Game Theory for Linguists - uni-tuebingen.deroland/GTL16/folien/GTL09.pdfExercises II. . . . . . Introduction to Evolutionary Game Theory Game Theory for Linguists Fritz Hamm, Roland

. . . . . . . . . . . . .Exercises II

. . . . . .Introduction to Evolutionary Game Theory

Game Theory for Linguists

Fritz Hamm, Roland Mhlenbernd

27. Juni 2016


. . . . . . . . . . . . .Exercises II


Overview

▶ Exercises II▶ Introduction to Evolutionary Game Theory


. . . . . . . . . . . . .Exercises II


Exercise 1: Signaling Game Properties

Exercise 1

The introduced type of a signaling game has a probabilityfunction and a denotation function. How are these functionsdefined 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


. . . . . . . . . . . . .Exercises II



Exercise 1

What is the fundamental difference between the Wine-Choiceand the Some-All scenario in respect of the way both scenariosare modeled as a game?

▶ both have the same number of states, messages andactions and the same utility tables

▶ but there is a fundamental difference in the way thedenotation function is given

▶ Wine-Choice: ∥mbeef∥ = {tbeef}, ∥mfish∥ = {tfish}▶ Some-All: ∥mall∥ = {t∀}, ∥msome∥ = {t∀, t∃¬∀}▶ only the Some-All scenario has a message with a

semantic/literal meaning encompassing more than oneinformation state


. . . . . . . . . . . . .Exercises II



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 yx pure sender strategies and zy pure receiver

strategies▶ and therefore yx × zy pure strategy combinations▶ thus both games have 22 = 4 sender strategies and 22 = 4

receiver strategies and 22 × 22 = 16 pure strategycombinations


. . . . . . . . . . . . .Exercises II



Exercise 1In session 4 the signaling games was introduced with a utilityfunction that was defined over combinations of states t ∈ T andactions a ∈ A, thus as U : T × A → R. Why does the utilityfunction of the signaling same defined in session 7 also takethe messages into consideration (U : T × M × A → R)?

▶ there are additional message costs that diminishes theutility value

▶ the utility value does not only represent the communicativesuccess, but is also influenced by the complexity of theexpression used to communicate the content

▶ note: the message costs should be minute in comparisonto the content that is communicated


. . . . . . . . . . . . .Exercises II


Exercise 2: Game Modeling & Interpretation

Exercise 2What 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 informationstate tcmk in comparison to tgmk , cf. revealed from corporaanalysis/google hits..., or b) the strength of prototypicality of aninformation state (in a given culture)

Pr acmk agmk mmk mcmk mgmk

tcmk 0.8 1, 1 0, 0√ √

-tgmk 0.2 0, 0 1, 1

√-

√

0.01 0.02 0.02

Tabelle : the milk game


. . . . . . . . . . . . .Exercises II



Exercise 2How many possible sender strategies does the game have? Depictthem. Which of them are appropriate to language use that considerssemantic/literal meaning as determined by the denotation function?

..

σ1

.

tcmk

.tgmk.

mcmk

.mmk

. mgmk ..

σ2

.

√

.

tcmk

.tgmk.

mcmk

.mmk

. mgmk ..

σ3

.

√

.

tcmk

.tgmk.

mcmk

.mmk

. mgmk

..

σ4

.

tcmk

.tgmk.

mcmk

.mmk

. mgmk ..

σ5

.

√

.

tcmk

.tgmk.

mcmk

.mmk

. mgmk ..

σ6

.

√

.

tcmk

.tgmk.

mcmk

.mmk

. mgmk

..σ7

.

tcmk

.tgmk.

mcmk

.mmk

. mgmk ..

σ8

.

tcmk

.tgmk.

mcmk

.mmk

. mgmk ..

σ9

.

tcmk

.tgmk.

mcmk

.mmk

. mgmk


. . . . . . . . . . . . .Exercises II



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∃¬∀ msome mall msbnat∀ 0.5 1,1 0,0

√ √-

t∃¬∀ 0.5 0,0 1,1√

-√

0.01 0.01 0.04

Tabelle : the extended Some-All game


. . . . . . . . . . . . .Exercises II



Exercise 2

Given these entities: ⟨ adult, boy, child, girl, human, man, woman ⟩How does the hyperonym/hyponym structure of these entities lookslike (note: it forms a binary tree!)

..

Human

.

Adult

.

Child

. Woman. Man. Girl. Boy


. . . . . . . . . . . . .Exercises II



Exercise 2Model a game of this structure as follows:

▶ only the leaves of the binary tree can be member of set T and set A, but everyentity 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 aw am ag ab mh ma mc mw mm mg mbtw 0.4 1, 1 0, 0 0, 0 0, 0

√ √-

√- - -

tm 0.4 0, 0 1, 1 0, 0 0, 0√ √

- -√

- -tg 0.1 0, 0 0, 0 1, 1 0, 0

√-

√- -

√-

tb 0.1 0, 0 0, 0 0, 0 1, 1√

-√

- - -√

0.02 0.02 0.01 0.02 0.01 0.01 0.01


. . . . . . . . . . . . .Exercises II


Exercise 3

Exercise 3

What is the Focal Meaning Assumption?

“Semantic meaning is focal in the sense thatpragmatic deliberation – to be identified as asequence of best responses – departs from semanticmeaning as a psychological attraction point ofinterlocutors’ attention.”

Franke 2009, pp. 47–48


. . . . . . . . . . . . .Exercises II


Exercise 3

Exercise 3What kind games reach a fix-point in the IBR-sequence, andwhat kind of games produce a cycle (of length>1)

▶ since the number of possible behavioral ∆e strategies iscountable for any game, there must be a recurrence of strategiesat one point. Thus, each game produces a circle in theIBR-sequence.

▶ a fix point is a cycle of length 1 and will be reached if thestrategy pair is a mutual best response, thus forms a Nashequilibrium over expected utilities

▶ therefore game with aligned interests – where players manageto coordinate on – produces a fix point, whereas games withnon-aligned interests produce a cycle of length > 1

▶ i.o.w. signaling games that represent an intentional violation ofthe maxim of quality do not produce a fix point!


. . . . . . . . . . . . .Exercises II


Exercise 3

Exercise 3Pr aω¬τ aτ¬β aβ mwant mtry msuc

tω¬τ 1/3 1,1 0,0 0,0√

- -tτ¬β 1/3 0,0 1,1 0,0

√ √-

tβ 1/3 0,0 0,0 1,1√ √ √

Tabelle : want-try-succeed game

..

σ0

.

tω¬τ

.

tτ¬β

.tβ.

mwant

.

mtry

. msuc.

ρ1

.

aω¬τ ,tω¬τ

.

aτ¬β,tτ¬β

. aβ,tβ.

σ2

.

mwant

.

mtry

. msuc

▶ Note: ρ3 = ρ1: ⟨σ2, ρ1⟩ is a fix point of IBR


. . . . . . . . . . . . .Exercises II


Exercise 3

Pr acmk agmk accmk asmk mmk mcmk mgmk mccmk msmktcmk 0.7 1,1 0,0 0,0 0,0

√ √- - -

tgmk 0.1 0,0 1,1 0,0 0,0√

-√

- -tccmk 0.1 0,0 0,0 1,1 0,0

√- -

√-

tsmk 0.1 0,0 0,0 0,0 1,1√

- - -√

0.01 0.02 0.02 0.02 0.02

Tabelle : the extended milk game

..

σ0

.

tgmk

.

tcmk

.tccmk

.tsmk.

mgmk

.

mcmk

.

mmk

.mccmk

. msmk.

ρ1

.

agmk ,tgmk

.

acmk ,tcmk

.accmk ,tccmk

. asmk ,tsmk.

σ2

.

mgmk

.

mcmk

.

mmk

.mccmk

. msmk.

ρ3

.

agmk ,tgmk

.

acmk ,tcmk

.accmk ,tccmk

. asmk ,tsmk

▶ Note: ρ3 = ρ1: ⟨σ2, ρ1⟩ is a fix point of IBR


. . . . . . . . . . . . .Exercises II


What is Evolutionary Game Theory?

▶ mathematical theory of games applied in a biologicalcontext

▶ evolved from the point of view that frequency-dependentfitness gives a strategic aspect to evolution

▶ subsequent work also reconsiders ’non-biological’ (mostlycultural) evolution


. . . . . . . . . . . . .Exercises II


Basic Concepts

Evolutionary Game Theory: Basic Concept▶ population of individuals (players,

agents)

▶ individuals are (genetically)programmed for a specific behavior(strategy)

▶ individuals replicate and theirstrategy is inherited to offspring

▶ replication success (fitness)depends on the average utility ofthe strategy against the otherstrategies of the population(essence of game theory)


. . . . . . . . . . . . .Exercises II


Basic Concepts

Replicator Dynamics

The replicator dynamics realizes a simple dynamics:

▶ a strategy that is better than average increases inproportion of population

▶ a strategy that is worse than average decreases inproportion of population

▶ note: since a strategie represent a hard-coded behavior, itcan be interpreted as type/species/breed


. . . . . . . . . . . . .Exercises II


Basic Concepts

Replicator DynamicsExample 1: The better survives

sA sBsA 1,1 1,1sB 1,1 0,0

Tabelle : A- & B-pigeon

Abbildung : replicator dynamics with mutation:proportion of A-pigeons p(sA) in the populationfor different initial proportions


. . . . . . . . . . . . .Exercises II


Basic Concepts

Replicator DynamicsExample 2: The ecological equilibrium I

sA sTsA 1,1 7,2sT 2,7 3,3

Tabelle : Hawk & Dove

Abbildung : replicator dynamics withoutmutation: proportion of eagles p(sA) in thepopulation for different initial populations


. . . . . . . . . . . . .Exercises II


Basic Concepts

Replicator DynamicsExample 3: The ecological equilibrium II

sR sP sSsR 0,0 -1,1 1,-1sP 1,-1 0,0 -1,1sS -1,1 1,-1 0,0

Tabelle : Rock, Paper, Scissors

Abbildung : replicator dynamics:proportion of Rock p(sR) andScissors p(sS)


. . . . . . . . . . . . .Exercises II


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 specificlinguistic domain

▶ is analyzed with the framework of EGT to explain stabilityaspect of different systems of that domain

▶ next session: evolutionary aspects of case markingsystems (Jäger 2007)


Documents

Game Theory for Linguists - uni-tuebingen.deroland/GTL16/folien/GTL09.pdfExercises II. . . . . . Introduction to Evolutionary Game Theory Game Theory for Linguists Fritz Hamm, Roland