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Unlearnable Games and “Satisficing” Decisions: A Simple Model for a Complex World

Jérôme Garnier-Brun1,2,*, Michael Benzaquen1,2,3, and Jean-Philippe Bouchaud1,3,4

  • 1Chair of Econophysics and Complex Systems, École polytechnique, 91128 Palaiseau Cedex, France
  • 2LadHyX, CNRS, École polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France
  • 3Capital Fund Management, 23 Rue de l’Université, 75007 Paris, France
  • 4Académie des Sciences, 23 Quai de Conti, 75006 Paris, France

  • *jerome.garnier-brun@polytechnique.edu
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Phys. Rev. X 14, 021039 – Published 6 June, 2024

DOI: https://doi.org/10.1103/PhysRevX.14.021039

Abstract

As a schematic model of the complexity economic agents are confronted with, we introduce the “Sherrington-Kirkpatrick game,” a discrete time binary choice model inspired from mean-field spin glasses. We show that, even in a completely static environment, agents are unable to learn collectively optimal strategies. This is either because the learning process gets trapped in a suboptimal fixed point or because learning never converges and leads to a never-ending evolution of agent intentions. Contrarily to the hope that learning might save the standard “rational expectation” framework in economics, we argue that complex situations are generically unlearnable and agents must do with satisficing solutions, as argued long ago by Simon [Q. J. Econ. 69, 99 (1955)]. Only a centralized, omniscient agent endowed with enormous computing power could qualify to determine the optimal strategy of all agents. Using a mix of analytical arguments and numerical simulations, we find that (i) long memory of past rewards is beneficial to learning, whereas overreaction to recent past is detrimental and leads to cycles or chaos; (ii) increased competition (nonreciprocity) destabilizes fixed points and leads first to chaos and, in the high competition limit, to quasicycles; (iii) some amount of randomness in the learning process, perhaps paradoxically, allows the system to reach better collective decisions; (iv) nonstationary, “aging” behavior spontaneously emerges in a large swath of parameter space of our complex but static world. On the positive side, we find that the learning process allows cooperative systems to coordinate around satisficing solutions with rather high (but markedly suboptimal) average reward. However, hypersensitivity to the game parameters makes it impossible to predict ex ante who will be better or worse off in our stylized economy. The statistical description of the space of satisficing solutions is an open problem.

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Popular Summary

Much of economic theory relies on the idea that individuals are rational and make optimal decisions using all the information at their disposal. However, even simple problems may have an overwhelmingly large number of nearly optimal solutions, as demonstrated by the physics of complex systems such as spin glasses. To assess whether agents can really behave rationally in similarly complex settings, we introduce a spin-glass-inspired multiplayer binary game, understood as an idealization of the economic world. We find that although agents are in a static environment where payoffs do not evolve, they cannot reach collectively optimal strategies even when they learn from past outcomes. Contrary to the hope that learning might save the “rational expectation” framework, we argue that complex situations are generically unlearnable.

The model is characterized by only three parameters: a memory loss rate controlling the learning, an intensity of choice describing stochastic decision making, and a nonreciprocity parameter quantifying competition between agents. Using tools from disordered systems, we find that (i) long memory is beneficial to learning, whereas overreaction to the recent past is detrimental; (ii) increased competition destabilizes the system; (iii) some amount of randomness in the decision process paradoxically allows the system to reach better outcomes; and (iv) a stationary state is never reached in a large swath of parameter space. Nonetheless, we find that the learning process allows cooperative systems to reach satisfactory and sufficing—or “satsificing”—solutions with rather high (but markedly suboptimal) average rewards.

Thanks to its simple yet extremely rich nature, our model could provide a unifying framework for the study of complex systems beyond the socioeconomic context, such as neural networks.

Article Text

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Phys. Rev. X 14, 021039– Published 6 June, 2024
  • Received 3 January 2024
  • Revised 15 February 2024
  • Accepted 29 April 2024
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