Abstract
This paper is concerned with how a simple metalanguage might coevolve with a simple descriptive base language in the context of interacting Skyrms–Lewis signaling games Lewis (Barrett and Skyrms in Br J Philos Sci, 2015; Lewis in Convention, Harvard University Press, Cambridge, 1969; Skyrms in Signals evolution, learning, & information, Oxford University Press, New York, 2010). We will first consider a metagame that evolves to track the successful and unsuccessful use of a coevolving base language, then we will consider a metagame that evolves a truth predicate for expressions in a coevolving base language. We will see how a metagame that tracks truth provides an endogenous way to break the symmetry between indicative and imperative interpretations of the base language. Finally, we will consider how composite signaling games provide a way to characterize alternative pragmatic notions of truth.
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Notes
See Barrett and Skyrms (2015) for a discussion of how simpler signaling games may combine to form more complex games by cue-reading, template transfer, and modular composition.
See Herrnstein (1970) for a discussion of simple reinforcement learning and Roth and Erev (1995), Erev and Roth (1998) and Huttegger et al. (2014) for discussions of more subtle forms of reinforcement learning and other options. In the case of the simplest varieties of reinforcement learning, one might imagine the agents learning by adjusting the contents of urns on the basis of their experience as described here.
See Skyrms (2010) for a discussion of the precise sense in which the evolved signals communicate information.
See Barrett (2006) for further details regarding the behavior of this and closely related signaling games.
Again, see Skyrms (2010) for a characterization of how one might understand information transfer in the context of signaling games.
The composite system might be taken to model agents observing the evolving language use of other other agents or agents observing their own evolving language use. See Barrett and Skyrms (2015) for a discussion of such composite systems in nature and how such complex games might self-assemble from simpler dispositions by way of evolutionary processes. Such models explain how it is possible for relatively sophisticated linguistic competences to evolve in the context of modest dispositional resources. They also support the view that one might individuate alternative pragmatic notions, represented here in the coevolving metagame language, by the games that evolves them.
Note that we are not assigning indicative content to the evolved signals of the base game or the metagame in this case. Given the symmetry of the two games, the expressions in either game might be interpreted as indicatives or imperatives. The signals in the metagame coevolve to communicate information concerning the success or failure of plays of the base game to the metagame receiver and this information is reflected in his successful actions given the state of nature. We will consider how the symmetry between indicative and imperative interpretations of the signals might be broken in the next section.
More specifically, the magnitude of the dispositions that individuate signals nearly always differ by better than two orders of magnitude, and typically significantly more. The metagame agents do yet better when they learn by way of a faster dynamics like reinforcement with punishment or forgetting or a dynamics like win-state/lose-shift. See Roth and Erev (1995), Erev and Roth (1998) and Barrett and Zollman (2009) for descriptions of learning dynamics that are both faster and more reliable in the context of such games.
There are a number of things that can prevent the metagame from evolving a sharp distinction between successful and unsuccessful action. If the metagame agents learn by simple reinforcement, the more biased the success rate in the base game early on and the longer this bias lasts, the more likely the metagame will evolve to a suboptimal pooling equilibrium. Such a bias may occur if the base agents learn too fast (if they are playing, say, a \(2 \times 2 \times 2\) game using win-stay/lose-shift learning on conditional acts, one should expect the base game to exhibit perfect signaling in a mean of about 14 plays.) or if they learn too slowly (if they are playing an \(8 \times 8 \times 8\) game using simple reinforcement learning, the cumulative success rate is less than 0.8 about 60 % of the time even after \(10^6\) plays). But note that even if the base game exhibits a strong bias in its cumulative success rate the matagame may still evolve to distinguish between success and failure in the base game if the metagame agents learn by means of a fast, flexible dynamics (something like win-stay/lose shift) or if they are simply lucky enough to avoid the suboptimal equilibrium on a slower, less flexible dynamics like simple reinforcement.
See Harms (2004a), Harms (2004b) and Millikan (2005) for discussions of the propositional content of evolved languages and Huttegger (2007) and Zollman (2011) for discussions of alternative approaches for breaking the symmetry between indicative and imperative interpretations of the evolved signals in Skyrms–Lewis signaling games. In short, Huttegger’s approach is to introduce deliberation as an additional primitive option in the game then individuate interpretations of the evolved signals by whether the sender or receiver choose to deliberate, and Zollman’s is to introduce a second receiver then consider a game where the sender might assert to both receivers or direct each separately. The present proposal, rather, is to break the symmetry by allowing for the evolution of a metalanguage that tracks the sender’s use of the base language given the current state of nature and how the base language has evolved. It is likely that the symmetry between indicatives and imperative is broken in multiple, context-dependent ways in the evolution and use of natural languages.
When they fail to do so, both metagame signals evolve to indicate that the base-game sender failed to send the signal most often used in the current situation. Since the metagame uses simple reinforcement learning, one would expect this suboptimal equilibrium from time to time on runs where the base-game sender is slow to converge to a set of stable, surefire dispositions. One would also expect such suboptimal behavior to be less likely if the metagame agents were to learn by something like win-stay/lose-shift or a form of reinforcement with punishment or forgetting. See Roth and Erev (1995), Erev and Roth (1998) and Barrett and Zollman (2009) for descriptions of such learning dynamics.
Note that if the base game gets stuck in a suboptimal pooling equilibrium and hence does not evolve a signaling system, then the customary signal for a particular state may not always lead to successful action. In this case, the metagame agents would evolve a correspondingly suboptimal truth predicate.
Note that while there is nothing here that breaks the symmetry of the interpretation of metagame signals, however one understands the metagame signals, when the base game evolves a signaling system, the metagame signals communicate information, in the sense characterized by Skyrms (2010), concerning whether the base-game sender’s signal faithfully represents the current state of nature.
The evolved distinction between true and false is available to represent possible failures in future plays of the base-game agents due to a broken or deceptive sender. Such use would further reinforce the metagame distinction.
A metagame like the one described here that is associated with the base game described in Barrett (2013) might, for example, coevolve to track a primitive sort of truth for very basic arithmetic statements.
Kevin Zollman, for example, suggested a natural extension of the present metagame where the metagame receiver uses the metagame signal to decide whether to use the base-game signal as a basis for action.
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Acknowledgments
I would like to thank Andrew Bollhagen, Brian Skyrms, Simon Huttegger, Cailin O’Connor, and Kevin Zollman for helpful discussions. I would also like to thank two anonymous reviewers for their very helpful comments on an early version of this paper.
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Barrett, J.A. On the Evolution of Truth. Erkenn 81, 1323–1332 (2016). https://doi.org/10.1007/s10670-015-9797-z
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DOI: https://doi.org/10.1007/s10670-015-9797-z