**1. Introduction**

Economic agents are often faced with partial information and make decisions under pressure, ye<sup>t</sup> many canonical economic models assume perfect information and perfect rationality. To address these challenges, Simon [1] introduced bounded rationality as an alternate attribute of decision-making. Bounded rationality aims to represent partial access to information, with possible acquisition costs, and limited computational cognitive processing abilities of the decision-making agents.

Information theory offers several natural advantages in capturing bounded rationality, interpreting the economic information as the source data to be delivered to the agen<sup>t</sup> (receiver) through a noisy communication channel (where the level of noise is related to the "boundedness" of the agent). This representation has spurred the creation of informationtheoretic approaches to economics, such as Rational Inattention (R.I.) [2], and more recently, the application of R.I. to discrete choice [3]. Another approach represents decision-making as a thermodynamic process over state changes and employs the energy-minimisation principle to derive suitable decisions [4].

These approaches have shown how one can incorporate a priori knowledge into decision-making, but place no consideration to inferring these decisions based on observed macroeconomic outcomes (e.g., a distribution of profit rates within a financial market) and

**Citation:** Evans, B.P.; Prokopenko, M. A Maximum Entropy Model of Bounded Rational Decision-Making with Prior Beliefs and Market Feedback. *Entropy* **2021**, *23*, 669. https://doi.org/10.3390/e23060669

Academic Editors: Ryszard Kutner, Christophe Schinckus and Eugene Stanley

Received: 21 April 2021 Accepted: 21 May 2021 Published: 26 May 2021

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market feedback loops. Independently, another recent information-theoretic framework, Quantal Response Statistical Equilibrium (QRSE) [5], was developed aiming to infer least biased (i.e., "maximally noncommittal with regard to missing information" [6]) decisions through the maximum entropy principle, given only the macroeconomic outcomes (e.g., when the choice data is unobserved). However, the ways to incorporate prior knowledge into such a system remain mostly unexplored.

In this work, we provide a unification of these approaches, showing how to incorporate prior beliefs into QRSE in a generic way. In doing so, we provide a least biased inference of decision-making, given an agent's prior belief. Specifically, we show how the incorporation of prior beliefs affects the agent's resulting decisions when their individual choices are unobserved (as is common in many real-world economic settings). The proposed information-theoretic approach achieves this by considering a cost of information acquisition (measured as the Kullback-Leibler divergence), where this cost controls deviations from an agent's prior knowledge on a discrete choice set. When the cost of information acquisition is prohibitively high (i.e., when an agen<sup>t</sup> is faced with limitations through time, cognition, cost, or other constraints), the agen<sup>t</sup> falls back to their prior beliefs. When information acquisition is free, the agen<sup>t</sup> becomes a perfect utility maximiser. The cost of information acquisition therefore measures the boundedness of the agent's decision-making.

The proposed approach is general, allowing the incorporation of any form of prior belief, while separating the agents' current expectations from their built-up beliefs. In particular, we show how incorporating prior beliefs into the QRSE framework allows for modelling decisions in a rolling way, when previous decisions "roll" into becoming the latest beliefs. Furthermore, we place the original QRSE in the context of related formalisms, and show that it is a special case of the general model proposed in our study, when the prior preferences (beliefs) are assumed to be uniform across the agen<sup>t</sup> choices. Finally, we verify and demonstrate our approach using actual Australian housing market data, in terms of agen<sup>t</sup> buying and selling decisions.

The remainder of the paper is organised as follows. Section 2 provides a background of information-theoretic approaches to economic decision-making, Section 3 describes QRSE and relevant decision-making literature. Section 4 outlines the proposed model, and Section 5 applies the developed model to the Australian housing market. Section 6 presents conclusions.

#### **2. Background and Motivation**

The use of statistical equilibrium (and more generally, information-theoretic) models remains a relatively new concept in economics [7]. For example, Yakovenko [8] outlines the use of statistical mechanics in economics. Scharfenaker and Semieniuk [9] detail the applicability of maximum entropy for economic inference, Scharfenaker and Yang [10] give an overview of maximum entropy and statistical mechanics in economics outlining the benefits of utilising the maximum entropy principle for rational inference, and Wolpert et al. [11] outline the use of maximum entropy for deriving equilibria with bounded rational players in game theory. Earlier, Dragulescu and Yakovenko [12] showed how in a closed economic system, the probability distribution of money should follow the Boltzmann-Gibbs law [13]. Foley [14] discusses Rational expectations and boundedly rational behaviour in economics. Harré [15] gives an overview of information-theoretic decision-theory and applications in economics, and Foley [16] analyses information-theory and results on economic behaviour.

Ömer [17] provides a comparison of "conventional" economic models and newly proposed ideas from complex systems such as maximum entropy methods and Agent-based models (ABM), which deviate from the assumption of homo economicus—a perfectly rational representative agent. Yang and Carro [18] discuss how a combination of agent-based modelling and maximum entropy models can be complementary, leveraging the analytical rigour of maximum entropy methods and the relative richness of agent-based modelling.

One of the key developments in this area is Quantal Response Statistical Equilibrium (QRSE) proposed by Scharfenaker and Foley [5]. This approach enabled applications of the maximum entropy method [6,19,20] to a broad class of economic decision-making. The QRSE model was further explored in [21], arguing that "any system constrained by negative feedbacks and boundedly rational individuals will tend to generate outcomes of the QRSE form". The QRSE approach is detailed in Section 3.1.

Ömer [22–24] applies QRSE to housing markets (which we also use as a validating example), modelling the change in the U.S. house price indices over several distinct periods, and explaining dynamics of growth and dips. Yang [25] applies QRSE to a technological change, modelling the adoption of new technology for various countries over multiple years and successfully recovering the macroeconomic distribution of rates of cost reduction. Wiener [26–28] applies QRSE to labour markets, modelling the competition between groups of workers (such as native and foreign-born workers in the U.S.), and capturing the distribution of weekly wages. Blackwell [29] provides a simplified QRSE for understanding the behavioural foundations. Blackwell further extends this in [30], introducing an alternate explanation for skew, which arises due to the agents having different buy (enter) and sell (exit) preferences. Scharfenaker [31] introduces Log-QRSE for income distribution, and importantly, (briefly) mentions informational costs as a possible cause for asymmetries in QRSE. This is captured by measuring utility *U* as a sum *<sup>U</sup>*[*<sup>a</sup>*, *x*] + *<sup>C</sup>*(*a*|*x*), allowing for higher costs ( *C*) of entrance or exit into a market, where *a* is an action and *x* is a rate. Such a separation allows for an "alternative interpretation of unfulfilled expectations".

These developments show the usefulness of maximum entropy methods, where we have placed particular focus on QRSE, for inferring decisions from only macro-level economic data. However, these approaches do not consider the contribution of a priori knowledge to the resulting decision-making process. The key objective of our study is to generalise the QRSE framework by the introduction of the prior beliefs, as well as the information acquisition costs as a measure of deviation from such priors.

#### **3. Underlying Concepts**

Two main concepts form the basis for the proposed model. The first is the QRSE approach developed by [5], and the second is a thermodynamics-based concept of decisionmaking derived from minimising negative free energy, proposed by [4].
