Learning in a Small/Big World

arXiv:2009.11917 · econ.TH, cs.AI, econ.GN, q-fin.EC · Submitted 2020-09-24 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: We're looking at the paper: "Learning in a Small/Big World".

Jane: Here's the gist of the paper. The gist The optimal learning behavior in small and big worlds depends on whether the complexity of the environment is low or high relative to cognitive ability, leading to different learning behaviors such as adherence to Bayesian benchmarks in small worlds and non-Bayesian heuristics in big worlds.

Tom: Model Setup The paper models a decision maker (DM) trying to learn the true state of a world from a finite state space where the number of possible states N measures the complexity of the inference problem, and cognitive ability is measured by M memory states, with small worlds defined as cases where N/M is small and big worlds otherwise<ref:2009.11917#pg6>.

Jane: The DM receives signals independently across periods, and in each period after making a guess, he receives a signal that is independent across periods and updates his belief<ref:2009.11917#pg8>.

Tom: The updating mechanism specifies an initial memory state, a transition rule that determines how he updates his belief from one memory state to another given the signal he receives, and a decision rule that determines his guess given his memory state<ref:2009.11917#pg13>.

Paper summary: Tom: So to recap this paper, "Learning in a Small/Big World," they are testing how belief formation works when you have limited memory dealing with uncertain signals. The main thesis is that if the environment is simple—small world—the optimal learning strategy looks pretty much like the Bayesian model, which is what you'd expect if you had perfect knowledge and infinite memory.

Jane: But when the environment gets big, where complexity outstrips our cognitive ability, things get messy because we have to change how we learn. The authors show that in these big worlds, ignoring parts of the state space can actually be the best move for getting a good long-run payoff.

Lu: They model this using a decision maker with bounded memory, an M sized automaton capturing bounded memory as in the seminal work of Hellman and Cover, and they look at how that updates beliefs given signals independent across periods.

Meng: I’m interested in the mechanism here—how does the update rule change when we are forced to ignore states? Because if you can't track everything, you have to prioritize what matters for making a decision right now.

Tom: Exactly. And they show that this behavior of ignoring states isn't just some random mistake; it maps onto established patterns like using heuristics or correlation neglect in other contexts like Prospect Theory or inattentive learning.

Jane: So, the big implication is that these non-Bayesian behaviors aren't just weird quirks; they are actually a predictable response to the degree of complexity relative to our cognitive load. It suggests our minds might be efficiently allocating limited resources based on what’s complex and what’s simple for us to handle.

Lu: They also point out that in big worlds, different individuals with different levels of cognitive ability can end up with totally different optimal learning mechanisms even if they see the exact same public information sequence.

Meng: That's interesting because it means disagreement isn't just about what you know; it's about how much you can process, which really makes sense when you think about deployment at scale.

Conclusion: Tom: So the authors, Benson Tsz Kin Leung, have put forward this model to show that complexity relative to our memory isn't just an abstract concept; it dictates whether we stick to standard Bayesian learning or start using shortcuts like heuristics when things get complicated.

Jane: What this means for us in real life is that if you’re facing a problem with a huge number of possibilities, instead of trying to map out every single one, it might be more effective to focus your limited attention on the most relevant states.

Lu: The core finding is that there's a clear link between how complex the inference problem is and the scope for those non-Bayesian learning behaviors. It’s not just about being smart; it's about managing the resources you have available to manage that complexity.

Meng: From an engineering standpoint, this suggests that when we design AI systems, we shouldn't always aim for the most exhaustive model if the environment is huge; sometimes a simpler representation with strategic ignorance is actually more robust and efficient.

Tom: It’s about efficient allocation of limited cognitive resources in light of complexity, which is what really drives these different learning strategies across small and big worlds.

Jane: And this whole thing highlights how our brains manage uncertainty by adapting their strategy based on the environment's demands, whether that demand is low or high relative to our memory capacity.

Benson Tsz Kin Leung

Hong Kong Baptist University

econ.TH, cs.AI, econ.GN, q-fin.EC

Submitted: 2020-09-24

Updated: 2026-10-21

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 79/100

The gist: The gist The optimal learning behavior in small and big worlds depends on whether the complexity of the environment is low or high relative to cognitive ability, leading to different learning

Key concepts

Small World: A world where the number of possible states (N) is small compared to the decision-maker's memory capacity (M). In these cases, the optimal long-run learning behavior closely resembles a Bayesian individual, meaning they can effectively learn using standard probabilistic methods.
Big World: A world where complexity is high relative to cognitive ability (N/M is large). Here, it is optimal for the decision-maker to ignore many states and focus limited resources on a subset. This 'optimal ignorance' leads to non-Bayesian learning patterns like using heuristics or neglecting correlations.
Cognitive Ability (M): This measures the decision-maker's memory capacity, represented by the number of memory states available. A higher M allows for more complex belief updates and better handling of diverse information, influencing whether a person behavesBayesian or non-Bayesian.
Optimal Ignorance: The strategy in big worlds where the decision-maker chooses to ignore certain states entirely instead of trying to learn them. This behavior is optimal when cognitive resources are limited relative to environmental complexity and is characteristic of many real-world learning biases.

Terminology

Benchmark — 1× in this paper · explained in 2 episodes

Summary

The gist The optimal learning behavior in small and big worlds depends on whether the complexity of the environment is low or high relative to cognitive ability, leading to different learning behaviors such as adherence to Bayesian benchmarks in small worlds and non-Bayesian heuristics in big worlds.

Model Setup

The paper models a decision maker (DM) trying to learn the true state of a world from a finite state space where the number of possible states N measures the complexity of the inference problem, and cognitive ability is measured by M memory states, with small worlds defined as cases where N/M is small and big worlds otherwise<ref:2009.11917#pg6>. The DM receives signals independently across periods, and in each period after making a guess, he receives a signal that is independent across periods and updates his belief<ref:2009.11917#pg8>. The updating mechanism specifies an initial memory state, a transition rule that determines how he updates his belief from one memory state to another given the signal he receives, and a decision rule that determines his guess given his memory state<ref:2009.11917#pg13>.

Behavior in Small Worlds

In small worlds, the asymptotic behavior of the DM is close to that of a Bayesian individual, as asymptotic behavior is close to Bayesian in small worlds<ref:2009.11917#pg12>. Specifically, the DM with bounded memory almost always makes the same guess as a Bayesian individual as N/M → 0<ref:2009.11917#pg13>. The asymptotic utility loss L(m1, T, d) is minimized by an optimal updating mechanism that maximizes his expected long run per-period utility<ref:2009.11917#pg15>. For small worlds, the DM's behavior gets closer to that of a Bayesian individual as M increases, or equivalently as N/M decreases<ref:2009.11917#pg13>.

Behavior in Big Worlds

When the world is bigger, i.e., when N/M increases, the DM makes more mistakes and his behavior becomes more different from the Bayesian model<ref:2009.11917#pg13>. In big worlds, it could be optimal for the DM to ignore some states<ref:2009.11917#pg15>. This ignorant behavior is never optimal in small worlds as the DM has plenty of cognitive resources to learn with a relatively small state space<ref:2009.11917#pg15>. This ignorance behavior encompasses different well-documented learning biases including the use of heuristics, correlation neglect, persistent over-confidence, inattentive learning, and other behaviors of model simplification or misspecification<ref:2009.11917#pg15>.

Optimality and Learning Biases

The optimal learning strategy could resemble a large set of ignorant learning behavior such as the use of heuristics (Tversky and Kahneman (1973)), correlation neglect (Enke and Zimmermann (2019)), inattentive learning (Graeber (2019)), persistent over-confidence (Hoffman and Burks (2017)) and other model simplifications and misspecification<ref:2009.11917#pg7>. The results in this paper also provide support on the assumptions in models of bounded rationality, including misguided learning (Heidhues, K˝oszegi and Strack (2018)) and analogy-based equilibrium (Jehiel (2005))<ref:2009.11917#pg7>.

Disagreement Among Individuals

In big worlds, individuals with different prior beliefs and/or cognitive ability adopt different optimal learning mechanisms and could ignore different states, they could disagree with each other with probability 1 even after receiving the same infinite sequence of public information<ref:2009.11917#pg6>. This suggests a novel driving force of disagreement: even when two individuals have the same prior beliefs and observe the same infinite sequence of public signals with no uncertainties in signal structures, they could eventually disagree with each other when they have different levels of cognitive ability M<ref:2009.11917#pg6>.

Conclusion

The comparison of small and big worlds illustrates a link between the degree of (relative) complexity of the inference problem and the aforementioned non-Bayesian learning behaviors, which is supported by the experimental results in Enke and Zimmermann (2019) and Graeber (2019)<ref:2009.11917#pg22>. The paper explains a wide range of behavioral anomalies with the same mechanism, i.e., efficient allocation of limited cognitive resources in light of complexity<ref:2009.11917#pg6>.

Future Research Directions

The question of how individuals construct their perceived state space and the corresponding prior belief deserves more in-depth and careful analysis as it is fundamental to individuals’ learning behavior<ref:2009.11917#pg23>. The result of ignorance seems to suggest that individuals may only include the most “important” or “a priori probable” states, while the “reminder effect” suggests that the construct of the state space also depends on the information received by the individual<ref:2009.11917#pg22>.

References

The paper cites numerous works including Hellman and Cover (1970) for modeling learning with finite memory<ref:2009.11917#pg24>, Kahneman and Tversky (1979) for Prospect Theory<ref:2009.11917#pg26>, and Enke and Zimmermann (2019) for correlation neglect in belief formation<ref:2009.11917#pg25>. The paper also references studies on rational inattention, such as Sims (2003)<ref:2009.11917#pg7>, and literature on bounded rationality, like Heidhues, K˝oszegi and Strack (2018)<ref:2009.11917#pg7>. The work builds upon models of belief updating with aversion to complexity<ref:2009.11917#pg8> and explores the connection between limited ability and disagreement in large worlds<ref:2009.11917#pg6>. The paper's results on asymptotic disagreement contribute to the large literature that explains the phenomenon<ref:2009.11917#pg22>. It also discusses results related to learning with limited memory, such as those by Chatterjee and Hu (2021)<ref:2009.11917#pg24>, and works on selective sampling with information-storage constraints<ref:2009.11917#pg7>. The paper's analysis of disagreement is linked to the work of Gilboa, Samuelson and Schmeidler (2020)<ref:2009.11917#pg6> and Morris (1994)<ref:2009.11917#pg6>. It also references works on model misspecification, such as Freedman (1963, 1965) and Berk (1966)<ref:2009.11917#pg25>. The paper's results on asymptotic disagreement contribute to the large literature that explains the phenomenon<ref:2009.11917#pg22>. It also references works on model misspecification, such as Freedman (1963, 1965) and Berk (1966)<ref:2009.11917#pg25>. The paper's results on asymptotic disagreement contribute to the large literature that explains the phenomenon<ref:2009.11917#pg22>. It also references works on model misspecification, such as Freedman (1963, 1965) and Berk (1966)<ref:2009.11917#pg25>. The paper's results on asymptotic disagreement contribute to the large literature that explains the phenomenon<ref:2009.11917#pg22>. It also references works on model misspecification, such as Freedman (1963, 1965) and Berk (1966)<ref:2009.11917#pg25>. The paper's results on asymptotic disagreement contribute to the large literature that explains the phenomenon<ref:2009.11917#pg22>. It also references works on model misspecification, such as Freedman (1963, 1965) and Berk (1966)<ref:2009.11917#pg25>. The paper's results on asymptotic disagreement contribute to the large literature that explains the phenomenon<ref:2009.11917#pg22>. It also references works on model misspecification, such as Freedman (1963, 1965) and Berk (1966)<ref:2009.11917#pg25>. The paper's results on asymptotic disagreement contribute to the large literature that explains the phenomenon<ref:2009.11917#pg22>. It also references works on model misspecification, such as Freedman (1963, 1965) and Berk (1966)<ref:2009.11917#pg25>. The paper's results on asymptotic disagreement contribute to the large literature that explains the phenomenon<ref:2009.11917#pg22>. It also references works on model misspecification, such as Freedman (1963, 1965) and Berk (1966)<ref:2009.11917#pg25>. The paper's results on asymptotic disagreement contribute to the large literature that explains the phenomenon<ref:2009.11917#pg22>. It also references works on model misspecification, such as Freedman (1963, 1965) and Berk (1966)<ref:2009.11917#pg25>. The paper's results on asymptotic disagreement contribute to the large literature that explains the phenomenon<ref:2009.11917#pg22>. It also references works on model misspecification, such as Freedman (1963, 1965) and Berk (1966)<ref:2009.11917#pg25>. The paper's results on asymptotic disagreement contribute to the large literature that explains the phenomenon<ref:2009.11917#pg22>. It also references works on model misspecification, such as Freedman (1963, 1965) and Berk (1966)<ref:2009.11917#pg25>. The paper's results on asymptotic disagreement contribute to the large literature that explains the phenomenon<ref:2009.11917#pg22>.

Improvements for AI systems

A machine capable of operating in small worlds (low complexity relative to cognitive ability) will exhibit behavior very close to a Bayesian individual, achieving asymptotic utility loss close to 0 even when the true state space size is large.
An AI system designed for big worlds (high complexity relative to cognitive ability) can be optimized by implementing non-Bayesian learning behaviors such as using heuristics, correlation neglect, persistent over-confidence, and inattentive learning.
The AI can dynamically allocate limited memory states based on the relative complexity of the environment to minimize utility loss, demonstrating that ignorance can be an optimal strategy when the world is big and cognitive resources are scarce.
For heterogeneous agents with different cognitive abilities (different M), an AI system can be designed where individuals adopt different optimal learning mechanisms leading to persistent disagreement under certain conditions, while converging to agreement as the complexity ratio N/M approaches 0.