By Professor Sudip Patra

The write up provides a short discussion on the quantum-like model of probability updates in case of costly signaling phenomenon relevant for financial markets, or economic decision making in general.

Recently (as summarised by Haven, Khrennikov, and Robinson, 2017 Quantum Methods in Social Science: A First Course (275 Pages) (worldscientific.com),  Quantum-like modelling of common knowledge based on quantum probability updating rules, rather than standard Bayesian updating rules, has demonstrated that rational agents can continue disagreeing on the posterior probabilities of any event even if they have a common prior and common knowledge about posterior probabilities, or beliefs.  Hence the Aumann’s(1976, Agreeing to Disagree (projecteuclid.org) famous and central theorem in standard decision-making theory is challenged. Standard neoclassical finance theory is based on information asymmetry, where one party to any transaction has a greater degree of private information about the quality of the asset than the other party has, which generates screening (adverse selection) or monitoring (moral hazard) problems. However, deep uncertainty is a scenario when all parties to a transaction may have a common prior belief state which is non-trivial to describe in the standard adverse selection/moral hazard set up. Quantum-like modelling can prove to be a useful alternative in describing and predicting agents’ behavior in such an uncertainty scenario, hence extending the signaling theory. 

Another such feature of  quantum-like modelling or quantum decision theory modelling is that there is no need to introduce ‘irrational’ or ‘inductive reasoning’ for demonstrating violations of neoclassical theory predictions. Even rational and deductive logic-based reasoning can also generate violations.

Introduction 

The previous decade has witnessed a significant upsurge in the research area of Quantum-like modelling (9781107012820_frontmatter.pdf (cambridge.org) in social sciences. There is a growing body of literature  which provides both theoretical and empirical justifications of reformulating decision-making theory based on the mathematical and logical formulation of Quantum theory. Authors have formulated decision-making models and found empirical support of the predictions in cognitive psychology studies, or in real life financial markets, or in general voting scenarios. However, application or extension of such quantum-like models in financial economics is still in its  nascent stage.

Haven and Khrennikov, as cited above , Khrennikov (2015, Quantum version of Aumann’s approach to common knowledge: Sufficient conditions of impossibility to agree on disagree – ScienceDirect) have formulated a general framework of ‘common knowledge’ among agents based on Quantum-like modelling. Common knowledge formation is the central theory for economics, mainly financial economics. The behavior of rational agents in any kind of equilibrium model  is based on the premise of common knowledge formation at the equilibrium. However, Quantum-like formulation of common knowledge has challenged and generalized the central Aumann (1976) theorem.  Hence, a few important questions arise, for example, how can this new formulation  be extended to agent-based decision-making theory in Economics? Primarily  when we try to explain decision-making under uncertainty? In mainstream neoclassical theory, costly signaling models have been proposed to explain agent’s behavior in such scenarios. However, as already noted in numerous studies,  the standard model fails to describe behavior under true randomness or deep uncertainty.

Therefore, this paper is an early attempt to build a costly signalling model under uncertainty based on Quantum-like formulation of common knowledge. The paper also discusses the contrasting features of the standard Bayesian probability updating model. Overall, the Quantum-like model is a more general model where under special conditions, a typical signalling solution can be achieved even if the agents update their belief states based on Quantum probability rule.

Costly signalling literature developed in the 1970s mainly for resolving information asymmetry problems, for example, the widely researched adverse selection problems in different markets, e.g., job market, insurance market, credit market, equity market and other related areas. Recent papers  emphasize that such signalling models can be extended to many other scenarios such as  commodity markets.  

The central problem remains information asymmetry, where one or some parties in a transaction or a contract have less than perfect information, or less information than the other parties, on different features, for example, the quality of products, or a firm’s future profitability or investment profile, and similar issues.  In such a scenario, rational agents may perceive markets to be occupied by different categories of firms or producers, for example, high and low types, or further finer discrete or continuous spectrum. Again in such a scenario, the genuine firms would find in its own utility maximization behaviour to send a sufficiently costly signal so that rational agents can update their beliefs regarding the quality of firms, and a separating equilibrium is created such that agents can screen one type from the other, and price them accordingly in markets. Hence, solving the embedded adverse selection problem. 

In the standard signaling literature in financial economics  fewer studies have been done on how rational agents learn about the true nature of the signals, it is always assumed that the agents are ‘Bayesian’ rational, or they start with a common prior belief and update according to Bayesian rule and form a common posterior belief based on the possibility of common knowledge formation, hence creating a separating equilibrium. Thus, Aumann’s (1976) ‘not agreeing to disagree’ theorem is a central assumption, which is not explicitly mentioned at times. However, there are some critical issues which are not resolved, and these issues are both theoretical and practical so to say. 

The theoretical issue is with the nature of uncertainty and the rule of belief updating, and the practical issue is with the inability of the standard signaling theory to explain divergent beliefs or behavior of rational agents around costly signals (or behavior of agents given noisy signals). Costly signaling theory is based on the classical set theory or measure theory of probability where there is no inherent randomness or uncertainty, or in other words, it’s theoretically possible to gain perfect knowledge about a system. However, such a theory is ill-equipped to explain true uncertainty that  agents face in the market scenarios. Hence, quantum-like modelling is based on deeper or true randomness which is described as a pure state or superposition of orthogonal basis states, which can describe an initial belief state or a common prior as in the minds of the agents.

A brief review of financial economics literature in the context 

Financial economics literature covers private information issues in detail (Tirole,2006, www.library.fa.ru/files/Tirole-Theory-corporate.pdf). Modern asset pricing literature is based on the problems of adverse selection, and/or moral hazard. Equity premium puzzle is based on the assertion that there is a presence of information asymmetry between the less informed and more informed agents which has a significant impact on the equity returns. Dispersion of investors beliefs  (Takeovers and Divergence of Investor Opinion | The Review of Financial Studies | Oxford Academic (oup.com)Chatterjee et al, 2012) is another strong strand of literature which is related to private information. Even the capital structure of the firms is significantly affected by the degree of information asymmetry in the market. Here again, we observe that deep uncertainty in financial markets, which is beyond simple asymmetry of information, is not fruitfully described in standard signalling theory. 

The practical problem, or the empirical problem generates from the fact that real choice data (for example in the scenario of signaling by firms: investing or not investing in assets) under uncertainty demonstrates some ‘contextual’ or deviant features (Haven and Khrennikov, 2009 Quantum-like model of cognitive decision making and information processing – ScienceDirect) which cannot be explained by the Bayesian rationality model. Specifically, in the case of signaling phenomena, the existence of divergence of opinions of agents (shareholders, or analysts via analyst forecasts) cannot be explained by the central Aumann theorem. Such divergence of opinions has a significant impact on asset prices (Chatterjee et al as cited above) which may deviate such values way beyond so-called fair price, thereby, undermining the very purpose of costly signaling. We observe that in standard theory there is no consensus on the explanation of divergence of opinions, which make the asset prices deviate from the equilibrium or fair price. 

No doubt there has been a strong response from the Behavioural Finance school to explain such divergent behaviours, but here we just remark that the diverse behavioural school is less coherent at times, and the explanations are mainly based on heuristics, thumb rules and the same Bayesian learning models. Therefore, the emerging quantum-like paradigm attempts to formulate a more general probability updating or decision-making theory with deep consequences for Aumann’s theorem or similar propositions.

Mutual knowledge can be defined as a state where everyone in a group knows about a specific event. Common knowledge is rather a stricter version where Alice and Bob know a specific event E, again, Alice knows that Bob knows the event E, again… hence, there are many orders of common knowledge.

To repeat, then, Aumann’s theorem states rational agents starting with common prior information about an event, if it finds that posteriors about that event are a common knowledge then posteriors must be equal, they just cannot agree to disagree. 

Here certainly by rational, it is meant that agents are Bayesian rational, or in other words, they update belief states based on the Bayesian probability update scheme. Aumann’s theorem has been an integral part of rational decision-making theory, specifically, in game theory, asset pricing models, rather than any standard financial market modelling. For example, the commonly used CAPM model has the basic assumption of homogeneous expectations of rational agents in equilibrium, which is based on the common knowledge theorem. Here, we show that common knowledge can be conceptualized both from the perspectives of classical set theory (or Kolgomorovian measure theory) and quantum logic. 

Further insights from Deductive and Inductive Reasoning

Another supposed contribution of quantum-like modelling of common knowledge is that we necessarily do not  need to assume that agents are acting based on complex inductive reasoning. Behavioral finance, as well as the emerging theory of complexity economics or finance (Arthur, 2013 for example, Complexity and the Economy – W. Brian Arthur – Google Books), do place a central importance on inductive reasoning, where agents who are bounded rational update their beliefs and hypotheses about the real world as more information is disclosed. Such inductive reasoning also calls for heuristics and at times, abandons any kind of probability-based decision-making theory altogether.

However, one can hold based on quantum-like modelling that rational agents who update beliefs according to quantum decision theory rules, can still remain committed to deductive reasoning and rationally disagree on the outcomes, for example asset prices. Hence in the current paper, as one of the main outcomes of the modified common knowledge model, we will have asset price dispersion rather than homogenous expectations generating single equilibrium asset price. However, for such a dispersion we need not hold agent’s irrational or inductive traders, so to speak.

Professor (Dr) Sudip Patra is Executive Deputy Director, CEASP, JSGP.