Models of Language Evolution Evolutionary game theory & - PowerPoint PPT Presentation

Models of Language Evolution Evolutionary game theory & signaling games Michael Franke

Topics for today 1 (flavors of) game theory 2 signaling games (& conversion into symmetric form) 3 Nash equilibrium (in symmetric games) 4 evolutionary stability 5 meaning of signals

Game Theory Signaling games Population Games Game Theory Signaling games Population Games 6 / 29

Game Theory Signaling games Population Games (Rational) Choice Theory Decision Theory: a single agent’s solitary decision Game Theory: multiple agents’ interactive decision making 7 / 29

Game Theory Signaling games Population Games Game Theory • abstract mathematical tools for modeling and analyzing multi-agent interaction • since 1940 : classical game theory (von Neumann and Morgenstern) • perfectly rational agents ::: Nash equilibrium • initially promised to be a unifying formal foundation for all social sciences • Nobel laureates: Nash, Harsanyi & Selten ( 1994 ), Aumann & Schelling ( 2006 ) • since 1970 : evolutionary game theory (Maynard-Smith, Prize) • boundedly-rational agents ::: evolutionary stability & replicator dynamics • first applications in biology, later also elsewhere (linguistics, philosophy) • since 1990 : behavioral game theory (Selten, Camerer) • studies interactive decision making in the lab • since 1990 : epistemic game theory (Harsanyi, Aumann) • studies which (rational) beliefs of agents support which solution concepts 8 / 29

Game Theory Signaling games Population Games Games vs. Behavior Game: abstract model of a recurring interactive decision situation • think: a model of the environment Strategies: all possible ways of playing the game • think: a full contingency plan or a (biological) predisposition for how to act in every possible situation in the game Solution: subset of “good strategies” for a given game • think: strategies that are in equilibrium, rational, evolutionarily stable, the outcome of some underlying agent-based optimization process etc. Solution concept: a general mapping from any game to its specific solution • examples: Nash equilibrium, evolutionary stability, rationalizability etc. 9 / 29

Game Theory Signaling games Population Games Kinds of Games uncertainty choice points simultaneous in sequence no strategic / static dynamic / sequential with complete info yes Bayesian dynamic / sequential with incomplete info 10 / 29

Game Theory Signaling games Population Games Game Theory Signaling games Population Games 11 / 29

Game Theory Signaling games Population Games t ∈ T a ∈ A ? receiver chooses act sender knows state, but receiver does not State-Act Payoff Matrix m ∈ M t ∈ T a 1 a 2 . . . t 1 1 , 1 0 , 0 t 2 1 , 0 0 , 1 . . . sender sends a signal (Lewis, 1969 ) 12 / 29

Game Theory Signaling games Population Games Signaling game A signaling game is a tuple �{ S , R } , T , Pr, M , A , U S , U R � with: Talk is cheap iff for all t , m , m ′ , a and { S , R } set of players X ∈ { S , R } : T set of states U X ( t , m , a ) = U X ( t , m ′ , a ) . Pr prior beliefs : Pr ∈ ∆ ( T ) Otherwise we speak of costly signaling . M set of messages A set of receiver actions U S , R utility functions : T × M × A → R . model of the context/environment/world 13 / 29

Game Theory Signaling games Population Games Example ( 2 - 2 - 2 Lewis game) 2 states, 2 messages, 2 acts Pr ( t ) a 1 a 2 t 1 p 1 , 1 0 , 0 t 2 1 − p 0 , 0 1 , 1 14 / 29

Game Theory Signaling games Population Games Example (Alarm calls) � 1 , 1 � � 0 , 0 � � 0 , 0 � � 1 , 1 � a 1 a 2 a 1 a 2 R R m 1 m 1 t 1 t 2 S N S p 1 − p m 2 m 2 R R a 1 a 2 a 1 a 2 � 1 , 1 � � 0 , 0 � � 0 , 0 � � 1 , 1 � 15 / 29

Game Theory Signaling games Population Games Strategies Pure s ∈ M T r ∈ A M fixed contingency plan Mixed s ∈ ∆ ( M T ) r ∈ ∆ ( A M ) ˜ ˜ uncertainty about plan Behavioral σ ∈ ( ∆ ( M )) T ρ ∈ ( ∆ ( A )) M probabilistic plan 16 / 29

Game Theory Signaling games Population Games Pure sender strategies in the 2 - 2 - 2 Lewis game t 1 m a t 1 m a “ m a m b ”: “ m b m a ”: m b m b t 2 t 2 t 1 m a t 1 m a “ m b m b ”: “ m a m a ”: m b m b t 2 t 2 17 / 29

Game Theory Signaling games Population Games Pure receiver strategies in the 2 - 2 - 2 Lewis game m a a 1 m a a 1 “ a a a b ”: “ a b a a ”: m b m b a 2 a 2 m a a 1 m a a 1 “ a b a b ”: “ a a a a ”: m b m b a 2 a 2 18 / 29

Game Theory Signaling games Population Games All pairs of sender-receiver pure strategies for the 2 - 2 - 2 Lewis game 4 1 2 3 5 7 6 8 9 10 11 12 13 14 15 16 19 / 29

Game Theory Signaling games Population Games Game Theory Signaling games Population Games 20 / 29

Game Theory Signaling games Population Games (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 21 / 29

Game Theory Signaling games Population Games Mixed strategies in symmetric games A mixed strategy in a symmetric game is a probability distribution σ ∈ ∆ ( A ) . Utility of mixed strategies defined as usual: U ( σ , σ ′ ) = ∑ σ ( a ) × σ ( a ′ ) × U ( a , a ′ ) a , a ′ ∈ A 22 / 29

Game Theory Signaling games Population Games Nash Equilibrium in Symmetric Games A mixed strategy σ ∈ ∆ ( A ) is a symmetric Nash equilibrium iff for all other possible strategies σ ′ : U ( σ , σ ) ≥ U ( σ ′ , σ ) . It is strict if the inequality is strict for all σ ′ � = σ . 23 / 29

Game Theory Signaling games Population Games Examples Prisoner’s Dilemma Hawk & Dove � � � � 2 0 1 7 U = U = 3 1 2 3 symmetric ne : � 0 , 1 � symmetric ne : � . 8 , . 2 � 24 / 29

Game Theory Signaling games Population Games 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 � , 25 / 29

Game Theory Signaling games Population Games 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 , strict symmetric ne 26 / 29

Game Theory Signaling games Population Games 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 27 / 29

Reading for Next Class Brian Skyrms ( 2010 ) “Information” Chapter 3 of “Signals” OUP.

References Lewis, David ( 1969 ). Convention. A Philosophical Study . Cambridge, MA: Harvard University Press.