Game Theory for Linguists

Transcription

1 Fritz Hamm, Roland Mühlenbernd 4. Juli 2016

2 What is? 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

3 Basic Concepts : 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)

4 Basic Concepts EGT-Setting and Replicator Dynamics Given: a large (practically infinite) population P of agents, which play pairwise a game G = S, U against each other, whereby: S = {s 1, s 2,..., s n} a set of strategies s i U : S S R a utility function over strategy pairs Further definition: p(s i ): proportion of individuals that play s i (p (S)) EU(s i ) = s j S p(s j)u(s i, s j ): expected utility (fitness) for playing s i AU = s i S p(s i)eu(s i ): average utility value of the whole population replicator dynamics: the replicator dynamics is defined by the following differential equation: dp(s i ) dt = p(s i )[EU(s i ) AU]

5 Basic Concepts Replicator Dynamics The replicator dynamics dp(s i ) dt = p(s i )[EU(s i ) AU] 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

6 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

7 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

8 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 )

9 Evolutionary Stable Strategy Evolutionary Stable Strategy Given a game G = S, U. A strategy s i S is evolutionary stable, iff the following wto conditions are fulfilled: 1. U(s i, s i ) U(s i, s j ) for all s j S \ {s i } 2. If U(s i, s i ) = U(s i, s j ) for some s j, then U(s i, s j ) > U(s j, s j ) Compare the following definitions: Nash Equilibrium: U(s i, s i ) U(s i, s j ) for all s j S \ {s i } Strict NE: U(s i, s i ) > U(s i, s j ) for all s j S \ {s i } An ESS has the following properties: SNE ESS NE it has an invasion barrier

10 Evolutionary Stable Strategy Evolutionary Stable Strategy An evolutionary stable strategy is an important concept under evolutionary dynamics, since it has the following properties: an ESS has an invasion barrier as far as a ESS has conquered a sufficient large part of the population, it cannot invaded by a proportional small number of another strategy Thus a population with a ESS is protected against invaders or mutants

11 Evolutionary Stable Strategy Evolutionary Stable Strategy Given is a game G = S, U. A strategy s i S is evolutionary stable, iff the following two conditions are fulfilled: 1. U(s i, s i ) U(s i, s j ) for all s j S \ {s i } 2. If U(s i, s i ) = U(s i, s j ) for some s j, then U(s i, s j ) > U(s j, s j ) Find the ESS: s A s B s A 1,1 1,1 s B 1,1 0,0 Tabelle: A- & B-pigeons s H s D s H 1,1 7,2 s D 2,7 3,3 Tabelle: hawks & doves s C s D s C 3,3 0,5 s D 5,0 1,1 Tabelle: P-Dilemma

12 Attraction Points & Equilibria Attraction Points & Equilibria Further important properties of population states: attraction point (basin of attraction): a state to which the system falls back by small deviation equilibrium: a state in which the system stays stable (without mutation) unstable equilibrium: a equilibrium that will be left by the smallest deviation evolutionary stable state: equilibrium and attraction point at the same time (stable equilibrium) Examples: states and dynamics A- & B-pigeon hawk & dove Note: pure ESS evolutionary stable state of a homogeneous population, mixed ESS one of a heterogeneous population

13 Attraction Points & Equilibria Attraction Points & Equilibria 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 Abbildung: Dynamics of the RPS-Spiel

14 Attraction Points & Equilibria Attraction Points & Equilibria Question: which aggregate dynamics belongs to which game? A L 1 L 2 L 3 L L L B L 1 L 2 L 3 L L L C L 1 L 2 L 3 L L L Spiel A - Dynamik 2; Spiel B - Dynamik 3; Spiel C - Dynamik 1

15 Attraction Points & Equilibria Summary 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 application: evolutionary aspects of case marking systems (Jäger 2007)

16 Introduction Introduction Among all logically conceivable case-marking systems, a surprisingly small number occurs very often among the languages of the world: 1. accusative systems with differential object marking 2. ergative systems with differential subject marking 3. combination of both the frequent systems are assumed to be well-adapted to their function in language use functional adaptation might be result of shaping by the forces of evolution

17 Introduction Case Marking Systems We consider the roles S (subject of intransitive sentence), A (agent, subject of transitive sentence) and O (object) Case marking helps the hearer to identify syntactic functions of an NP: A (agent) or O (object) Note there are four possible binary splits: 1. {S, A, O} vs. (absence of case distinction) 2. {S, A} vs. {O} (nominative/accusative) 3. {S, O} vs. {A} (ergative/absolutive) 4. {A, O} vs. {S} (virtually non-existent) there are tripatite splits, but very rare

18 Introduction Differential Case Marking Differential object marking (DOM): all NP s from the top section of the definiteness hierarchy are case marked: 1 personal pronoun > proper noun > definite full NP > indefinite specific NP > non-specific indefinite NP Diffrential subject marking (DSM): NP s from the lower section of the hierarchy are case marked split ergative: DOM and DSM co-occur in one language 1 Some languages consider a animacy hierarchy: human > animate > inanimate

19 Introduction Differential Case Marking NP s can be characterized as prominent p or nonprominent n (we assume pronouns VS. full NP s) there are four possible clause types: we get 9 plausible case marking strategies then A/p e e z z e e e z e A/n e z e z e e e e z O/p a a a a a z z a z O/n a a a a z a z z a Which strategies use minimal case morphology?

20 Introduction Differential Case Marking strategies with minimal case morphology 1. zzaa: O/p a, O/n a (1.0) 2. eezz: A/p e, A/n e (1.0) 3. zeaz: A/n e, O/p a (0.3) 4. ezza: A/p e, O/n a (1.7) Note: frequency of clause types also plays a role Abbildung: Data from CHRISTINE corpus of spoken English Which system minimizes the average/expected number of case morphemes per clause? the Split ergative system turns out to be optimal, since pn >> np

21 Signaling Game for Case Marking Signaling Game In last section the notion of optimality was defined in an ad hoc fashion Let s use Game theory to make things more precise The game G = {(S, H), M, F, P, C, U S, U H } is a situation between speaker and hearer, whereby M is a set of meanings F is a set of forms S : m f is a speaker strategy H : f m is a hearer strategy U S : S H R is the speaker s utility function U H : S H R is the hearer s utility function more precise: { 1 iff H(S(m)) = m δ m(s, H) = 0 else U S (S, H) = m P(m) (δm(s, H) k cost(s(m)))2 U H (S, H) = m P(m) δm(s, H) 2 k represents the speaker s laziness: importance of successful communication VS effort of marking

22 Signaling Game for Case Marking Signaling Game & Case Marking Let s apply the game on the case marking problem: M = {A/p-O/p, A/p-O/n, A/n-O/p, A/n-O/n, O/p-A/p, O/p-A/n, O/n-A/p, O/n-A/n} F: tuple of a, e and z tuple of p and n probabilities P(m) are computed by corpus frequencies costs C(f ) are defined by number of markings S contains 10 plausible coding strategies: eezz, zzaa, ezaz, zeza, zeaz, ezzz, zezz, zzaz, zzza, zzzz H standard strategy: interpret e as agent and a as object But how to deal with a unmarked form zz? This can be reduced to 4 encoding strategies: AO: agent before object (default) OA: object before agent pa: if default in doubt, more prominent is agent po: if default in doubt, more prominent is object

23 Signaling Game for Case Marking Signaling Game & Case Marking Let s play a round: Nature chooses: A/p-O/n Sender plays strategy: zeza What form does the sender actually send? za pn Hearer plays standard strategy and interprets: A/p-O/n Let s play another round: Nature chooses: A/p-O/n Sender plays strategy: zezz What form does the sender actually send? zz pn Hearer plays po What meaning does the receiver actually construe? O/p-A/n

24 Signaling Game for Case Marking Asymmetric Game & Case Marking to fit to evolutionary dynamics we will abstract from the sequential nature of the game the game can be expressed as static by reconsidering expected utilities: Abbildung: The static asymmetric game of case marking for k =.1

25 Evolutionary Analysis most natural language grammars are expected to be evolutionary stable (EGT) can help to analyze the stability of systems EGT is not to explain or predict language change ergo this approach is relevant to typology rather than to historical linguistics

26 Evolutionary Analysis The tools: since we have a asymmetric game, we consider a two-population model (speaker & hearer) replicator dynamics: speaker: d dt p(s i) = p(s i ) ( j p(h j) u(s i, H j ) AU) hearer: d dt p(h i) = p(h i ) ( j p(s j) u(s j, H ) AU) an evolutionary stable state for asymmetric games is defined as follows: Theorem 1 (Selten 1980): (S, H) is evolutionary stable if and only if it is a strict Nash Equilibrium.

27 Evolutionary Analysis Asymmetric Game & Case Marking What are the evolutionary stable states? the split ergative pattern zeaz/pa note that e.g. zzaa/so (accusative marking + SO word order) is a Nash equilibrium, but not evolutionary stable Abbildung: The static asymmetric game of case marking for k =.1

28 Evolutionary Analysis Asymmetric Game & Case Marking Let s analyze the following sub-game: SO pa zzaa zeaz zzaa/so is an attractor state, but not an ESS

29 Evolutionary Analysis Asymmetric Game & Case Marking Abbildung: The static asymmetric game of case marking for k =.1 the only ESS is split ergative zeaz/pa, that is very common among Australian aborigenes languages note that the values are influenced by the value k What will happen if we change this value?

30 Evolutionary Analysis Asymmetric Game & Case Marking What are the evolutionary stable states for k = 0.45? Abbildung: The static asymmetric game of case marking for k =.45 the differential object marking (DOM) pattern zzaz/pa; e.g. English, Dutch the inverse differential subject marking (inverse DSM) ezzz/po; e.g. Wahki (Iranian, Northern Pakistan)

31 Evolutionary Analysis Asymmetric Game & Case Marking What are the evolutionary stable states for k = 0.55? Abbildung: The static asymmetric game of case marking for k =.55 the differential subject marking (DSM) pattern zezz/pa; e.g. several caucasian languages the inverse differential object marking (inverse DOM) zzza/po; e.g. Nganasan (North Siberia)

32 Evolutionary Analysis Asymmetric Game & Case Marking What are the evolutionary stable states for k = 1? Abbildung: The static asymmetric game of case marking for k = 1 no case marking zzzz/pa; e.g. Bantu languages (south to middle Africa)

33 Evolutionary Analysis Results only very few languages are not evolutionarily stable in this sense: zzaa: Hungarian ezza: Arrernte (Central Australia) eeaa: Wangkumara (Australia, already extinct) curious asymmetry: if there are two competing stable states, one is common and the other one rare it can be shown: the rare ones are not stochastically stable states (temporal information) the only stochastically stable strategies: split ergative zeaz differential object marking zzaz differential subject marking zezz no case marking zzzz

34 Evolutionary Analysis Results the configuration of evolutionary strategies depend on the case marking costs k and the split point between prominent and non-prominent arguments a low split point makes pp > nn more probable, a high split point nn > pp

35 Evolutionary Analysis Conclusion out of 4 16 = 64 possible case marking patterns only four are stochastically stable vast majority of all languages that fit into this categorization are stochastically stable linguistic universals need not be based on innate language instinct but can be result of evolutionary pressure in the sense of cultural evolution