The good regulator theorem and its critics

Keywords: good regulator theorem, Conant & Ashby, internal model, Virgo, free energy principle

Detail page for the cheat sheet. Legend: ✓ read · ◐ secondary/abstract · † unread · ✗ no source.

Introduction. Conant, R. C. & Ashby, W. R. (1970). Every good regulator of a system must be a model of that system. Int. J. Systems Sci. 1(2): 89–97. doi:10.1080/00207727008920220.

  • Optimality criterion (p. 91). “successful regulation” means H(Z) is minimal. A regulator that pins the system to one catastrophic outcome scores perfectly. The good regulator is reliable, not desirable ✓.
  • Three concessions after the proof (p. 96) ✓:
    1. “not all optimal regulators are models of their regulands”: the universal quantifier in the title is false on the authors’ own account.
    2. The proof “avoid[s] all mention of the inputs to the regulator R and its opponent S”: no sensing, channel, noise or dynamics.
    3. The disturbance distribution may change only “slowly with time”. This excludes the non-stationary environments System Four exists for.

Critics and repairs

  • Baez, J. (2016). “The Internal Model Principle”, Azimuth blog, 27 Jan 2016. The lemma is “so easy that I can’t imagine the theorem… is very helpful”, and the step from “regulator state is a function of system state” to “contains a model” is unargued ✓.
  • Wentworth, J. (2021). “Fixing the Good Regulator Theorem”, AI Alignment Forum, 9 Feb 2021. He calls it the “most misleading title and summary I have ever seen on a math paper”, because the identity function qualifies as a model. His repair adds expected utility, information arriving over time and a storage cost. The minimal optimal regulator is then isomorphic to the Bayesian posterior ✓.
  • Virgo, Biehl, Baltieri & Capucci (2025). “A ‘Good Regulator Theorem’ for Embodied Agents”, arXiv:2508.06326, accepted for ALIFE 2025. C&A “doesn’t strictly succeed, even in its own terms”. They replace being a model with having one, through an interpretation map, and the attributed model may be trivial ✓. They credit Baez and Wentworth.

Where the theorem travels outside VSM

  • Friston, K. (2013). Life as we know it. J. R. Soc. Interface 10: 20130475. It cites Conant & Ashby and calls its own result “exactly consistent with the ‘good regulator theorem'” ✓. It does not mention requisite variety.
  • Seth, A. K. (2015). The cybernetic Bayesian brain. In Metzinger & Windt (eds), Open MIND 35(T). doi:10.15502/9783958570108. It quotes Ashby and C&A (“must instantiate a model”) and recasts predictive processing as serving control ✓.
  • Bettinger, J. S. & Friston, K. J. (2023). Conceptual foundations of physiological regulation incorporating the free energy principle and self-organized criticality. Neurosci. Biobehav. Rev. 155: 105459. They say: “Ashby’s Law of Requisite Variety underpins the Good Regulator Theorem” ◐.

Important authors. Roger C. Conant (University of Illinois at Chicago Circle) extended Ashby’s programme with information-theoretic measures. His later paper, Conant 1976, “Laws of information which govern systems”, IEEE SMC 6(4): 240–255, is relevant for any quantitative channel study †.

Importance for cybernetics / VSM. System Four and the Three–Four homeostat are taught as the good regulator theorem “in organisational dress”.

Importance for the chapter. It carries the sections “The theorem that is not a theorem” and “From a complaint into a claim”. The Virgo reconstruction is what turns the Ch 2 grievance (“scanning produces material, nothing moves”) into a proposition that can be disputed with evidence: the attributable model is trivial.

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