Game Theory for Linguists

Transcription

1 Fritz Hamm, Roland Mhlenbernd 27. Juni 2016

2 Overview Exercises II

3 Exercise 1: Signaling Game Properties Exercise 1 The introduced type of a signaling game has a probability function and a denotation function. How are these functions defined and what do they represent? The probability function Pr is a probability distribution over T, Pr (T ) is defined as Pr : T R, whereas t T Pf (t) = 1 and t T : Pr(t) > 0 represents frequency/prototypicality of information states The denotation function is defined as : M P(T )\ represents the predefined semantic/literal meaning of a message

4 Exercise 1: Signaling Game Properties Exercise 1 What is the fundamental difference between the Wine-Choice and the Some-All scenario in respect of the way both scenarios are modeled as a game? both have the same number of states, messages and actions and the same utility tables but there is a fundamental difference in the way the denotation function is given Wine-Choice: m beef = {t beef }, m fish = {t fish } Some-All: mall = {t }, m some = {t, t } only the Some-All scenario has a message with a semantic/literal meaning encompassing more than one information state

5 Exercise 1: Signaling Game Properties Exercise 1 How many different pure strategies has the Wine-Choice game, how many the Some-All game? Note: A game with x states, y messages and z actions has y x pure sender strategies and z y pure receiver strategies and therefore y x z y pure strategy combinations thus both games have 2 2 = 4 sender strategies and 2 2 = 4 receiver strategies and = 16 pure strategy combinations

6 Exercise 1: Signaling Game Properties Exercise 1 In session 4 the signaling games was introduced with a utility function that was defined over combinations of states t T and actions a A, thus as U : T A R. Why does the utility function of the signaling same defined in session 7 also take the messages into consideration (U : T M A R)? there are additional message costs that diminishes the utility value the utility value does not only represent the communicative success, but is also influenced by the complexity of the expression used to communicate the content note: the message costs should be minute in comparison to the content that is communicated

7 Exercise 2: Game Modeling & Interpretation Exercise 2 What linguistic phenomenon does the milk-game represent? What is your interpretation of probability p in this game? the game represents a) a hypernym/hyponym relationship, or b) a communicative situation that might trigger a I-implicature probability p might represent a) the frequency of information state t cmk in comparison to t gmk, cf. revealed from corpora analysis/google hits..., or b) the strength of prototypicality of an information state (in a given culture) Pr a cmk a gmk m mk m cmk m gmk t cmk 0.8 1, 1 0, 0 - t gmk 0.2 0, 0 1, Tabelle : the milk game

8 Exercise 2: Game Modeling & Interpretation Exercise 2 How many possible sender strategies does the game have? Depict them. Which of them are appropriate to language use that considers semantic/literal meaning as determined by the denotation function? t cmk σ 1 m mk σ 2 m mk σ 3 m mk. m gmk. m gmk. m gmk t gmk m cmk t cmk t gmk m cmk t cmk t gmk m cmk t cmk σ 4 m mk σ 5 m mk σ 6 m mk. m gmk. m gmk. m gmk t gmk m cmk t cmk t gmk m cmk t cmk t gmk m cmk t cmk m cmk t cmk m cmk t cmk m cmk σ 7 m mk σ 8 m mk σ 9 m mk. m gmk. m gmk. m gmk t gmk t gmk t gmk

9 Exercise 2: Game Modeling & Interpretation Exercise 2 Model a signaling game for the scalar implicature some, all, whereby there are three messages possible: some, all, and some but not all. Pr a a m some m all m sbna t 0.5 1, 1 0, 0 - t 0.5 0, 0 1, Tabelle : the extended Some-All game

10 Exercise 2: Game Modeling & Interpretation Exercise 2 Given these entities: adult, boy, child, girl, human, man, woman How does the hyperonym/hyponym structure of these entities looks like (note: it forms a binary tree!) Human Adult Child. Woman Man Girl Boy

11 Exercise 2: Game Modeling & Interpretation Exercise 2 Model a game of this structure as follows: only the leaves of the binary tree can be member of set T and set A, but every entity can be a message of set M the utility function is defined as usual (1 if t matches a, else 0) the probability Pr(t) of being an adult is 4 times as high as being a child, whereas being male or female has the same probability the denotation function represents the structure of the binary tree the message costs C(m) are set to: 0.01 (number of syllables of message m) Pr a w a m a g a b m h m a m c m w m m m g m b t w 0.4 1, 1 0, 0 0, 0 0, t m 0.4 0, 0 1, 1 0, 0 0, t g 0.1 0, 0 0, 0 1, 1 0, t b 0.1 0, 0 0, 0 0, 0 1,

12 Exercise 3 Exercise 3 What is the Focal Meaning Assumption? Semantic meaning is focal in the sense that pragmatic deliberation to be identified as a sequence of best responses departs from semantic meaning as a psychological attraction point of interlocutors attention. Franke 2009, pp

13 Exercise 3 Exercise 3 What kind games reach a fix-point in the IBR-sequence, and what kind of games produce a cycle (of length>1) since the number of possible behavioral e strategies is countable for any game, there must be a recurrence of strategies at one point. Thus, each game produces a circle in the IBR-sequence. a fix point is a cycle of length 1 and will be reached if the strategy pair is a mutual best response, thus forms a Nash equilibrium over expected utilities therefore game with aligned interests where players manage to coordinate on produces a fix point, whereas games with non-aligned interests produce a cycle of length > 1 i.o.w. signaling games that represent an intentional violation of the maxim of quality do not produce a fix point!

14 Exercise 3 Exercise 3 Pr a ω τ a τ β a β m want m try m suc t ω τ 1/3 1, 1 0, 0 0, t τ β 1/3 0, 0 1, 1 0, 0 - t β 1/3 0, 0 0, 0 1, 1 Tabelle : want-try-succeed game t ω τ σ 0 m want ρ 1 a ω τ,t ω τ σ 2 m want t τ β m try a τ β,t τ β m try t β. m suc a β,t β m suc Note: ρ 3 = ρ 1 : σ 2, ρ 1 is a fix point of IBR

15 Exercise 3 Pr a cmk a gmk a ccmk a smk m mk m cmk m gmk m ccmk m smk t cmk 0.7 1, 1 0, 0 0, 0 0, t gmk 0.1 0, 0 1, 1 0, 0 0, t ccmk 0.1 0, 0 0, 0 1, 1 0, t smk 0.1 0, 0 0, 0 0, 0 1, Tabelle : the extended milk game t gmk t cmk t ccmk t smk σ 0 m gmk ρ 1 σ 2 m gmk a m gmk,t gmk a cmk m gmk,t gmk a m cmk,t cmk cmk a mk m cmk,t cmk m a ccmk,t mk ccmk ccmk m a ccmk,t ccmk m a smk,t ccmk. smk smk m a smk,t smk smk Note: ρ 3 = ρ 1 : σ 2, ρ 1 is a fix point of IBR ρ 3

16 What is Evolutionary Game Theory? mathematical theory of games applied in a biological context evolved from the point of view that frequency-dependent fitness gives a strategic aspect to evolution subsequent work also reconsiders non-biological (mostly cultural) evolution

17 Basic Concepts Evolutionary Game Theory: Basic Concept population of individuals (players, agents) individuals are (genetically) programmed for a specific behavior (strategy) individuals replicate and their strategy is inherited to offspring replication success (fitness) depends on the average utility of the strategy against the other strategies of the population (essence of game theory)

18 Basic Concepts Replicator Dynamics The replicator dynamics realizes a simple dynamics: a strategy that is better than average increases in proportion of population a strategy that is worse than average decreases in proportion of population note: since a strategie represent a hard-coded behavior, it can be interpreted as type/species/breed

19 Basic Concepts Replicator Dynamics Example 1: The better survives s A s B s A 1,1 1,1 s B 1,1 0,0 Tabelle : A- & B-pigeon Abbildung : replicator dynamics with mutation: proportion of A-pigeons p(s A ) in the population for different initial proportions

20 Basic Concepts Replicator Dynamics Example 2: The ecological equilibrium I s A s T s A 1,1 7,2 s T 2,7 3,3 Tabelle : Hawk & Dove Abbildung : replicator dynamics without mutation: proportion of eagles p(s A ) in the population for different initial populations

21 Basic Concepts Replicator Dynamics Example 3: The ecological equilibrium II s R s P s S s R 0,0-1,1 1,-1 s P 1,-1 0,0-1,1 s S -1,1 1,-1 0,0 Tabelle : Rock, Paper, Scissors Abbildung : replicator dynamics: proportion of Rock p(s R ) and Scissors p(s S )

22 Basic Concepts Outlook: Evolutionary Game Theory and Linguistics language change as an entity of cultural evolution linguistic items get reproduces in dependence of communicative success (fitness) idea: the signaling game is used as an decoding/encoding model for a specific linguistic domain is analyzed with the framework of EGT to explain stability aspect of different systems of that domain next session: evolutionary aspects of case marking systems (Jäger 2007)