Evolutionary Game Theory and Linguistic
Typology: a Case Study
Gerhard J¨ager
Abstract
The paper deals with the typology of the case marking of semantic core roles.
The competing economy considerations of hearer (disambiguation) and speaker
(minimal effort) are formalized in terms of evolutionary game theory. It will be
shown that the case marking patterns that are attested in the languages of the
world are those that are evolutionary stable for different relative weightings of
speaker economy and hearer economy, given the statistical patterns of language
use that were extracted from corpora of naturally occurring conversations.
1 The frequencies of clause types
Consider all (logically) possible case marking types that only use case splits
induced by the contrast between pronouns and full NPs. I will restrict attention
to possible grammars where the morphological form of the intransitive
subject (nominative/absolutive) is less complex than ergative and accusative
(if present). Which language types are functional and which aren’t? The main
function of case marking is of course to disambiguate, i.e. to enable the hearer
to identify the semantic role of the denotation of an NP. More particular, case
should uniquely identify the argument roles “A” (agent, i.e. the transitive subject)
and “O” (the direct object). We can assume without loss of generality
that the hearer always interprets an ergative morpheme as A if there is one,
and likewise an accusative morpheme as O, so ambiguity can safely be avoided
if at least one NP per clause is case marked. For the sake of brevity, I will
denote case marking patterns from now on as a quadruple of case forms, in
the order: case of 1. pronominal agents, 2. non-pronominal agents, 3. pronominal
objects, and 4. non-pronominal objects. Ergative marking is abbreviated as
“e”, accusative as “a”, and zero marking (i.e. nominative/absolutive) as “z”.
For instance, a language like English where only pronominal objects are case
marked would thus follow the pattern zzaz, while a language like Basque with
obligatory ergative marking of all agents is eezz.
Ambiguity will only arise if a grammar admits clause types without any
case marking. However, this need not lead to ambiguity if one of the two unmarked
arguments is prominent and the other isn’t. Then the hearer may em-
Institute for Linguistics, University of Potsdam, e-mail: jaeger@ling.uni-potsdam.de
ploy a default rule to the effect that in such a case the more prominent NP is A
(or vice versa). This taken into account, the speaker strategies zeaz and ezza
also avoid ambiguity in the sense that there is a corresponding hearer strategy
that always correctly identifies semantic roles. One might assume that word order
is a good predictor of syntactic roles too, but even in languages with fixed
word order there may occur elliptical expressions which are, without the aid of
case morphology, ambiguous. Let us assume that disambiguation is the main
priority of the speaker, but he has the secondary priority to use as few case
morphemes as possible. It depends on the relative frequencies of clause types
which patterns minimize the average number of case morphemes per clause. We
only have to consider four clause types – both A and O may be p (pronominal)
or n (non-pronominal). The percentages in figure 1 are extracted from Geoffrey
Sampson’s CHRISTINE corpus of spoken English, and I took pronouns to be
prominent and full NPs to be non-prominent. The set of all clauses comprising
a subject and a direct object amount to 100%.
O/p O/n
A/p 19.70% 71.24%
A/n 1.59% 7.46%
Figure 1: clause
type frequencies
I will refer to the four cells of this table with
pairs pp, pn, np, and nn, where the first element
gives the specification of A and the second of O. The
concrete figures of course depend on the corpus under
investigation and the choice of the prominence
split. However, for the results reported below, the
only thing that matters is that pn > np, and this
inequality robustly holds for all corpora (including
both spoken and written corpora in English, German and Swedish) I investigated
and for all split points along the definiteness hierarchy or the animacy (to
the degree that the corpora investigated were annotated for animacy) hierarchy.
2 Game Theory
Game Theory is well-suited to make the possibly conflicting priorities of speakers
and hearers more precise. Let us assume that a fixed set of meanings M
and forms F is given. A speaker strategy is any function S from M to F, i.e.
a production grammar. Likewise, a hearer strategy is a comprehension grammar,
i.e. a function from F to M. In an utterance situation, the speaker has to
decide what to say and how to say it. Only the latter decision is a matter of
grammar; the decision about what meaning the speaker tries to communicate
is related to other cognitive domains. Let us thus assume that in each game,
nature presents the speaker with a meaning m, and the speaker only has to
choose how to express m. Communication is successful if the hearer recovers
the intended meaning from the observed form. It is measured by the -function:
m(S,H) =
(
1 iff H(S(m)) = m
0 else
Forms differ with respect to their complexity. I take it that the complexity can
be measured numerically, i.e. cost is a function from F to the non-negative real
numbers. The speaker has two possibly conflicting interests: he wants to communicate
the meaning as accurately as possible while simultaneously minimizing
the complexity of the form used. This is captured by the following definition of
speaker utility:
us(m, S,H) = m(S,H) − k × cost(S(m))
Here k is some positive coefficient that formalizes the priorities of the speaker.
A low value for k means that communicative success is more important than
minimal effort and vice versa. The hearer tries to recover the intended meaning
as accurately as possible. So the hearer utility can be identified with the -
function:
uh(m, S,H) = m(S,H)
Nature presents meanings to the speaker according to a certain probability
distribution x. The average utilities of speaker and hearer in a game can thus
be given as
us(S,H) =
X
m
xm × ( m(S,H) − k × cost(S(m)))
uh(S,H) =
X
m
xm × m(S,H)
We are only concerned with elementary transitive clauses. So we are dealing
with two NPs. One is A and the other O, and both may be either p or n. I am
not concerned with the effect of word order or head marking on argument linking
in this paper. Therefore I take it that nature chooses the word orders A − O
and O − A with a 50% probability each, and that this choice is stochastically
independent from the specifications of the NPs as p or n. Furthermore nature
specifies which of the two NPs is A and which is O, and whether they are n
or p. This gives a total of eight meanings. x is a probability distribution over
these eight meanings. It is plausible to assume that the prominence of an NP
is always unambiguously encoded in its form. This leaves us with 36 possible
forms — each of the two NPs may be p or n, and either one may be marked with
ergative, accusative, or zero case. The cost function simply counts the number
of case morphemes per clause.
I will restrict attention to just a small subset of simple strategies. First,
word order effects are kept out of considerations. Furthermore, I take it that
the case morphology of a given NP only depends on its own prominence value
and syntactic function, not on the prominence value of the other NP. Among
these strategies, I will restrict attention to those where the two marked forms
are reserved for one syntactic role each while the unmarked form is in principle
ambiguous between A and O. This leaves us, modulo renaming of e and a, with
16 case marking patterns, eeee, eeaz, eeza, · · · , zzza, zzzz. Of these 16 strategies,
6 are strictly dominated (i.e. they are never optimal, no matter what the
hearer does), namely those that sometimes use two case morphemes per clause,
and the inverse split ergative pattern ezza.
A hearer strategy is a mapping from forms to meanings. If ergative is only
used to mark A by the speaker and accusative only for O, it would obviously
be unreasonable by the hearer to interpret the case morphemes otherwise. I
will call the hearer strategies that interpret ergative as A and accusative as O
“faithful.” There are only 16 faithful strategies. Thus only the interpretation of
clauses without case morphology is undetermined. There are four such clause
types (depending on the prominence features of the two NPs), each of which
may receive two possible interpretations. If both NPs in a form f have the
same prominence value, both interpretation strategy classes have actually the
same expected payoff because by assumption, the speaker strategies exclude
correlations between word order and meaning, and the prominence values give
no clue. So we may safely identify any pair of hearer strategies which only differ
in their interpretation of p/z−p/z or n/z−n/z. Now we are down to four hearer
strategies — they differ with respect to the meaning they assign to p/z − n/z
and n/z − p/z. I will denote these strategies as AA, AO, OA and OO, where
the first component is the interpretation of the first NP in p/z − n/z, and the
second component the interpretation of the first component of n/z − p/z.
The configuration of Nash Equilibria (NEs henceforth) depends on the
value of k. For small values of k, the split ergative pattern zeaz/AO is a strict
NE (i.e. each component strategy is the unique best response to the opponent’s
strategy). Besides, each combination of a pure ergative (eezz) or pure accusative
(zzaa) speaker strategy with any hearer strategy 6= AO is a non-strict NE. For
larger values of k, two strict NEs coexist, either differential object marking
(zzaz/AO) and inverse differential subject marking (ezzz/OA), or differential
subject marking (zezz/AO) and inverse differential object marking (zzza/OA).
Finally, for very large values of k, the system without case marking zzzz/AO
is the unique (and hence also strict) NE.
Let us take stock. Of the sixteen case marking strategies that we considered,
only eight give rise to an NE in some configuration. The eight strategies
that were excluded are in fact typologically unattested or at least very rare.
There is apparently only one language with a full-blown tripartite system, i.e.
with the strategy eeaa, namely the Australian language Wangkumara. Inverse
split ergative systems — ezza in my notation — are also very rare. It is a bit
tricky to decide whether languages of the type zeaa or the like exist. There
are several split ergative languages where the split points for ergative and accusative
differ, and where there is an overlap in the middle of the hierarchy
with a tripartite paradigm. Since the system I use here implicitly assumes that
the two split points always coincide, such languages cannot really be accommodated;
they are a mixture of eeaz, zeaa and zeaz. To my knowledge, clearcut
instances of eeaz or zeaa do not exist, and the combinations ezaa and eeza
are unattested as well. There are no languages which would have a tripartite
paradigm for all and only the prominent or all and only the non-prominent NPs.
Hence zeza and ezaz are correctly excluded. So the concept of a Nash Equilibrium
proves fairly successful in identifying possible case marking systems.
Conversely, we expect to find instances of languages with an NE pattern. This
is certainly the case for zzaz (like English), zezz (for instance the Circassian
languages Adyghe and Kabardian), zeaz (like Dyirbal), and zzzz (as in several
Bantu languages). However, the concept is still too inclusive. I know of only
one language of the types zzza and ezzz each, namely Nganasan (see [3], p. 90)
as instance of the former and (according to [1]) Wakhi of the latter. The pure
accusative systems — eezz — do exists (Hungarian is an example), but they
are also very rare. Most accusative languages have DOM, and most ergative
languages DSM. Besides, the rationalistic approach has the same conceptual
problem as any functional explanation of grammatical patterns: natural languages
are not consciously designed, and it is a priori not clear at all why we
should expect to find functionally plausible patterns.
3 Evolutionary Game Theory
In Evolutionary Game Theory (EGT), we are dealing with populations of players
that are programmed for a certain strategy. Players replicate and pass on
their strategy to their offsprings. The number of offsprings is directly related to
the average payoff of the parent strategy.
How can this model be applied to linguistics? If the strategies in the EGT
sense are identified with grammars (as done in the previous section), games
should be identified with utterances. However, grammars are not transmitted
via genetic but via cultural inheritance. Therefore, imitation dynamics is
more appropriate here than the replicator dynamics that is used in applications
of EGT to theoretical biology. According to the imitation dynamics, players are
not mortal and have no offsprings. However, every so often, a player is offered
the opportunity to pick out some other player and to change his own strategy
against the strategy of the other player. The probability that a certain strategy
is adopted for imitation is positively correlated to the gain in average utility
that is to be expected by this strategy change. So here as well as in the standard
model, successful strategies will tend to spread while unsuccessful strategies die
out. Moreover, exactly the same strategies are evolutionary stable under the
replicator dynamics and under the imitation dynamics. Several sources of mutations
are conceivable here, ranging from plain speech errors to socio-linguistic
factors like language contact. We expect that most natural language grammars
are evolutionary stable because unstable grammars do not persist. The Game
of Case that was introduced in the last section is an asymmetric game. In
a population dynamic setting, this means that we are dealing with two separate
populations. So rather than with evolutionary stable strategies, we have
to deal with evolutionary stable strategy pairs here. In multi-population dynamics,
evolutionary stability can be characterized quite easily in rationalistic
terms. Briefly put, a strategy pair is evolutionary stable iff it is a Strict Nash
Equilibrium (SNE henceforth).
Let us apply the analytical tools of EGT to the different instantiations of
the Game of Case. The NEs using a pure case marking strategy (eezz or zzaa)
are never strict and thus not evolutionary stable. The remaining 6 NEs are
strict though. Of these 6 strategy pairs, two are very rare among the languages
of the world: zzza/OA and ezzz/OA. Put differently, it is important to note
that these two “wrong” SNE each coexist with a well-attested SNE, namely
inverse differential subject marking (ezzz/OA) with differential object marking
(zzaz/AO), and inverse differential object marking zzza/OA with differential
subject marking zezz/AO. In both scenarios, the typologically attested SNE is
Pareto-optimal, i.e. it has a higher average utility than the competing SNE.
The standard approach to EGT assumes that populations are infinite.
If we assume instead that the populations are finite but large, every invasion
barrier is occasionally broken, no matter how low the mutation rate is. (With
increasing population size, the likelihood of such an incident converges towards
0.) It can be shown that the Pareto-optimal SNE always has a higher invasion
barrier than the other SNE. In a finite population, it is thus more probable to
switch to than from the Pareto-optimal SNE. In a population of finite populations,
the unique attractor state is the one where the majority of population
is in the Pareto-optimal SNE, and as the size of the single finite populations
increases, the probability of the non-Pareto-optimal SNE converges towards 0.
(See [4] for a similar explanation of the asymmetry between multiple evolutionary
stable strategies.)
To sum up, under the assumption of a population of finite but large
populations of speakers/hearers, only four strategies are evolutionary stable:
split ergativity, differential subject marking, differential object marking, and
absence of case marking. This fits the typological findings rather well. While
the majority of languages is in an evolutionary stable state, there are some exceptions.
Evolutionary Game Theory predicts that such language types should
be diachronically unstable. This is an empirically testable claim that should be
tackled in future research.
References
[1] Elena Bashir. Beyond split-ergativity: Subject marking in Wakhi. In Papers
from the 22nd regional meeting of the Chicago Linguistic Society, pages 14–
35. CLS, Chicago, 1986.
[2] Barry J. Blake. Case. Cambridge University Press, 2001.
[3] R. M. W. Dixon. Ergativity. Cambridge University Press, Cambridge, 1994.
[4] Robert van Rooy. Signaling games select Horn strategies. manuscript, University
of Amsterdam, 2002.
from http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.94.8693&rep=rep1&type=pdf
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