Quantum(-Like) Decision Making: On Validity of the Aumann Theorem

Abstract

Through set-theoretic formalization of the notion of common knowledge, Aumann proved that if two agents have the common priors, and their posteriors for a given event are common knowledge, then their posteriors must be equal. In this paper we investigate the problem of validity of this theorem in the framework of quantum(-like) decision making.

Appendices

Appendix 1: Biased Decision Making and Violation of the Aumann Theorem

An important source of possible violation of the Aumann theorem is the presence of various biases in the “heads of agents”. Roughly speaking any bias may destroy the purity of the Bayesian update.

As a widely discussed example of the anti-Aumann bias, we consider the so called SSA-bias. A Self Sampling Assumption (SSA) says you are more likely to be present in worlds where a greater proportion of agents which are like you, see N. Bostrom for the detailed discussion on SSA [35]. Except that “agents” can be any set of things you could have been in some sense, even if you currently know you are not some of them. This group is called a reference class. Agents basing their reasoning on the SSA and having different reference classes need not come to the same posterior probabilities, even if the assumptions of the Aumann theorem, about the common prior and common knowledge, hold true. And this is clear why. Such an agent can ignore some of her/his information in forming her/his reference classe, since it asks for the proportion of her/his reference class of whom all of her/his information is true. This can lead to simple ignorance of a part of information presented in common knowledge. It is often argued that the decision making based on the selection of an appropriate reference class is irrational. And we agree with such evaluation of the SSA decision makers. However, we do not assign negative valuation to “irrationality” in the decision making. As was demonstrated in [35], the SSA-operating is quite common phenomenon. Since this happens and happens in many contexts, such a behavior of agents has to be modeled mathematically. And if in the classical probabilistic framework this is impossible (as signed in the violation of the Aumann theorem which is heavily based on Kolmogorov probability), then it is natural to explore other probabilistic models, e.g., quantum probability. In this short note we are not able to discuss quantum modeling of SSA in more details, it will be done in one of further articles. We finish this discussion on SSA with the following remark on the interpretation of the wave function, quantum state. The SSA approach to decision making matches well with the many world interpretation of the quantum state. A SSA-agent position her/him self as belonging to a few possible reference classes, which play here the roles of the worlds.

Appendix 2: On the Logical Structure of the Aumann Argument

As ia well known, he Aumann argument on the impossibility of agree on disagree is based on the special systems of axioms of the modal logic, the system S5. And, of course, any deviation from this system might lead to a violation of the classical Aumann theorem. In this paper it was shown that the usage of quantum logic can generate a possibility to agree on disagree. This is a good place to point out that our emphasize on similarity between the classical (S5) and quantum knowledge operators is a bit provocative, since this similarity is only formal, operational, and from the logical viewpoint these are very different representations of knowledge.

In fact, understanding of “what quantum logic is from semantic viewpoint” is a complex problem by itself, see, e.g., works of Garola [36] and of Garola and Sozzo [37] and the recent paper of Khrennikov and Schumann [38] for details. One of still debated problems is whether one can really assign to propositions a special “quantum truth” value or it is even possible to proceed with the classical truth value. In [38] it was motivated that the essence of logical nonclassicality is the performative part of quantum mechanics and at the theoretical level one can still proceed with classical logic. Thus it was motivated that even in quantum physics logical nonclassicality is only due to the language representation. Such a discussion is helpful to come closer to understanding the following fundamental problem: whether quantumness is in the world or in the mind. It seems that the argument presented in [38] supports the latter, i.e., that violations of classical logic and “quantum logical effects” are generated by the performative structure used for the interpretation of some natural and mental phenomena.

In this context it may useful to use the scientific methodology in which any scientific representation has two level, the ontic level and the epistemic level, see, e.g., Atmanspacher and Primas [39]. In such an approach violation of classical logic happens at the level of epistemic description.

Appendix 3: Classical Formalization for the Aumann Argument

Aumann’s considerations are applicable for a finite number of agents, call them $i=1,2,..., N.$ These individuals are about to learn the answers to various multi-choice questions, to make observations.

Mathematically the situation is represented with the aid of classical probability space (based on the Kolmogorov axiomatics, 1933). Typically it is assumed that the state space $\varOmega$ representing all possible states of the world is finite. Events are subsets of $\varOmega .$

Each agent creates its information representation for possible states of the world based on its own possibilities to perform measurements, “to ask questions to the world.” Mathematically these represetations are given by partitions of $\varOmega : \mathcal{P}^{(i)}= (P_j^{(i)}),$ where $\cup _j P_j^{(i)}= \varOmega$ and $P_j^{(i)} \cap P_k^{(i)} \emptyset , j\not =k.$ Thus an agent cannot get to know the state of the world $\omega$ precisely; she can only get to know to which element of its information partition $P_j^{(i)}= P_j^{(i)}(\omega )$ this $\omega$ belongs. The agent $i$ knows an event $E$ in the state of the world $\omega$ if

$$\begin{aligned} P_j^{(i)}(\omega ) \subset E. \end{aligned}$$

(17)

It is assumed that on $\varOmega$ there is defined probability $p,$ the common prior of all agents. In the accordance with the measure-theoretic model of probability theory (Kolmogorov, 1933) there is given a $\sigma$-algebra, say $\mathcal{F},$ of subsets of $\varOmega ,$ its elements represent events (“propositions” in some interpretations), and there is given a probability measure $p$ defined on $\mathcal{F}.$ In the knowledge models it is typically assumed that $\mathcal{F}$ is generated by agents’ partitions, i.e., this is the minimal $\sigma$-algebra containing all systems of set $\mathcal{P}^{(i)}, i=1,...,N.$

We consider the systems of sets $\tilde{\mathcal{P}}^{(i)}=\{ \cup _m P^{(i)}_{j_m}\}$ consisting of finite unions of the elements of the systems $\mathcal{P}^{(i)}$ and the system $\tilde{\mathcal{P}} = \cap _i \tilde{\mathcal{P}}^{(i)}.$ We recall that the meet of the partitions $\mathcal{P}^{(i)},$ denoted by the symbol $\wedge _i \mathcal{P}^{(i)},$ is the finest common coarsening of $\mathcal{P}^{(i)}.$ In particular, $\wedge _i \mathcal{P}^{(i)} \subset \tilde{\mathcal{P}}.$

As was proven in [29], an event $E$ is common knowledge at $\omega$ if $E$ contains that element of $\mathcal{P}^{(1)} \wedge \mathcal{P}^{(2)}$ (the meet) containing $\omega .$ (See footnote 3 on the definition of common knowledge.)

This result implies that, for each $i,$ the set of all states of the world for which $E$ is common knowledge, denoted by the symbol $\kappa E,$ can be represented (in the case $\kappa E \not = \emptyset$) in the form:

$$\begin{aligned} \kappa E= \cup _m P^{(i)}_{j_m}. \end{aligned}$$

(18)