The grammaticalization cycle of the progressive

A game-theoretic analysis


A well-attested phenomenon in morpho-semantic change is known as the progressive cycle, which depicts a directed and cyclic pathway of a grammatical progressive marker through its emergence and disappearance inside the imperfective domain. Deo (2015) offers a model within the framework of evolutionary game theory to study the evolutionary dynamics of four preselected types of progressive-imperfective grammars. Based on her basic game-theoretic model, we investigate which types of grammars would emerge from the first principles in a population of agents under reinforcement learning. In our computational model, the actual progressive-imperfective cycle can be reconstructed from such atomic interactions between learner agents after the addition of several simple assumptions to the basic game-theoretic model.

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  1. 1.

    The Swahili-specific glossing abbreviations are taken from Londfors (2003): NPX = Noun prefix, NC = Noun class, IND = Indicative.

  2. 2.

    Here a fast shift means that it takes a short time in comparison to the time a system stays in a stable state.

  3. 3.

    Note that these labels of forms and meanings differ from Deo’s Imperfective Game, which, however, does not change the structure of the game.

  4. 4.

    A contextual cue is a variable for any kind of additional information helping to suggest one reading apart from the form itself. This might be additional linguistic material, the type of the verb itself, or the situation of conversation. The model here abstracts from the concrete materialization of the cue.

  5. 5.

    Note that since the communicative success between speaker and hearer is context-independent, the δ-function can easily be applied for the context-related strategies by abstracting from contexts: \(\delta_{m}(\mathcal{S},\mathcal{H}) = 1\), iff \(\mathcal{H}(\mathcal{S}(m,c),c') = m\) for any \(c,c' \in C\). The utility functions \(U_{s}\) and \(U_{h}\) can be defined in the same way for context-related strategies.

  6. 6.

    Deo (2015) calls this system partially context dependent (pcd), due to the fact that the contextual cue \(c_{p}\) is still helpful for disambiguation, whereas the contextual cue \(c_{s}\) is not needed anymore, since here both meanings are disambiguated by both forms.

  7. 7.

    Admittedly, we have no proof that languages with an optional progressive marker actually use the progressive form according to the strategy outlined here, thus we are not aware of any study that analyzes in what contexts a speaker of a language with an optional progressive marker actually uses the progressive form. We believe that this gap in research is due to the fact that it is not easy to judge if a given context is more likely to license structural or phenomenal readings. Note that the contextual cues of our model are theoretical constructs for encompassing a complex mixture of all possible external linguistic and extra-linguistic cues licensing such a reading. But nevertheless, the OP system as defined in our model follows a particular line of thought: let us assume that we have a ZP language that uses solely contextual cues to disambiguate structural and phenomenal reading inside the imperfective domain. And then a new form appears that is used more and more frequently to phenomenal readings, but optionally next to the old form (ZP → OP shift). When would it be most useful to apply this new form? Admittedly, in situations with contextual cues that is more likely not to license a phenomenal reading, since in those situations with contextual cues that are likely to actually license a phenomenal reading, it is not necessary to use the new form: since the contextual cue helps to disambiguate successfully. Furthermore, it is known from several studies that emerging forms of a grammaticalization process are considered as marked forms (a good example is the German ‘am-Progressive’, which appears to be highly marked and barely considered as a grammatical form of Standard German). Such marked forms are generally used to express a non-prototypical meaning (note that this strategy follows a more general principle in pragmatics and language use: ‘Division of Pragmatic Labor’ (cf. Horn 1984)), and the non-prototypical case in a situation with a contextual cue that licenses a structural reading is the phenomenal meaning.

  8. 8.

    Admittedly, this is not the case for all types of games. But particularly for signaling games multiple studies exhibit that replicator dynamics and reinforcement learning approximate in the long run for diverse configurations, such as game parameters or learning parameters (cf. Barrett 2006; Argiento et al. 2009; Skyrms 2010; Huttegger and Zollman 2011; Mühlenbernd 2013).

  9. 9.

    This condition ensures that there was no change for the last 10 generations; this indicates that an evolutionary stable state is reached.

  10. 10.

    Note that strategies that involve pooling – e.g. speaker strategies that assign the same form to multiple meanings, or hearer strategies that assign the same meaning to multiple forms – are accordingly called pooling strategies.

  11. 11.

    The result is in line with Huttegger (2007), who showed that in binary signaling games, where states are not equiprobable, the pooling strategies have a positive basin of attraction. Furthermore, Enke et al. (2016), showed that in a setting where states are equiprobable, cf. with a cue probability \(c_{p} = 0.5\), all agents learn perfect signaling for both contextual cues.

  12. 12.

    That \(S_{2}\) and \(S_{10}\) emerge more often that \(S_{6}\) and \(S_{14}\) is due to the fact that the allocation \(m_{s} \rightarrow f_{old}\) is biased by being the most common one for context \(c_{s}\).

  13. 13.

    Additional tests showed that for any reduction of contextual cues above 7% a categorical system eventually emerged.

  14. 14.

    For example, both William Shakespeare (1564–1616) and the Irish novelist Laurence Sterne (1713–1768) used OP.

  15. 15.

    Note that for full cue access, as in Experiment I, the optional progressive system \(\langle S_{2}, H_{1} \rangle\) emerged at least for a part of the population.

  16. 16.

    The α-parameter is already used by Deo (2015) who references to Jäger (2007), who used a similar model for analyzing case marking systems with evolutionary game theory. Jäger interprets the α-parameter in terms of speakers priorities: how highly the speaker values linguistic clarity over signal costs. In the given model this parameter can then be interpreted as follows: when α is low then disambiguation by two explicit forms is highly valued, because there are no other means for disambiguation in that language, whereas when α is high, disambiguation by two explicit forms is not highly valued, because there is stronger support for disambiguation by other means. It can be assumed that due to language change the support of such ‘other means’ can vary, and so does the α-parameter in Experiment III. Note furthermore that the α-parameter is not defined for single agents, but a global parameter. In this way it represents changes in the linguistic system as a global construct. Admittedly, more realistic models might consider breaking the α-parameter down to an individual feature of agents which might be part of horizontal and vertical transmission. But we chose to abstract from that, especially since in a complete network structure (such as the one we use in our model) we highly expect individual α-parameters to align and therefore eventually behave as a global value.

  17. 17.

    Note that each simulation run eventually reached a population-wide one form system due to two factors: (i) when the alternating cost parameter exceeds a particular threshold, it makes a one form system more attractive than a two form system; and (ii) the input asymmetry – here implemented as childhood asymmetry – increases the total average probability \(P_{C}(c_{p})\) (note that without input asymmetry it was consistently very small: \(P_{C}(c_{p}) = 0.05\)), and therefore mitigates the difference between \(P_{C}(c_{p})\) and \(P_{C}(c_{s})\). As further experiments showed: such a mitigation supports the emergence of a homogeneous population where only one of both one form systems is used, contrasting with the result of Experiment III, where a mixed population emerged where both one form systems are used.

  18. 18.

    Note that the progressive cycle cannot always be reconstructed, cf. Fig. 11.


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Correspondence to Roland Mühlenbernd.

Additional information

We express our thanks to the audiences of the Tübingen, Munich, and Stanford seminars, of the 17th International Morphology Meeting (Vienna), PLC 40 (Philadelphia), Evolang XI (New Orleans), Germanic Linguistics Roundtable (Berkeley), Diskussionsforum Linguistik (Zurich), and SLE 49 (Naples). Special thanks are owed to Christopher Ahern, Phillip Alday, Sara Sanchez Alonso, Christian Bentz, Heather Burnett, Ashwini Deo, Dieter Gunkel, Olav Hackstein, Gerhard Jäger, Guido Seiler, Thilo Weber, and Igor Yanovich. Any errors that remain are our own responsibility. We gratefully acknowledge the support of the ERC under project EVOLAEMP, (RM), the Department of Linguistics at Eberhard Karls University, Tübingen (RM) and the Institute of German Philology of LMU Munich (DE).

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Mühlenbernd, R., Enke, D. The grammaticalization cycle of the progressive. Morphology 27, 497–526 (2017).

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  • Morpho-semantic change
  • Progressive cycle
  • Evolutionary dynamics
  • Game theory
  • Reinforcement learning