Evolutionary Game Theory and Linguistic Typology: a Case Study

Gerhard Jager

CONSIDER all possible case marking types that only use case splits induced by the contrast between pronouns and full NPs

THE MAIN FUNCTION OF CASE MARKING is to disambiguate: enable the hearer to identify the argument roles A (agent, the transitive subject) and O (the direct object)

SPEAKER wants to disambiguate and use as few case morphemes as possible

SPEAKER STRATEGY:

any function S from M(eaning) to F(orm)

HEARER wants to correctly understand the utterance

HEARER STRATEGY:

any function H from F(orm) to M(eaning)

COMMUNICATION is successful if the hearer (H) recovers the intended meaning m from the observed form

SPEAKER UTILITY:

Here k formalizes the priorities of the speaker

HEARER UTILITY

WE ARE DEALING with elementary transitive clauses: two NPs, A(gent) and O(bject), both may be either p(ronoun) or n(oun)

NATURE chooses the word order A-O and O-A with a 50% probability each. Furthermore nature specifies which of the two NPs is A and which is O, and whether they are n or p

CASE MARKING patterns (speaker strategy):

1) Pronominal subjects

2) Non-pronominal subjects

3) Pronominal objects

4) Non-pronominal objects

EXAMPLES: zzaz (English), eezz (Basque)

16 case patterns

eeaz zzaz

eeza zzza

eezz zzzz

eeaa zzaa

zeaz ezaz

zeza ezza

zezz ezzz

zeaa ezaa

eeaz zzaz

eeza zzza

eezz zzzz

eeaa zzaa

zeaz ezaz

zeza ezza

zezz ezzz

zeaa ezaa

eeaz zzaz

eeza zzza

eezz zzzz

eeaa zzaa

zeaz ezaz

zeza ezza

zezz ezzz

zeaa ezaa

HEARER always interprets e -> A and a -> O

AMBIGIOUS clause types:

p/z – p/z

n/z – n/z

p/z – n/z

n/z – p/z

HEARER strategies only differ with respect to the meaning they assign to 3. and 4.

1. p/z – p/z (not relevant: the same payoff)

2. n/z – n/z (not relevant: the same payoff)

3. p/z – n/z

4. n/z – p/z

-> 4 possible strategies: AA AO OA OO

zzaz/AO zzaz/OA zzaz/OO zzaz/AA

zzza/AO zzza/OA zzza/OO zzza/AA

eezz/AO eezz/OA eezz/OO eezz/AA

zzzz/AO zzzz/OA zzzz/OO zzzz/AA

zzaa/AO zzaa/OA zzaa/OO zzaa/AA

zeaz/AO zeaz/OA zeaz/OO zeaz/AA

zezz/AO zezz/OA zezz/OO zezz/AA

ezzz/AO ezzz/OA ezzz/OO ezzz/AA

FOR small values of k

zzaz/AO zzaz/OA zzaz/OO zzaz/AA

zzza/AO zzza/OA zzza/OO zzza/AA

eezz/AO eezz/OA eezz/OO eezz/AA

zzzz/AO zzzz/OA zzzz/OO zzzz/AA

zzaa/AO zzaa/OA zzaa/OO zzaa/AA

zeaz/AO zeaz/OA zeaz/OO zeaz/AA

zezz/AO zezz/OA zezz/OO zezz/AA

ezzz/AO ezzz/OA ezzz/OO ezzz/AA

*strict NE *non-strict NE

FOR larger values of k

zzaz/AO zzaz/OA zzaz/OO zzaz/AA

zzza/AO zzza/OA zzza/OO zzza/AA

eezz/AO eezz/OA eezz/OO eezz/AA

zzzz/AO zzzz/OA zzzz/OO zzzz/AA

zzaa/AO zzaa/OA zzaa/OO zzaa/AA

zeaz/AO zeaz/OA zeaz/OO zeaz/AA

zezz/AO zezz/OA zezz/OO zezz/AA

ezzz/AO ezzz/OA ezzz/OO ezzz/AA

*strict NE

FOR larger values of k

zzaz/AO zzaz/OA zzaz/OO zzaz/AA

zzza/AO zzza/OA zzza/OO zzza/AA

eezz/AO eezz/OA eezz/OO eezz/AA

zzzz/AO zzzz/OA zzzz/OO zzzz/AA

zzaa/AO zzaa/OA zzaa/OO zzaa/AA

zeaz/AO zeaz/OA zeaz/OO zeaz/AA

zezz/AO zezz/OA zezz/OO zezz/AA

ezzz/AO ezzz/OA ezzz/OO ezzz/AA

*strict NE

FOR very large values of k

zzaz/AO zzaz/OA zzaz/OO zzaz/AA

zzza/AO zzza/OA zzza/OO zzza/AA

eezz/AO eezz/OA eezz/OO eezz/AA

zzzz/AO zzzz/OA zzzz/OO zzzz/AA

zzaa/AO zzaa/OA zzaa/OO zzaa/AA

zeaz/AO zeaz/OA zeaz/OO zeaz/AA

zezz/AO zezz/OA zezz/OO zezz/AA

ezzz/AO ezzz/OA ezzz/OO ezzz/AA

*strict NE

THE SYSTEMS that very excluded are typologically unattested or at least very rare

EIGHT SYSTEMS that give rise to an NE in some configuration are present

zzaz English zzza Nganasan (only!)

zezz Adyghe ezzz Wakhi (only!)

zeaz Dyrbal eezz Basque (rare)

zzzz Bantu zzaa (rare)

EVOLUTIONARY Game Theory (EGT): populations of players that are programmed for a certain strategy

PLAYERS replicate and pass on their strategy to their offsprings

But grammars are not transmitted via genetic but via cultural inheritance

replicator dynamics -> imitation dynamics

IMITATION DYNAMICS:

Players are not mortal and have no offsprings

The probability that a certain strategy is imitated is positively correlated to the gain in average utility -> successful strategies will tend to spread, unsuccessful will die out

A strategy pair is evolutionary stable

iff it is a Strict Nash Equilibrium

The NEs using a pure case marking strategy (zzaa, eezz) are never strict -> not stable

zzza/OA and ezzz/OA each coexist with a well-attested SNE, that have higher average utility

-> only four strategies are evolutionary stable: split ergativity, differential subject marking, differential object marking, and absence of case marking;

exceptions should be diachronicaly unstable