrizing asymmetric games Example: signaling game • big population of agents • every agent might be sender or receiver • an agent’s strategy is a pair � s , r � of pure sender and receiver strategies • utilities are defined as the average of sender and receiver role: s ′ , r ′ � ) = 1 / 2 ( U S ( s , r ′ ) + U R ( s ′ , r ))) � U ( � s , r � , 8 / 29
Population games Evolutionary Stability Meaning Evolution Example (Symmetrized 2 - 2 - 2 Lewis game) s 1 s 2 s 3 s 4 s 5 s 6 s 7 s 8 s 9 s 10 s 11 s 12 s 13 s 14 s 15 s 16 s 1 � m 1 , m 1 , a 1 , a 1 � . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 � m 1 , m 1 , a 1 , a 2 � s 2 . 5 . 5 . 5 . 5 . 75 . 75 . 75 . 75 . 25 . 25 . 25 . 25 . 5 . 5 . 5 . 5 s 3 � m 1 , m 1 , a 2 , a 1 � . 5 . 5 . 5 . 5 . 25 . 25 . 25 . 25 . 75 . 75 . 75 . 75 . 5 . 5 . 5 . 5 s 4 � m 1 , m 1 , a 2 , a 2 � . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 � m 1 , m 2 , a 1 , a 1 � s 5 . 5 . 75 . 25 . 5 . 5 . 75 . 25 . 5 . 5 . 75 . 25 . 5 . 5 . 75 . 25 . 5 s 6 � m 1 , m 2 , a 1 , a 2 � . 5 . 75 . 25 . 5 . 75 1 . 5 . 75 . 25 . 5 0 . 25 . 5 . 75 . 25 . 5 s 7 � m 1 , m 2 , a 2 , a 1 � . 5 . 75 . 25 . 5 . 25 . 5 . 25 . 75 . 5 . 75 . 5 . 75 . 25 . 5 0 1 � m 1 , m 2 , a 2 , a 2 � s 8 . 5 . 75 . 25 . 5 . 5 . 75 . 25 . 5 . 5 . 75 . 25 . 5 . 5 . 75 . 25 . 5 s 9 � m 2 , m 1 , a 1 , a 1 � . 5 . 25 . 75 . 5 . 5 . 25 . 75 . 5 . 5 . 25 . 75 . 5 . 5 . 25 . 75 . 5 s 10 � m 2 , m 1 , a 1 , a 2 � . 5 . 25 . 75 . 5 . 75 . 5 . 75 . 25 . 5 . 25 . 5 . 25 . 75 . 5 1 0 � m 2 , m 1 , a 2 , a 1 � s 11 . 5 . 25 . 75 . 5 . 25 0 . 5 . 25 . 75 . 5 1 . 75 . 5 . 25 . 75 . 5 s 12 � m 2 , m 1 , a 2 , a 2 � . 5 . 25 . 75 . 5 . 5 . 25 . 75 . 5 . 5 . 25 . 75 . 5 . 5 . 25 . 75 . 5 s 13 � m 2 , m 2 , a 1 , a 1 � . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 � m 2 , m 2 , a 1 , a 2 � s 14 . 5 . 5 . 5 . 5 . 75 . 75 . 75 . 75 . 25 . 25 . 25 . 5 . 5 . 5 . 5 . 5 s 15 � m 2 , m 2 , a 2 , a 1 � . 5 . 5 . 5 . 5 . 25 . 25 . 25 . 25 . 75 . 75 . 75 . 75 . 5 . 5 . 5 . 5 s 16 � m 2 , m 2 , a 2 , a 2 � . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 9 / 29
Population games Evolutionary Stability Meaning Evolution Population games Evolutionary Stability Meaning Evolution 10 / 29
Population games Evolutionary Stability Meaning Evolution Mean-Field Population • (nearly) infinite populations for each distinguishable role • each population is entirely homogeneous • agents play pure strategies • each agent interacts purely at random with other agents • strategy updates are rare 11 / 29
Population games Evolutionary Stability Meaning Evolution Evolutionary Stability (Intuition) A strategy s is evolutionarily stable if a population that consists entirely/mostly of s -agents (the incumbents) cannot be invaded by any minority of mutants/invaders playing strategy t . 12 / 29
Population games Evolutionary Stability Meaning Evolution Evolutionary Stability (Derivation) Intuition s cannot be invaded by a minority of mutants t fitness of incumbent > fitness of mutant ( 1 − ǫ ) U ( s , s ) + ǫ U ( s , t ) > ( 1 − ǫ ) U ( t , s ) + ǫ U ( t , t ) • if ǫ is infinitesimal, this holds when U ( s , s ) > U ( t , s ) • but if U ( s , s ) = U ( t , s ) , then it also holds when U ( s , t ) > U ( t , t ) 13 / 29
Population games Evolutionary Stability Meaning Evolution Evolutionarily Stable Strategy (Definition) A strategy s is evolutionarily stable iff for all t : (i) U ( s , s ) > U ( t , s ) , or (ii) U ( s , s ) = U ( t , s ) and U ( s , t ) > U ( t , t ) . Connection with ne • strict- ne s ⊂ ⊂ ess s ne s 14 / 29
Population games Evolutionary Stability Meaning Evolution Example (Symmetrized 2 - 2 - 2 Lewis game) s 1 s 2 s 3 s 4 s 5 s 6 s 7 s 8 s 9 s 10 s 11 s 12 s 13 s 14 s 15 s 16 s 1 � m 1 , m 1 , a 1 , a 1 � . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 � m 1 , m 1 , a 1 , a 2 � s 2 . 5 . 5 . 5 . 5 . 75 . 75 . 75 . 75 . 25 . 25 . 25 . 25 . 5 . 5 . 5 . 5 s 3 � m 1 , m 1 , a 2 , a 1 � . 5 . 5 . 5 . 5 . 25 . 25 . 25 . 25 . 75 . 75 . 75 . 75 . 5 . 5 . 5 . 5 s 4 � m 1 , m 1 , a 2 , a 2 � . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 � m 1 , m 2 , a 1 , a 1 � s 5 . 5 . 75 . 25 . 5 . 5 . 75 . 25 . 5 . 5 . 75 . 25 . 5 . 5 . 75 . 25 . 5 s 6 � m 1 , m 2 , a 1 , a 2 � . 5 . 75 . 25 . 5 . 75 1 . 5 . 75 . 25 . 5 0 . 25 . 5 . 75 . 25 . 5 s 7 � m 1 , m 2 , a 2 , a 1 � . 5 . 75 . 25 . 5 . 25 . 5 . 25 . 75 . 5 . 75 . 5 . 75 . 25 . 5 0 1 � m 1 , m 2 , a 2 , a 2 � s 8 . 5 . 75 . 25 . 5 . 5 . 75 . 25 . 5 . 5 . 75 . 25 . 5 . 5 . 75 . 25 . 5 s 9 � m 2 , m 1 , a 1 , a 1 � . 5 . 25 . 75 . 5 . 5 . 25 . 75 . 5 . 5 . 25 . 75 . 5 . 5 . 25 . 75 . 5 s 10 � m 2 , m 1 , a 1 , a 2 � . 5 . 25 . 75 . 5 . 75 . 5 . 75 . 25 . 5 . 25 . 5 . 25 . 75 . 5 1 0 � m 2 , m 1 , a 2 , a 1 � s 11 . 5 . 25 . 75 . 5 . 25 0 . 5 . 25 . 75 . 5 1 . 75 . 5 . 25 . 75 . 5 s 12 � m 2 , m 1 , a 2 , a 2 � . 5 . 25 . 75 . 5 . 5 . 25 . 75 . 5 . 5 . 25 . 75 . 5 . 5 . 25 . 75 . 5 s 13 � m 2 , m 2 , a 1 , a 1 � . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 � m 2 , m 2 , a 1 , a 2 � s 14 . 5 . 5 . 5 . 5 . 75 . 75 . 75 . 75 . 25 . 25 . 25 . 5 . 5 . 5 . 5 . 5 s 15 � m 2 , m 2 , a 2 , a 1 � . 5 . 5 . 5 . 5 . 25 . 25 . 25 . 25 . 75 . 75 . 75 . 75 . 5 . 5 . 5 . 5 s 16 � m 2 , m 2 , a 2 , a 2 � . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 non-strict symmetric ne , ESS 15 / 29
Population games Evolutionary Stability Meaning Evolution All pairs of sender-receiver pure strategies for the 2 - 2 - 2 Lewis game 1 2 3 4 5 7 6 8 9 10 11 12 13 14 15 16 ESSs 16 / 29
Population games Evolutionary Stability Meaning Evolution Population games Evolutionary Stability Meaning Evolution 17 / 29
Population games Evolutionary Stability Meaning Evolution Meaning in Lewis games Signaling systems of the 2 - 2 - 2 Lewis game m a a 1 m a a 1 t 1 t 1 m b a 2 m b a 2 t 2 t 2 Fix an n - n - n Lewis game with S ig S ys � s , r � (i.e., ESS), and define: indicative meaning imperative meaning [[ m ]] T = { t ∈ T | s ( t ) = m } [[ m ]] A = { a ∈ A | r ( m ) = a } (Lewis, 1969 ) 19 / 29
Population games Evolutionary Stability Meaning Evolution Natural vs. non-natural meaning Natural meaning E.g.: smoke means fire Non-natural meaning E.g.: this gesture meant that the party is boring Non-natural meaning: Grice’s definition “ A meant NN something by x ” is roughly equivalent to “ A uttered x with the intention of inducing a belief [in his audience] by means of the recognition of this intention.” (Grice, 1957 ) 20 / 29
Population games Evolutionary Stability Meaning Evolution The Herod examples ( 1 ) Herod presents Salome with the head of St. John the baptist. ( 2 ) Herod says to Salome “He’s dead.” ( 3 ) Herod leaves the head somewhere; Salome happens to see it. ( 4 ) Herod leaves the head where he knows Salome will see it, correctly supposing she will not realize he left it for her to see. ( 5 ) Herod leaves the head where Salome will see it, mistakenly supposing she will not realize he left it for her to see. (Grice, 1957 ) 21 / 29
Population games Evolutionary Stability Meaning Evolution Meaning NN in signaling systems Behavior in a S ig S ys is compatible with common belief in rationality. We can then construct an infinite chain of rational inten- tion recognition based on S ig S ys -behavior. Meaning in S ig S ys s can be construed as meaning NN if the ascription of relevant mental states to agents is warranted. (Lewis, 1969 ) 22 / 29
Population games Evolutionary Stability Meaning Evolution Practical reasoning justification in a S ig S ys “Suppose I am the communicator and you are the audience (. . . ) and having observed that t 1 holds, I give m a in conformity to our convention. (. . . ) The intention with which I do m a can be established by examining the practical reasoning that justifies me in doing it. I need not actually go through that reasoning to have an intention; actions done without deliberation are often done with definite intentions. (. . . ) My decision to do m a , having observed t 1 , is premised on my expectation that I can thereby produce a 1 and on my desire to produce a 1 . So I do m a with the intention to produce a 1 . I expect you to infer t 1 upon observing that I do m a . I expect you to recognize my desire to produce a 1 , conditionally upon t 1 . I expect you to recognize my expectation that I can produce a 1 by doing m a . So I expect you to recognize my intention to produce a 1 , when you observe that I do m a . (. . . )” (Lewis, 1969 , p. 155 ) 23 / 29
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