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Common Knowledge: A Quantum Decision Theory Discussion

By Sudip Patra

In the ongoing quantum decision theory series, we have seen some basic probability modeling of human behavior. The crucial differences between classical set-theoretic or Bayesian probability updating model and quantum probability updating model is that the concept of uncertainty in the later is deeper and fundamental, one can not avoid uncertainty since there is an inherent randomness in nature which is expressed through uncertainty relations, whereas in the former uncertainty is not fundamental but is an illusion created by lack of information.
Certainly then, the rules of computing and updating probabilities are different in both these models, which makes a significant difference in results. In this section we will turn to a very central concept in all of decision/probability/ game theory models: common knowledge. We will try to see that what can be significant differences once quantum probability model is adopted for decision making given the common knowledge about some event.
Say there is a belief among people /agents about a certain event, but no one is telling that to anyone else, then suddenly a credible person among the group announces the same thing, in that case, the common belief becomes a piece of common knowledge immediately. Hence, every one knows that everyone knows that event, again everyone knows that everyone knows that everyone knows that event, etc….till infinitum. This is the formal concept of common knowledge.
Common knowledge is the central assumption of any so-called rational decision-making model. Take the standard game theory, that the other player is a rational being is common knowledge among all the players, which is why you can formulate rational strategies.  Aumann the famous Nobel laureate and game theorist had a breakthrough paper (1976), which proved that if the rational agents have common prior knowledge about any event and they update their beliefs in a classical Bayesian way such that the posterior probabilities of that event is common knowledge then such probabilities should be the same! Or in other words, we can not agree to disagree forever if we are Bayesian rational beings.
The very theorem is a big influence on fundamental models in neoclassical economics. However, in reality, we do not always see rational agents agreeing on posterior probabilities even if there are favorable scenarios of common knowledge formation, which again impacts significantly on various social and economic outcomes: asset pricing to opinion polls in elections. Rational agents keep on disagreeing. The basic question is why?
Quantum probability modeling can provide an answer. It can be shown (Andrei Khrennikov, 2017) that if agents are not typically Bayesian rational but update probabilities according to quantum rules (in the earlier sections we have discussed very briefly about the Born’s rule of probability computation and probability updates which is a completely different and in many ways a more general model than Kolgomorov’s measure theory formulation), then even if posterior probabilities are a common knowledge, they may not be same. Certainly there will be conditions when quantum modeling will converge into classical results, but in general, they would differ.
Continuing to disagree on a known event is one of the basic reasons for social outcomes of any type. If the Quantum modeling finds good empirical support (which we are up to) then this will be a fundamental shift from standard probability theory used in social sciences.

Professor Sudip Patra is an Assistant Professor of Management Practice at O.P. Jindal Global University. His research interest encompasses dividend signaling theory under information asymmetry, game theory for applied corporate finance, econometric modeling, and allied areas.

Image Source- Deposit Photos

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