OR Forum—Quantum Mechanics and Human Decision Making

OPERATIONS RESEARCH Vol. 61, No. 1, January–February 2013, pp. 1–16 ISSN 0030-364X (print) — ISSN 1526-5463 (online) http://dx.doi.org/10.1287/opre.1120.1068 © 2013 INFORMS CROSSCUTTING AREAS OR Forum—Quantum Mechanics and Human Decision Making Paras M. Agrawal, Ramesh Sharda William S. Spears School of Business, Institute for Research in Information Systems, Oklahoma State University, Stillwater, Oklahoma 74078 {[email protected], [email protected]} In physics, at the beginning of the twentieth century it was recognized that some experiments could not be explained by the conventional classical mechanics, but the same could be explained by the newly discovered quantum theory. It resulted in a new mechanics called quantum mechanics that revolutionized scientific and technological developments. Again, at the beginning of the twenty-first century, it is being recognized that some experiments related to the human decision-making processes could not be explained by the conventional classical decision theory but the same could be explained by the models based on quantum mechanics. It is now recognized that we need quantum mechanics in psychology as well as in economics and finance. In this paper we attempt to advance and explain the present understanding of applicability of quantum mechanics to the human decision-making processes. Using the postulates analogous to the postulates of quantum mechanics, we show the derivation of the quantum interference equation to illustrate the quantum approach. The explanation of disjunction effect experiments of Tversky and Shafir (Tversky A, Shafir E (1992) The disjunction effect in choice under uncertainty. Psych. Sci. 3(5):305–309) has been chosen to demonstrate the necessity of a quantum model. Further, to suggest the possibility of application of the quantum theory to the business-related decisions, some terms such as price operator, state of mind of the acquiring firm, etc., are introduced and discussed in context of the merger/acquisition of business firms. The possibility of the development in areas such as quantum finance, quantum management, application of quantum mechanics to the human dynamics related to healthcare management, etc., is also indicated. Subject classifications: decision analysis: theory; quantum decision model; quantum information processing; human decision making; quantum interference. Area of review: OR Forum. History: Received July 2010; revisions received April 2011, November 2011; accepted February 2012. Published online in Articles in Advance December 13, 2012. 1. Introduction nanotechnology, femto-chemistry, molecular biology, cosmology, high-energy physics, quantum mechanics is valuable and indispensible. In recent years, one notes a growing interest in the application of quantum mechanics to areas such as quantum cryptography (e.g., Bennett and Brassard 1984, Bennett et al. 1992, Chung et al. 2008) as well as quantum computation (e.g., Shor 1997, Lo et al. 2000, Hand 2009). As regards the application of quantum mechanics beyond physical sciences, Bohr (1929) attempted to show the similarity between the mental processes and the quantum mechanical phenomena. In his writings, he also discussed the similarities between quantum mechanics and the functions of the brain (e.g., Bohr 1933). In recent decades, there have been various notable attempts to ascribe the quantum mechanical properties to brain, mind, and consciousness (e.g., Chalmers 1996, Lockwood 1989, Penrose 1989, Penrose et al. 2000, Pessa and Vitiello 2003, Satinover 2001). Recently there has been some new work to explore applicability of quantum models in better understanding nuances of human decision making. The purpose of this paper is to introduce and explain the quantum concepts through simple terms and notations, to apply these concepts in better understanding the recent applications of quantum mechanics to It is well known that quantum mechanics leads to many peculiar results such as quantum interference, uncertainty principle, quantum nature of light, quantum theory of measurement, tunneling, etc., which are in contradiction with our common sense and classical mechanics. Physicist Niels Bohr, who won the 1922 Nobel Prize in Physics chiefly for his work on atomic structure, once remarked, “If quantum mechanics hasn’t profoundly shocked you, you haven’t understood it yet” (Bohr n.d.). Despite its strange behavior, the quantum mechanics is considered as the most successful theory of physics. Stenholm and Suominen (2005, p. 1), in their book on quantum approach to informatics, write: “Quantum theory has turned out to be the most universally successful theory of physics. 0 0 0 Without the understanding offered by quantum theory, our ability to build integrated circuits and communication devices would not have emerged.” A large number of scientists (e.g., Planck, Einstein, Bohr, de Broglie, Heisenberg, Schrödinger, Born, Dirac, Pauli, Pauling) have been awarded Nobel Prizes for their contributions related to quantum mechanics. In various walks of modern science and technology, including electronics, nuclear technology, 1 Agrawal and Sharda: Quantum Mechanics and Human Decision Making 2 human decision making, and to suggest applicability of models based on quantum mechanics to some new areas of research. 1.1. Quantum Mechanics and Human Decision Making Regarding the human mind, economics Nobel Laureate Herbert Simon wrote in collaboration with Newell (Simon and Newell 1958, p. 9): “The revolution in heuristic problem solving will force man to consider his role in a world in which his intellectual power and speed are outstripped by the intelligence of machines. Fortunately, the new revolution will at the same time give him a deeper understanding of the structure and working of his own mind.” It is interesting to note the significance attached by these authors to quantum mechanics in the same paper in these words: “In dealing with the ill-structured problems of management we have not had the mathematical tools we have needed—we have not had ‘judgment mechanics’ to match quantum mechanics” (p. 6). The expectations of Simon and Newell, expressed half a century ago, regarding the necessity of understanding of our own mind and a mechanics of the decision-making process are not yet fulfilled. However, we are now gaining momentum in the direction of understanding the human decision processes even through quantum mechanics. Thus, the objective of this paper is to introduce these concepts and review the recent progress to stimulate more exploratory research on applications of quantum mechanics concepts in decision making. Kahneman, Tversky, and Shafir have made notable contributions in the area of judgment under uncertainty and the influence of heuristics and biases on the cognitive system (Tversky and Kahneman 1974, Tversky and Shafir 1992, Shafir and Tversky 1992). The significance of the work is attested to by the fact that Kahneman was awarded the Nobel Prize in 2002. Results of several experiments related to the judgment under uncertainty, as noted by Tversky and Shafir (Tversky and Shafir 1992, Shafir and Tversky 1992), in the area of human psychology could not be explained by the classical statistics. The disjunction effect experimentally observed by Tversky and Shafir (1992) is a typical example of the intricacies of the human mind that could not be understood by the classical decision theory. For example, in an experiment of Tversky and Shafir (1992), a participant is offered to play a gamble (by tossing a coin) with a 50% chance of winning $200 and a 50% chance of losing $100. After the first play, the participant is offered to play the second identical game with or without the knowledge of the outcome of the first gamble. It has been observed that a majority of participants are ready to accept the second gamble after knowing that they have won the first one, and a majority of participants are also ready to accept the second gamble after knowing that they have lost the first one, but only a small fraction of participants are ready to accept the second gamble if they do not know the outcome of the first gamble. The question arises: if they prefer to accept Operations Research 61(1), pp. 1–16, © 2013 INFORMS the second gamble in case they win or lose the first gamble, then according to the sure-thing principle of Savage (1954), they should prefer to accept the second gamble even when they do not know the outcome of the first gamble. However, the experiment contradicts such logical expectations. Why? We could not yet get the answer of this “why” from the conventional (classical) theories. Such a violation of the sure-thing principle of Savage (1954) has also been observed by Tversky and Shafir (1992) in another experiment related to buying an attractive vacation package. The successful studies to explain some of these paradoxes by incorporating mathematical equations related with quantum mechanics into the psychology (Aerts 2009, Busemeyer et al. 2006, Pothos and Busemeyer 2009, Khrennikov 2009, Yukalov and Sornette 2009a) clearly reveal that some aspects of the human behavior, which could not yet be explained by the classical decision theory, can be explained by quantum mechanical equations. The investigations of Bordley (1998) and Bordley and Kadane (1999) also suggest the importance of quantum mechanical notions and equations in explaining some aspects of human decision making. It may be noted that classical mechanics and quantum mechanics differ ideologically as well as mathematically; and for macrosystems the approximate form of mathematical equations of quantum mechanics agrees with the equations of classical mechanics. If the decision-making processes of the human mind follow the probabilistic behavior of quantum mechanics, then one can expect the applicability of the same in other areas, which are directly affected by human decision making. Thus, it is not surprising that researchers in economics and finance have explored application of quantum mechanics (Baaquie 2004, Bordley 2005, Kondratenko 2005, Baaquie 2009a). The application of quantum mechanics to economics and finance can be seen in various areas, such as a price dynamics model (Choustova 2007), stock price (Schaden 2003, Bagarello 2009), interest rate (Baaquie 2009b), incorporation of private information (Ishio and Haven 2009, Haven 2008), etc. As an example of the value of quantum mechanics in the field of economics, one can refer to the study conducted by Segal and Segal (1998). In this study they consider quantum effects to explain extreme irregularities in the evolution of prices in financial markets. In the concluding paragraph of this study, Segal and Segal (1998, p. 4075) write: “The quantum extension of Black-Scholes-Merton theory provides a rational, scientifically economical, and testable model toward the explanation of market phenomena that show greater extreme deviations than would be expected in classical theory 0 0 0 0” An electronic companion to this paper is available as part of the online version at http://dx.doi.org/10.1287/opre .1120.1068. In Appendix-A in the e-companion, we have described some facts related to the historical development of quantum mechanics. At the beginning of the twentieth century, it was recognized that some experimental Agrawal and Sharda: Quantum Mechanics and Human Decision Making 3 Operations Research 61(1), pp. 1–16, © 2013 INFORMS results could not be explained by the conventional classical mechanics, but the same could be explained by the newly discovered quantum physics. It resulted in a new mechanics called quantum mechanics that revolutionized scientific and technological development. The uncertainty principle, the quantum theory of measurement, the statistical significance of the wave functions of quantum mechanics, the mathematics involving operators and the abstract vector space, quantum statistics, etc., are some of the various features of quantum mechanics that are not available in the conventional classical mechanics. Therefore, in case of difficulty in explaining some aspects of psychology, or economics, or any other branch of knowledge through the classical mechanics, one may expect that some special features of quantum mechanics may be helpful. As mentioned earlier, we have already noted some success in this direction in the area of human decision making. In view of such aspects, it becomes valuable to be familiar with some basics of quantum mechanics. By the phrase “understanding human decision process through quantum mechanics,” we mean the application of some aspects, such as mathematical framework, of quantum mechanics. For example, we may consider some states of mind in an abstract space that mathematically behave as the quantum states in the Hilbert space (von Neumann 1983, Messiah 1970), and the decision-making process as a process statistically governed by the formulation based on the postulates of quantum mechanics. This, however, does not mean that the human mind becomes a quantum mechanical object. Just as a quantum description of electrons, light quanta, etc., require the necessity of a constant known as Planck’s constant (h = 60626 × 10−34 Joule-Second), we do not need Planck’s constant for explaining the abovementioned disjunction effect or other paradoxes of psychology. Likewise, in quantum mechanics, Schrödinger’s time-independent and time-dependent wave equations contain Planck’s constant, but in the corresponding equations of quantum dynamics of human decision making (Busemeyer et al. 2006, Pothos and Busemeyer 2009), this constant occurring in the equations of quantum mechanics is replaced by another parameter. 1.2. Application of Quantum Models to Disjunction Effect and Other Decisions While explaining the disjunction effect and other paradoxes of psychology with the help of quantum models, Khrennikov (2009) considers the effect of quantum interference in the form of an equation that has an adjustable parameter called the coefficient of interference. Yukalov and Sornette (2008, 2009a, b, c) provide a detailed theory called quantum decision theory (QDT). Using the postulates analogous to the postulates of quantum mechanics, they derive the quantum interference equations that relate various experimental probabilities. To explain the same disjunction effect, Pothos and Busemeyer (2009) consider the evolution of the state of mind using an equation analogous to Schrödinger’s time-dependent wave equation of quantum mechanics. The duration of time and the interaction parameters have been considered as adjustable parameters. For comparison, they also study the evolution of the state of mind using the equivalent Markov (classical) model, and found that their Markov (classical) model could not explain the experimentally observed violations of the sure-thing principle of Savage (1954), whereas the quantum model could explain them. In a quantum decision model discussed in detail in §2, we employ various aspects of the quantum decision theory of Yukalov and Sornette (2009a) but consider more general and simpler kinds of operators to derive the same quantum interference equation as derived by Yukalov and Sornette (2009a), so that the range of applicability may widen and it becomes easier to apply to other related problems. Further, to demonstrate the possibility of application of the quantum approach to other decision problems, we consider an example of the problem of a merger of two business firms. It may be a long way to arrive at a successful and valuable outcome of the application of a quantum model to the problem of merger of two business firms. However, here we shall simply introduce the problem to familiarize the reader with the notations and application of the quantum models in this area. Thus, the purpose of this paper is to serve as a tutorial by illustrating current and future potential applications of quantum concepts to human decision making. In §2, the mathematical details of a sample quantum model are described. Application of the quantum approach in explaining the disjunction experiments, and the possibility of its application to the business-related problems, together with the discussion of the related studies in this area, are presented in §3. The final section provides the summary and concluding remarks. 2. Mathematical Details of a Sample Decision Model To simplify, we describe the model with reference to a practical example. We consider the two-stage gambling experiment of Tversky and Shafir (1992). In this experiment, subjects are first offered to play a gamble (by tossing a coin) with a 50% chance of winning $200 and a 50% chance of losing $100. After the first play, the subjects are offered to play the identical game with or without the knowledge of winning or losing the first gamble. We introduce several probabilities and the related notations associated with this experiment in Table 1. In this section, we shall describe the quantum decision model to explain the results of this experiment, which could not be explained by the conventional (classical) methods. For an understanding of the basic postulates of quantum mechanics, we consider an analogy: Suppose we ask our banker to provide us with information regarding the Agrawal and Sharda: Quantum Mechanics and Human Decision Making 4 Table 1. p4X1 5 p4X2 5 p4A — X1 5 p4AX1 5 p4A — X2 5 p4AX2 5 p4A5 p4B — X1 5 p4BX1 5 p4B — X2 5 p4BX2 5 p4B5 Operations Research 61(1), pp. 1–16, © 2013 INFORMS Notations regarding various probabilities. Probability of winning the first gamble = No. of participants winning the first gamble/No. of participants. Probability of not winning the first gamble (losing) = No. of participants not winning the first gamble/No. of participants. Probability of accepting the second gamble after knowing that he/she has won the first gamble. Joint probability of winning the first gamble and accepting the second gamble. It is equal to the product of p4X1 5 and p4A — X1 5. p4AX1 5 = p4X1 5p4A — X1 5. (1) Probability of accepting the second gamble after knowing that he/she has not won the first gamble. Joint probability of not winning the first gamble and accepting the second gamble. It is equal to the product of p4X2 5 and p4A — X2 5. p4AX2 5 = p4X2 5p4A — X2 5. (2) Probability of accepting the second gamble in absence of any knowledge of winning or losing the first gamble. Probability of not accepting the second gamble after knowing that he/she has won the first gamble. See the corresponding terms with A in this table. The difference between A and B symbols is in “accepting” and “not accepting.” In this regard, parallel to Equations (1) and (2), we shall have p4BX1 5 = p4X1 5p4B — X1 5, and (3) p4BX2 5 = p4X2 5p4B — X2 5. (4) It can be easily understood that the sum of probability of accepting and not accepting is 1, i.e., p4A5 + p4B5 = 10 (5) Notes. The events and activities are denoted as follows: X1 : Win first gamble. X2 : Not win the first gamble. A: Accept second gamble. B: Not accept the second gamble. amount of interest we have earned. To answer this question, the bank teller would first open our account, which would contain all information regarding our deposits, check withdrawals, etc. The details regarding our account in the register or on the computer screen would give the present status of our account. To answer our query regarding the amount of interest, the teller would perform some operations/ Table 2. calculations related to the amount of interest. Similarly, in quantum mechanics, analogous to the details of our account, there exists a state function or a state ket or a state vector or simply a state of the system for the system under consideration; and corresponding to “the amount of interest” in our analogy, in quantum mechanics we have an operator. For example, in physics there are operators corresponding to energy, momentum, position, etc., and in the quantum decision model we will see that there can be operators like price operator, buy operator, pass operator, win operator, accept operator, etc. Finally, from some operations of the operators on the state of the system we would get the desired result just as a bank teller arrives at the amount of interest earned. The example above describes the method of getting the desired information in the framework of quantum mechanics. We present such a framework in Table 2. We describe the postulates of quantum mechanics (Messiah 1970, White 1966, Agrawal 1989) and also introduce those for the quantum decision model. One would note that there is one-to-one correspondence between these two sets of postulates. Just like the existence of “our account” in the above-mentioned example, postulate #1 described in Table 2 asserts that there exists a state ket —–” that represents/describes the system. Next, corresponding to the “amount of interest” in the above analogy, here in Table 2 we have postulate #2 that says that there exists an operator Ô associated with a measurable O. Further, just like the method of determination of the amount of interest, postulate #3 described in the table provides the recipe to compute the average value of O from the knowledge of the state of the system —–” and the operator Ô. According to this postulate, the average value of O is “O” = “–—Ô—–”0 For the meaning of the matrix element “–—Ô—–”, Equation (EC-1) and Table 5 in the e-companion would be helpful. To provide confidence and clarity, its application at many places has been illustrated in this section. While going through applications, the reader would note that the mathematical treatment discussed in this work does not need any complicated calculus or algebra or trigonometry. Postulates of quantum mechanics and quantum decision model. Regarding (1) State corresponding to a system (2) Operator corresponding to a measurable (3) Result of measurement Quantum mechanics Quantum decision model The dynamical state of a system can be fully represented by a state ket —–”. With every physical quantity (dynamical variable) O, an operator Ô can be associated. The average result of measurement, “O”, of O in state —–” is given by The state of a system can be represented by a state ket —–”. With every prospect or a variable O an operator Ô can be associated. The average result of measurement, “O”, of O in state —–” is given by “O” = “–—Ô—–”/“– — –”. When —–” is normalized, i.e., for “– — –” = 1, the above relation becomes “O” = “–—Ô—–”/“– — –”. When —–” is normalized, i.e., for “– — –” = 1, the above relation becomes “O” = “–—Ô—–”. “O” = “–—Ô—–”. (6) (6a) Agrawal and Sharda: Quantum Mechanics and Human Decision Making 5 Operations Research 61(1), pp. 1–16, © 2013 INFORMS One needs to be familiar with only a few notations such as —–”, “–—, “– — ””, and “–—Ô—””. In this regard, the description of terms and notations of quantum mechanics given in Appendix-B and C in the e-companion would be helpful. A state may be denoted by different kinds of notations (see Appendix-B). We have adopted Dirac’s notations (Dirac 1958) to denote the state ket. In Dirac’s notation, a state is denoted by a label placed in the symbol — ” (see Table 3). As an example of the state of a system, we again draw our attention to the problem of the two-stage gambling experiment. In Table 3, we have described various states. The first entry in this table gives a state —A” corresponding to accepting the second gamble, i.e., in this state (state of mind) the probability of accepting the second gamble is 100%. We denote this state by notation —A”. Here, symbol — ” is used to specify that it is a state ket (see Appendix-B). Before proceeding further, it may be appropriate for a reader to be familiar with all other states described in Table 3. In Table 4, we describe two operators, ÔW and ÔA . ÔW operator corresponds to the probability of winning the first gamble, and it may be named as “win operator.” ÔA operator corresponds to the probability of accepting the second gamble, and it may be named as “accept operator.” 2.1. Experimental Data to Be Explained Before going further, it would be appropriate to be familiar with, in our notations, the experimental data we need to Table 3. —A” —B” —X1 ” —X2 ” —AX1 ” —AX2 ” —BX1 ” —BX2 ” Notations regarding quantum states. Represents a state corresponding to accepting the second gamble. In this state the probability of accepting the second gamble is 100%. Represents a state corresponding to not accepting the second gamble. In this state the probability of not accepting the second gamble is 100%. Represents a state in which the probability of winning the first gamble is 100%. Represents a state in which the probability of winning the first gamble is 0%. Represents a state in which the probability of winning the first gamble is 100% and the probability of accepting the second gamble is 100%. It is the tensor product (see Appendix-C in the e-companion) of —A” and —X1 ”, i.e., —AX1 ” = —A”—X1 ”. Represents a state in which the probability of losing the first gamble is 100% and the probability of accepting the second gamble is 100%. It is the tensor product of —A” and —X2 ”, i.e., —AX2 ” = —A”—X2 ”. See the corresponding terms with A in this table. The difference between A and B symbols is in “accepting” and “not accepting.” Note. For mathematical simplicity all states described in this table are taken as normalized. Table 4. Notations regarding operators. ÔW (Win operator) Operator corresponding to the probability of winning the first gamble. It has eigenvalues 1 and 0. Eigenvalue = 1 corresponds to the winning of the first gamble and 0 corresponds to losing the first gamble. ÔA Operator corresponding to the probability of accepting the second gamble. It has (Accept operator) eigenvalues 1 and 0. Eigenvalue = 1 corresponds to accepting the second gamble and 0 corresponds to not accepting the second gamble. explain. The experiments performed by Tversky and Shafir (1992) reveal the following: p4A — X1 5 = 00691 p4B — X1 5 = 1 − p4A — X1 5 = 00311 (7) p4A — X2 5 = 00591 p4B — X2 5 = 1 − p4A — X2 5 = 00411 (8) p4A5 = 00361 and p4B5 = 1 − p4A5 = 00640 (9) The interesting part of these data is as follows: 69% of participants are ready to accept the second gamble if they know that they have won the first gamble, and 59% of participants are ready to accept the second gamble, even if they know that they have lost the first gamble. However, when they do not know the result of the first gamble, only 36% of the participants are ready to accept the second gamble. We note here that a majority of participants prefer to accept the second gamble in either case of win or lose, but when they are uncertain about winning or losing, then only a small fraction of the participants are ready to accept the second gamble. In literature (Tversky and Shafir 1992), this is known as the disjunction effect in choice under uncertainty. The above data contradicts the sure-thing principle given by Savage (1954). According to this principle, if a prospect x is preferred to y knowing that event R happens, and if x is preferred to y with the knowledge that event R did not happen, then x should be preferred to y even when the result of happening of R is unknown. This principle is considered as one of the basic axioms of the rational classical theory of decision under uncertainty. 2.2. Eigenkets and Eigenvalues Using the hermitian nature of the operators in quantum mechanics, in general, it can be shown that if in state —”i ” the result of measurement of O is ai with 100% certainty, then the following relation holds well: Ô—”i ” = ai —”i ”0 Here Ô is an operator associated with O. In quantum mechanics, we call —”i ” satisfying above relation as eigenket of operator Ô having eigenvalue ai (see Appendix-B in the e-companion for more details). The reverse is also Agrawal and Sharda: Quantum Mechanics and Human Decision Making 6 Operations Research 61(1), pp. 1–16, © 2013 INFORMS true, i.e., if the state of the system is an eigenstate of an operator Ô, then the result of measurement of O in that eigenstate would be the corresponding eigenvalue and the uncertainty or variance in that result of measurement would be zero. In the quantum decision model, we assume that kets (—–”), bras (“–—), and operators belong to the Hilbert space in the same way that we consider in quantum mechanics. Therefore, we get the same concept of eigenkets and eigenvalues in the quantum decision model as well. The scalar product “– — –” and matrix elements “–—Ô—–” are also defined in the same way as they are in quantum mechanics (see Appendix-B in the e-companion). In view of this description, we can say that state —A” is an eigenstate of operator ÔA corresponding to eigenvalue 1, and state —B” is an eigenstate of the same operator corresponding to eigenvalue zero. Similarly, —X1 ” is an eigenstate of operator ÔW corresponding to eigenvalue 1 and —X2 ” is an eigenstate of operator ÔW corresponding to eigenvalue 0. In equation form we can write ÔA —A” = —A”1 ÔA —B” = zero—B” = —0”1 ÔW —X1 ” = —X1 ”1 (10) ÔW —X2 ” = zero—X2 ” = —0”0 (11) Here —0” refers to a ket such that its scalar product with any bra equals zero. Before proceeding further, we need to be familiar with the basics of the tensor product space as described in Appendix-C. In the foregoing description, we consider two types of spaces. (1) A two-dimensional win-lose space: the operator ÔW , and states —X1 ” and —X2 ” belong to this space. (2) A two-dimensional accept-reject space: the operator ÔA , and states —A” and —B” belong to this space. As explained in Appendix-C, we can have a four-dimensional tensor product space as a product of these two spaces. The states —AX1 ”, —AX2 ”, —BX1 ”, and —BX2 ” (here —AX1 ” ≡ —A”—X1 ”, and so on) would belong to such tensor product space. Further, as explained in Appendix-C, for an operator corresponding to ÔA (or ÔW 5 of two-dimensional space we can have another similar operator in this four-dimensional space and, for the sake of convenience and without causing any confusion, such an operator in the four-dimensional tensor product space can be denoted by the same notation ÔA (or ÔW 5 (Messiah 1970). Thus, we can write ÔA —AX1 ” = —AX1 ” Figure 1. and ÔW —AX1 ” = —AX1 ”0 (12) For other similar equations, one may refer to Appendix-D in the e-companion. Further, we know that eigenkets belonging to different eigenvalues of a given operator are orthogonal (for details one may refer to Appendix-B in the e-companion). We use the same concept here. Thus, we shall have “X1 — X2 ” = “X2 — X1 ” = 01 “A — B” = “B — A” = 00 “AX1 — AX2 ” = “AX2 — AX1 ” = 0 (13) and “BX1 — BX2 ” = “BX2 — BX1 ” = 00 (14) For more equations along these lines, please refer to Appendix-D in the e-companion. Here “X1 — X2 ” is called the scalar product of kets —X1 ” and —X2 ”; for details, one may refer to Appendix-B in the e-companion. 2.3. Determination of the State of the System We have operators and eigenstates related to this experiment. All we need is to obtain the ket representing the state of the system. Let —”” gives the state of win/lose of the system. This state —”” in win-lose space (Hilbert space) would be a linear combination (for an illustration, see Figure 1) of the related eigenket —X1 ” corresponding to win and eigenket —X2 ” corresponding to lose. Thus, we can write —”” = 1 —X1 ” + 2 —X2 ”0 (15) For the sake of mathematical convenience, we choose the expansion coefficients 1 and 2 such that the ket —”” is normalized, i.e., “” — ”” = 1. This normalization condition would lead to —1 —2 + —2 —2 = 10 (16) As the ÔW operator corresponds to the probability of winning the first gamble, for computing the probability of winning the first gamble we shall employ Equation (6a) and this operator ÔW . Using Equation (6a), for the probability of winning the first gamble, we can write p4X1 5 = “”—ÔW —””0 (17) In view of Equations (15) and (11) we can get ÔW —”” = 1 —X1 ”0 (18) For the bra vector “”—, Equation (15) leads to “”— = ∗1 “X1 — + ∗2 “X2 —0 (See Table 5 in the e-companion.) (19) Here superscript ∗ has been used to indicate the complex conjugate, i.e., ∗1 is complex conjugate of the number 1 . The win-lose state —”” of the first game is a linear combination of win state (—X1 ”) (“smiling face”) with probability amplitude 1 , and the lose state (—X2”) (“sad face”) with probability amplitude 2 (see Equation (15)). Win
1 | X1 〉 Lose +
2 | X2 〉 |〉 =
1 | X1〉 +
2 | X2 〉 Agrawal and Sharda: Quantum Mechanics and Human Decision Making 7 Operations Research 61(1), pp. 1–16, © 2013 INFORMS Because the sum of p4A — X1 5 and p4B — X1 5 equals 1, using Equation (24) and (26) one can obtain By combining Equations (17)–(19), we get p4X1 5 = “”—ÔW —”” = 6∗1 “X1 — + ∗2 “X2 —761 —X1 ”7 = 6∗1 1 “X1 — X1 ” + ∗2 1 “X2 — X1 ”70 (20) p4B—X1 ” = 1 − p4A — X1 5 = —b1 —2 0 Because —X1 ” is normalized, therefore, “X1 — X1 ” = 1. Equation (20) in combination with this normalization condition and Equation (13) leads to Similarly, for the state of mind —–2 ” (for an illustration, see Figure 3) associated with losing the first gamble, one can write p4X1 5 = ∗1 1 = —1 —2 0 (21) —–2 ” = a2 —AX2 ”+b2 —BX2 ”1 (22) Again, using operator ÔA , Equations (6a), (27), (EC-13), (EC-15), (EC-25), and the normalization property 6“AX2 — AX2 ” = 17, we can get From this result, one can obtain 2 p4X2 5 = 1 − p4X1 5 = —2 — 0 It is obvious that the win/lose is not a matter of decision by the players in this experiment. Further, we know that among the participants who have won the first gamble, some would accept the second gamble and some would not. As —AX1 ” corresponds to winning the first gamble and accepting the second gamble, and —BX1 ” corresponds to winning the first gamble and rejecting the second gamble, therefore, the state of mind —–1 ” associated with the winning of the first gamble can be expressed (for an illustration, see Figure 2) as a linear combination of —AX1 ” and —BX1 ”: where —a2 —2 +—b2 —2 = 10 p4A—X2 ” = “–2 —ÔA —–2 ” = —a2 —2 0 (27) (28) Further, because the sum of p4A — X2 5 and p4B — X2 5 equals 1, therefore, using Equations (27) and (28), one can have p4B—X2 ” = 1 − p4A—X2 ” = —b2 —2 0 The normalization condition requires “–1 — –1 ” = 1. This would lead to To arrive at the state of mind —–” representing all subjects (sum of those who are losing and those who are winning the first play), we would have linear combination of states —–1 ” and —–2 ”. The expansion coefficients for this combination must be the same as those occurring in Equation (15), i.e., —a1 —2 + —b1 —2 = 10 —–” = 1 —–1 ” + 2 —–2 ”0 —–1 ” = a1 —AX1 ” + b1 —BX1 ”0 (23) (24) For computing the probability of accepting the second gamble with the knowledge of winning the first gamble, we can use Equation (6a), operator ÔA , and state —–1 ”. Thus, we have (29) The above equation with the help of Equations (23), (12), (EC-11), (EC-22), and the normalization property 6“AX1 — AX1 ” = 17 leads to This equation implies that the state —X1 ” of Equation (15) in the win-lose space becomes state —–1 ” in the win-loseaccept-reject space; the state —X2 ” of Equation (15) in the win-lose space becomes state —–2 ” in the win-lose-acceptreject space; and the state —”” of the same equation in win-lose space becomes state —–” in win-lose-accept-reject space. On substituting the value of —–1 ” given by Equation (23) and of —–2 ” given by Equation (27) into Equation (29), we obtain p4A — X1 5 = “–1 —ÔA —–1 ” = —a1 —2 0 —–” = c1 —AX1 ” + c2 —AX2 ” + c3 —BX1 ” + c4 —BX2 ”1 p4A — X1 5 = “–1 —ÔA —–1 ”0 Figure 2. (25) (26) (30) Win state leads to win and accept state (rectangle and “smiling face”) with probability amplitude a1 , and reject-win state (triangle and “smiling face”) with probability amplitude b1 . The linear combination of these two states has been expressed as —–1 ” (see Equation (23)). The factor 1 in every block reminds the probability amplitude of the win state (see Figure 1). Accept Win
1a1 | AX1〉
1a1 | AX1〉 +
1b1 | BX1〉 =
1 | ψ1〉
1 | X1〉 Reject
1b1 | BX1〉 Agrawal and Sharda: Quantum Mechanics and Human Decision Making 8 Figure 3. Operations Research 61(1), pp. 1–16, © 2013 INFORMS Lose state leads to accept and lose state (rectangle and “sad face”) with probability amplitude a2 , and rejectlose state (triangle and “sad face”) with probability amplitude b2 . The linear combination of these two states has been expressed as —–2 ” (see Equation (27)). The factor 2 in every block reminds the probability amplitude of the lose state (see Figure 1). Accept
2 a2 | AX2〉 Lose
2 a2 |AX2〉 +
2 b2 |BX2〉 =
2 | ψ2〉
2 | X2 〉
2 b2 | BX2〉 Reject where and, using Equations (34) and (31) we can get c1 = 1 a1 1 c2 = 2 a2 1 c3 = 1 b1 1 and c4 = 2 b2 0 (31) p4AX2 5 = —c2 —2 0 (35) We can similarly have It may be noted that —AX1 ” represents a state corresponding to accepting the second gamble and winning the first gamble, —AX2 ” represents a state corresponding to accepting the second gamble and not winning the first gamble, —BX1 ” represents a state corresponding to not accepting the second gamble but winning the first gamble, and —BX2 ” represents a state corresponding to not accepting the second gamble and not winning the first gamble (for an illustration see Figure 4). These are four orthonormal eigenkets in four-dimensional win-lose-accept-reject space. p4BX1 5 = —1 —2 —b1 —2 = —c3 —2 1 2 2 and 2 p4BX2 5 = —2 — —b2 — = —c4 — 0 (36) Using Equations (32)–(36), (31), (27), (24), and (16), it can be seen that p4AX1 5 + p4AX2 5 + p4BX1 5 + p4BX2 5 = —c1 —2 + —c2 —2 + —c3 —2 + —c4 —2 = 10 (37) The above equation in combination with Equation (31) leads to An alternative way to arrive at these results may be to employ Equation (30) together with projection operators (Messiah 1970). —–” given by Equation (30) is a linear combination of four orthonormal kets —AX1 ”, —AX2 ”, —BX1 ”, and —BX2 ”. —AX1 ” represents a state in which the probability of winning the first gamble and accepting the second gamble is 100%. Therefore, for computing the joint probability of winning the first gamble and accepting the second gamble, one needs to determine the relative weight of state —AX1 ” in the combination given by Equation (30). This can be done by using Equation (6a) with Ô as the projection operator —AX1 ”“AX1 — (Messiah 1970). Thus, we get p4AX1 5 = —c1 —2 0 p4AX1 5 = “–—4—AX1 ”“AX1 —5—–” 2.4. Analysis of Probabilities For evaluating the joint probability of winning the first gamble and accepting the second gamble, p4AX1 5, we can employ Equations (1), (21), and (26) and obtain p4AX1 5 = —1 —2 —a1 —2 0 (32) (33) = 4“– — AX1 ”54“AX1 — –”5 = c1∗ c1 = —c1 —2 0 Again, using Equations (2), (22), and (28) we can have 2 2 p4AX2 5 = —2 — —a2 — 1 Figure 4. (34) (38) Similarly, for p4AX2 5, p4BX1 5, and p4BX2 5 we can get the results, in agreement with those given by Equations (35) The state of the system —–” given by Equation (30) is a linear combination of four states corresponding to win-accept, win-reject, lose-accept, and lose-reject states. Win-Accept
1 a1 | AX1〉 + Win-Reject
1 b1 | BX1〉 + Lose-Accept
2 a2 | AX2〉 + Lose-Reject
2 b2 | BX2〉 = |ψ〉 Agrawal and Sharda: Quantum Mechanics and Human Decision Making 9 Operations Research 61(1), pp. 1–16, © 2013 INFORMS and (36), by using Equation (6a) and the corresponding projection operators, —AX2 ”“AX2 —, —BX1 ”“BX1 —, and —BX2 ”“BX2 —, respectively. 2.5. Analysis of the Situation When Outcome of the First Game Is Unknown Equation (30) represents a state that contains the information regarding winning and losing. The experiments of Tversky and Shafir (1992), performed with 98 subjects to determine p4A—X1 5 and p4A—X2 5, can be described by this state together with Equation (15). Next, for the determination of probability of accepting the second gamble without any information regarding winning and losing the first gamble, p4A5, they performed the experiment after 10 days. In the experiments, there is no magic wand to wash the information of winning and losing from the mind of a participant. Therefore, Tversky and Shafir (1992) used time to wash the memory. However, in mathematics, one can wash the information regarding pass and fail contained in Equation (30) by deleting X1 and X2 labels. Thus, corresponding to the state of mind of another batch of participants considered by Tversky and Shafir (1992), we would have a state —– 0 ” that can be obtained by ignoring X1 and X2 in —–” given by Equation (30), i.e., —– 0 ” = c10 —A” + c20 —A” + c30 —B” + c40 —B” = 4c10 + c20 5—A” + 4c30 + c40 5—B”0 (39) Equation (39), we shall see that one gets p4A5 that equals to a sum of p4AX1 5, p4AX2 5, and an interference term. The existence of such an interference term is not possible in the classical framework. It is a special feature of the quantum framework. The difference in the expectation values of operator ÔA in the states given by Equations (30) and (39) can be compared by using an analogy of a double slit experiment of physics that is performed to study interference of electrons with and without a watch over electrons near the slits. We know that (see Feynman et al. 1966a) the interference pattern is not observed when the electrons are watched to know the slit through which they pass. However, when we do not have such a watch, then we get the interference pattern. We may also look at Equations (30) and (39) from the reverse perspective. In the double-slit experiment (see Feynman et al. 1966a), if we want to watch the electrons, then we need to put the detectors near slits. In the same way, here, if we want to insert the knowledge of outcome of the first gamble in Equation (39), then we need to put X1 and X2 in Equation (39) such that we get the values of p4AX1 5, p4AX2 5, p4BX1 5, and p4BX2 5 the same as given by Equation (30) (see Equations (33), (35), and (36)). This would be possible when the values of coefficients ci and ci0 4i = 1–45 are such that their magnitudes are equal (see Equation (40)). We can now compute p4A5 by using Equations (6a) and (39), and the operator ÔA : We would see that ci0 , (i = 1–4) in this equation, are the same as ci occurring in Equation (30) except that they may differ in the phase factor, i.e., p4A5 = “– 0 —ÔA —– 0 ”0 —c10 — = —c1 —1 —c20 — = —c2 —1 —c30 — = —c3 —1 and Using the above equation together with Equations (39), (10), and (13), and the normalization condition “A — A” = 1, we obtain —c40 — = —c4 —0 (40) Thus, we can express ci0 in terms of their absolute values and the phase angles, say ˆi , as follows: c10 = —c10 — exp4iˆi 5 = —ci — exp4iˆi 50 (It may be noted that exp4iˆ5 = cos4ˆ5 + i sin4ˆ5, and i2 = −1 (see Feynman et al. 1966b).) Regarding the phase angles ˆi , the values depend on the states of mind of the participants. For our purpose, here it may be sufficient to make sure that the values must be such that —– 0 ” described by Equation (39) is normalized so that the sum of p4A5 and p4B5 given by —– 0 ” satisfies Equation (5). It can easily be seen that if we compute the expectation value of operator ÔA to get the probability of accepting the second gamble with the information regarding winning and losing using Equation (30), we would have the absence of interference between the probability amplitude terms c1 and c2 , i.e., we would simply get this probability as a sum of p4AX1 5 and p4AX2 5. However, with the state given by p4A5 = —c10 + c20 —2 = —c10 —2 + —c20 —2 + c10 ∗ c20 + c20 ∗ c10 0 (41) In view of Equations (33), (35), and (40), the above equation leads to p4A5 = p4AX1 5 + p4AX2 5 + qint 4A51 (42) where qint 4A5 = c10 ∗ c20 + c20 ∗ c10 0 (43) Because c10 and c20 can be complex numbers, in view of Equations (33), (35), and (40), we can express c10 and c20 in terms of their absolute values, and the respective phase angles ˆ1 and ˆ2 as follows: c10 = —c10 — exp4iˆ1 5 = 6p4AX1 571/2 exp4iˆ1 51 and c20 = —c20 — exp4iˆ2 5 = 6p4AX2 571/2 exp4iˆ2 50 (44) (45) With these expressions, Equation (43) yields qint 4A5 = 26p4AX1 5p4AX2 571/2 cos4ˆ2 − ˆ1 50 (46) Agrawal and Sharda: Quantum Mechanics and Human Decision Making 10 Operations Research 61(1), pp. 1–16, © 2013 INFORMS Because the minimum and maximum possible values of the cosine are −1 and +1, respectively; therefore, using Equation (46) one can find that the interference term qint 4A5 satisfies the following relation: −26p4AX1 5p4AX2 571/2 ¶ qint 4A5 ¶ 26p4AX1 5p4AX2 571/2 0 (47) Similarly, using Equations (6a) and (39), and the operator [Î − ÔA ] for the probability of not accepting the second gamble, (here Î is a unit operator), one can get p4B5 = p4BX1 5 + p4BX2 5 + qint 4B51 (48) where1 qint 4B5 = c30 3.1. Explanation of the Two-Stage Gambling Experiment For the coin-tossing experiment performed by Tversky and Shafir (1992), one can take p4X1 5 = 005 and p4X2 5 = 005. Further, using Equations (1), and (2) and the data given in Equations (7) and (8), we can obtain p4AX1 5 = 00345 and p4AX2 5 = 00295. Classically, one expects the value of p4A5 as sum of p4AX1 5 and p4AX2 5 4= 00645. Against this expectation, we here get p4A5 = 0036 (see Equation (9)). Equation (42), however, shows that this anomalous behavior can be explained by the value of qint 4A5 equal to −0028. This value of qint 4A5 = −0028 is consistent with Equation (47), which gives −00638 ¶ qint 4A5 ¶ 006380 ∗ c40 + c40 ∗ c30 0 (49) By writing equations similar to (44) and (45) for c30 and c40 , we can get qint 4B5 = 26p4BX1 5p4BX2 571/2 cos4ˆ4 − ˆ3 50 (50) Further, as mentioned earlier, we need phase angles such that the sum of p4A5 given by Equation (42) and p4B5 given by Equation (48) satisfies Equation (5). Combining Equations (42) and (48), we obtain p4A5 + p4B5 = p4AX1 5 + p4AX2 5 + p4BX1 5 + p4BX2 5 + qint 4A5 + qint 4B50 (51) Equation (51) in association with Equations (37) and (5) gives qint 4A5 + qint 4B5 = 00 (52) Thus, the above equation, together with Equations (50) and (46), provides the relationship between cos4ˆ2 − ˆ1 5 and cos4ˆ4 − ˆ3 5, which must hold well to ensure the validity of Equation (5). It may be noted that the main results of this treatment, Equations (42), (46), (48), and (50), have also been derived by Yukalov and Sornette (2009a) using the postulates and states very similar to those described here. The treatment presented here mainly differs from their treatment in the consideration of a different kind of operators. 3. Application to Decision-Making Problems In this section, we discuss the applicability of the equations presented in the previous section to the experiments of Tversky and Shafir (1992). We also compare various models and describe the possibility of application of the quantum models to other decision-related problems. (53) This consistency suggests that this treatment based on the interference occurring in the quantum decision model cannot predict in advance the results of the experiment but can explain the results. It may be noted that these experimental results violate the classical axiom known as Savage’s surething principle (1954). Further, if we consider cos4ˆ2 − ˆ1 5 of Equation (46) as an adjustable parameter, then we can say that the experimental data can be explained by assigning cos4ˆ2 − ˆ1 5 = −004390 (54) This value of cos4ˆ2 − ˆ1 5 with Equation (46) leads to qint 4A5 = −0028, which can explain the experimental results for accepting the gamble. Similarly, for the probability of rejecting to play the second gamble, we can get following results: p4BX1 5 = 001551 qint 4B5 = 00281 p4BX2 5 = 002051 and p4B5 = 00641 (55) cos4ˆ4 − ˆ3 5 = 007850 (56) Yukalov and Sornette (2009a) argue that under uncertainty (the lack of knowledge of the outcome of the first game) the decision in favor of an “action” (accepting to play the second game) is more difficult than that in favor of an “inaction” (not to play). Therefore, with this assumption, out of two terms, qint 4A5 and qint 4B5, that add to 0 (see Equation (52)), we can say so much in advance that qint 4A5 would be negative and qint 4B5 would be positive. Can we think of any method of predicting the value of qint 4A5 or phase angles ˆ2 and ˆ1 in advance? The visualization of any experimental procedure to determine phase angles in advance without any knowledge of p4A5 or qint 4A5 seems to be beyond the scope of our present understanding. Probably there may be some link between the phase angles and the distribution of time taken by different subjects in making a particular decision. The problem, however, is very complex and may be a matter for future research. At present, we can only say that using quantum theory we are able to explain the results of the two-stage gambling experiment, which could not be explained by any classical formulation such as the sure-thing principle of Savage (1954) or classical Markov model studied by Pothos and Busemeyer (2009). Agrawal and Sharda: Quantum Mechanics and Human Decision Making 11 Operations Research 61(1), pp. 1–16, © 2013 INFORMS 3.2. Explanation of the Buy-or-Not-to-Buy Experiment In this experiment of Tversky and Shafir (1992), the subjects (undergraduate students at Stanford University) were asked to imagine that they had just taken a tough examination, and at the end of the fall quarter an attractive Christmas vacation package to Hawaii at a very low price was being offered to them. One group of 67 subjects (say group 1) was asked to imagine that they passed the examination, another group of 67 (say group 2) was asked to imagine that they failed the examination, and the third group of 66 (say group 3) was asked to imagine that the outcome of their examination was not known to them. In this experiment, 54% of subjects from group 1, 57% of subjects from group 2, and 32% of subjects from group 3 were ready to buy the vacation package. The analysis of these data has been presented and discussed in Appendix-E in the e-companion. From the analysis presented there, we note that these experimental results can be explained by relations similar to the interference equations derived and discussed in §2.5. Thus, again we see that by using a quantum model we are able to explain the results of buy-ornot-to-buy experiment of Tversky and Shafir (1992), which could not be explained by any classical formulation such as the sure-thing principle of Savage (1954). 3.3. Classical vs. Quantum Model In classical formulation, the interference term qint 4A5 given in Equation (42) remains absent. According to the classical statistics, a simple addition of probabilities p4AX1 5 and p4AX2 5 equals p4A5. To explain the results of twostage gambling experiment and other experiments, Pothos and Busemeyer (2009) have employed quantum as well as Markov (classical) models. While comparing different models in §2 of their work, Pothos and Busemeyer (2009, p. 2174) write: “Although cognitive dissonance tendencies can be implemented in both the Markov and quantum models, we shall see that it does not help the Markov model, and only the quantum model explains the sure thing principle violations.” The following essential differences between the classical and quantum formulations are worth noting: (a) In a quantum model, the probabilities are expressed as the squares of the probability amplitudes c1 1 c2 1 0 0 0 (see Equations (33), (35), and (36)). In classical formulation, we do not have such terms as probability amplitudes. (b) In a quantum model, for getting p4A5 instead of adding the respective probabilities, the respective probability amplitudes get added, and then p4A5 is obtained by squaring the probability amplitudes. Thus, the probability amplitude for p4A5 equals (c10 + c20 ), and p4A5 equals —4c10 + c20 5—2 (see Equation (41)). The interference term qint 4A5 automatically appears when we compute the value of —4c10 + c20 5—2 . However, in classical formulation, instead of addition of probability amplitudes the probabilities are added. Thus, classically, p4A5 equals to the sum of p4AX1 5 and p4AX2 5. (c) The probability amplitudes, in general, may be complex numbers (see Equations (44)–(45)). The phase factors affect the magnitude of the interference term, qint 4A5. In classical framework, we do not have such phase factors or complex numbers in connection with the probabilities. It may be added that the value of the interference term given by a quantum model may be 0 also. In such a case, the quantum results merge to the classical results. This happens when the value of cosine term of Equations (46) or (50) equals 0. Without an explanation of the disjunction effect with the help of a quantum interference term, one may only say that the disjunction effect, as observed in the two-stage gambling experiment, may be due to lack of sound thinking of the subjects under uncertainty. In this regard, Tversky and Shafir (1992, p. 305) write: “We suggest that, in the presence of uncertainty, people are often reluctant to think through the implications of each outcome and, as a result, may violate STP.” 3.4. Quantum Decision Theory of Yukalov and Sornette Various features presented here are the same as given by the quantum decision theory (QDT) of Yukalov and Sornette (2009a). The main difference is in the selection of operators Ô associated with Equation (6a). In §2, we have considered the operators characterized by their explicit operations (such as “win,” “accept”) through their eigenkets and eigenvalues. In QDT, the operators are expressed in terms of prospect states. For example, they consider an operator (—11 —2 —AX”1 “AX1 —) corresponding to the prospect state (11 —AX1 ”). By prospect state they mean the state in which one is interested in reaching from the given state of mind. It may be noted that in physics both kinds of operators (the operators associated with observables similar to what we have described here, and operators of kind —s”“s— (a ket multiplied by a bra on right) similar to what Yukalov and Sornette (2009a) have employed) are used. It is interesting to note that the final results given by Equations (42), (46), (48), (50), (53), (54), (EC-33), and (EC-34) are in total agreement with those given by Yukalov and Sornette (2009a). Regarding Equations (33), (35), (36), and (41), these results would also be in agreement with the QDT results, provided their parameters have the following values: —ij — = 11 for 4i = 11 21 and j = 11 250 (57) Here parameters ij correspond to the expansion coefficients occurring in Equations (27) and (28) of Yukalov and Sornette (2009a) (for a 2 × 2 dimensional case; i = 1 and 2 correspond to our A and B, and j = 1 and 2 correspond to our X1 and X2 , respectively). Agrawal and Sharda: Quantum Mechanics and Human Decision Making 12 In absence of the knowledge of the explicit four equations required for the determination of 11 , 12 , 21 , and 22 , it is not possible to know the various possible values of these parameters of Yukalov and Sornette (2009a). However, it can be verified that the values of these parameters given by Equation (57) do not disagree with the requirements described by Yukalov and Sornette (2009a). In view of the fact that Equation (57), which has been obtained by comparing the present results and those of QDT, does not disagree with the requirements described by Yukalov and Sornette (2009a), and because there are not a sufficient number of equations to determine these parameters ij , the present work also becomes valuable to the formulation of Yukalov and Sornette (2009a) in the sense that it provides a way (if not “the way”) to determine 11 , 12 , 21 , and 22 . In the present work, for determining the probability of accepting the second gamble when the result of the first gamble is not known, for the state of mind of the participants we employ Equation (39), not Equation (30). In addition, we use Equation (6a) and the accept operator ÔA to compute the probability of accepting the second gamble. Thus, we do not involve variables X1 and X2 related to win and lose in such a determination. However, Yukalov and Sornette (2009a) employ the state of mind given by Equation (30) and an operator that includes variables X1 and X2 . It is interesting to note that their final results are exactly the same as obtained here [Equations (42) and (46)]. Such an agreement creates scope for further advancement in this area. A search for the cause of arriving at the same result by two different paths may be helpful in gaining a finer understanding of the paths. 3.5. Khrennikov’s Coefficient of Interference In an attempt to explain some experiments related to the cognitive decision making and information processing, Khrennikov (2009) describes the necessity of quantum or quantum-like models. He explains that if classical mechanics holds well, then qint 4A5 of Equation (42) would be equal to zero, and in case of nonclassical behavior, ‹ = qint 4A5/826p4AX1 5p4AX2 571/2 9 can be called the coefficient of interference. In view of Equation (46), this definition of coefficient of interference shows that ‹ = cos4ˆ2 − ˆ1 5. For the data related to the experiment performed by Tversky and Shafir (1992), Khrennikov reports ‹ = −0044, in agreement with Equation (54). Khrennikov further adds that when the experimental data satisfy Equation (46) or (47), i.e., when —‹— ¶ 1, then we can say that the results are covered by the conventional quantum model, but when —‹— > 1, then we are outside the conventional quantum model. Khrennikov also discusses an example for which —‹— > 1 holds well. He calls such interference that is not covered by the quantum formalism as the hyperbolic (or cosh-type) interference. However, in his words (Khrennikov 2009, p. 186), “the conventional quantum formalism can be used as the simplest nonclassical model for mental and social modeling.” Operations Research 61(1), pp. 1–16, © 2013 INFORMS 3.6. Quantum Model of Pothos and Busemeyer The disjunction effect related to the two-state gambling experiment of Tversky and Shafir (1992) has also been explained by Pothos and Busemeyer (2009), using an alternative quantum model and a few adjustable parameters. Their quantum probability model is based on an equation parallel to the time-dependent Schrödinger equation of quantum mechanics (e.g., see Messiah 1970). They employ this equation to study the effect of external information on the evolution of the state of the mind with time. Besides the explanation of the results related to the psychology experiments, the concluding remarks of Pothos and Busemeyer (2009, p. 2177) regarding the quantum nature of human cognition are worth noting. They write, “Finally, recent results in computer science have shown quantum computation to be fundamentally faster compared with classical computation, for certain problems (Nielsen and Chung 2000). Perhaps the success of human cognition can be partly explained by its use of quantum principles.” 3.7. Application to Other Decision Problems The above description reveals that in many situations the decisions taken by a human mind cannot be understood by a classical model but can be explained by a quantum model. The usefulness of the equations related to quantum mechanics in psychology suggests that quantum models may also be useful to other disciplines where the human psychology plays an important role. One can argue that a successful and valuable application of quantum models to the mainstream of business-related decisions is not a matter of “if” but of “when,” “where,” and “how.” In this connection, the following remarks of Overman (1996, p. 88) presented a while back are also worth noting: “The experimentation and adoption of the metaphors and methods of chaos and quantum theory hold new promise for the management sciences in the next century. It is not so much that traditional social scientific methods have become obsolete; it is that we have a continuing need to expand the scope and power of our methods just to keep pace with our organizational realities.” Group decision making in virtually any setting (e.g., DeSanctis and Gallupe 1987), collaborative or extended supply chain management (e.g., Guide and Van Wassenhove 2009), merger and acquisition of business firms (e.g., DePamphilis 2010), are some examples of the decision-related problems of interest. Can quantum mechanics provide a new and valuable insight in such areas? The challenging task in tackling such problems is to find the operators and eigenkets corresponding to the key variables associated with the problem. It may be a long way from realizing the potential of a quantum model and arriving at a successful and valuable application of the quantum mechanical framework to the mainstream of business. However, at this stage, it may be worthwhile to see how some issues related to a business problem can be expressed in the notations of the quantum decision model. Agrawal and Sharda: Quantum Mechanics and Human Decision Making 13 Operations Research 61(1), pp. 1–16, © 2013 INFORMS With this objective, we consider an example of merger/ acquisition of a firm (say, firm B) by another firm A. First, in addition to the financial interests, the psychology of managers and board members of firm A and B plays a major role in arriving at the merger deal. Next, immediately after the announcement of the merger deal, the role of the public perception influenced by the comments and analysis by experts regarding the synergy, ego, and other aspects related with the merger influences the share price of firms A and B. However, here our interest is just to introduce the concept; therefore, we shall limit our discussion to the state of mind of the acquiring firm. This example would serve an additional purpose. It would illustrate that a quantum decision model has a wider applicability. One can use it even when interference effect, similar to that described by Equations (42) and (46), does not take place. —OA ” denotes a component of the state of the mind of firm A that may contain fear (fear of competitors acquiring B, in case this proposed merger of B with A does not take place), ego (ego of becoming a big firm), greed, and other factors, such as future synergy perceived by A but not currently perceived by the public, that do not contribute to the market price within a short duration after the announcement of the deal. The firm A may realize that the public would not be able to visualize some synergy factors in the near future. According to A, such synergy factors would not contribute to the market value of A immediately after the announcement of the merger deal. The effect of such synergy factors are also absorbed in the coefficient a2 . In Appendix-F in the e-companion, using the concept of a price operator P̂A and related eigenkets and eigenvalues, we have shown that the expected market value of the stock of A as perceived by A in the state of mind —ëA ” is 3.8. Merger and Acquisition Problem: State of Mind of the Acquiring Firm P A = “ëA —P̂A —ëA ” = 41 − —a2 —2 5pa KA 0 There is a large body of literature in finance and strategy that has studied mergers and acquisitions (e.g., Malmendier and Tate 2008, Morellec and Zhdanov 2005, Shleifer and Vishny 2003, Tichy 2001, Andrade et al. 2001). The purpose of this illustration here is only to show the potential application of quantum mechanics to analysis of this common phenomenon. Thus, we take a simplified view of mergers and acquisition issues. Let us denote the state of mind of the acquiring firm A by —ëA ”. Using the expansion postulate (see Appendix-B in the e-companion), it can be expressed as a linear combination of related eigenkets of the price operator: This equation is very interesting. If a2 = 0, then —ëA ” = —KA ”, and in this state of mind the firm A expects to see the price of their stock after the announcement of the merger to be pa KA . However, the presence of the nonzero value of a2 shows that firm A is ready to be satisfied if the market value of their stock is 41 − —a2 —2 5 times pa KA . In other words, in view of other possibilities, such as fear of competitors, demand of ego to be a big firm, greed, and future synergy perceived by them but not perceived by the public, they are ready to sacrifice the price of one share (immediately after announcement of the merger deal) by an amount equal to —a2 —2 pa KA in favor of firm B. The value of —a2 — may be unknown to them, but at the negotiation table —a2 — may evolve to the right size to match the state of mind of firm A with that of B, in case the merger deal is done. It may be noted that in some situations, P A given by Equation (60) may be less than pa . For KA = 1, it is certainly less than pa when —a2 —2 is nonzero. The presence of the a2 term in Equation (58) is not to be considered as a weakness of A. Its existence facilitates the merger. Due to this a2 term, gain to B becomes larger than that to A, and the deal becomes possible. If the negotiating team members are chosen such that in their mind the value of a2 is very small, then the chances of concluding a deal could be low. On the other hand, if it is very large, then the premium payable to B would be very large. In addition to the advantage of the synergy factor to B, firm B would have an additional gain due to the sacrifice made by A due to the presence of the a2 term. If the total number of shares of firm A is NA , then the sacrifice —a2 —2 KA pa per share by A would mean an additional gain equal to —a2 —2 KA pa 4NA /NB 5 to one share of B. Because 4NA /NB 5 is usually very large compared to 1, the gain per share may be very large to B for even a small value of a2 . It is interesting to note that the market value of stock of the acquiring firm usually falls on the announcement of the —ëA ” = a1 —KA ” + a2 —OA ”0 (58) Here a1 and a2 are expansion coefficients. The normalization condition 6“ëA — ëA ” = 1] gives —a1 —2 + —a2 —2 = 10 (59) —KA ” denotes a component of the state of the mind of firm A related to the financial factors and public perception (as viewed by the firm A) responsible for the stock price of A. Here, the symbol KA is chosen with an additional purpose. KA in —KA ” is such that the market value of stock A, as perceived by firm A, after the announcement of the merger is KA times the present market value (pa ) of stock A. The value of KA is usually close to unity. The firm A is of the view that the merger would be welcomed by the public such that the net value of all shares of A and B would be S 4A5 times the current market value of the same, where S 4A5 is a synergy dependent factor. The synergy factor S 4A5 as perceived by A is absorbed in KA . KA may be time dependent, but here we are confining our attention to the value of KA within a short duration after the announcement of the merger. (60) Agrawal and Sharda: Quantum Mechanics and Human Decision Making 14 merger. A study conducted by Andrade et al. (2001) over 3,688 completed mergers during 1973–1998 reveals that an acquiring firm on average loses 0.7% and the target firm gains 16% on announcement. The interesting point of this study is that these data are nearly the same for the three decades of the study: 1973–1979, 1980–89, and 1990–98. The simple model presented above illustrates the potential of applying a quantum mechanical framework to study phenomena encountered in practice where the state of mind of the players can be taken into account. Of course, measurement of parameters such as a2 remains a topic for further study. 4. Summary and Concluding Remarks The most successful theory of physics, quantum mechanics, has a wide range of applicability in physics, chemistry, biology, cosmology, etc. In recent years, its potential application to quantum cryptography and computation is also being explored. In social sciences, its application to economics and psychology appears very promising. Recently observed success of quantum models (e.g., Pothos and Busemeyer 2009, Khrennikov 2009, Yukalov and Sornette 2009a) in explaining some experimental results of psychology, such as the disjunction effect observed by Tversky and Shaffir (1992), that could not be explained by the conventional (classical) theories adds a quantum dimension to the human decision-making process. In view of such success, one may argue for the need to explore use of quantum mechanics in other branches of knowledge such as business where human decision making is involved. With an objective of introducing various important and basic concepts and mathematical equations associated with any quantum treatment, here we have described a quantum model with focus on the explanation of the disjunction effect experimentally observed by Tversky and Shaffir (1992). Due to basic differences between the physical objects and the human decision-making processes, one expects to see some differences between the quantum mechanics as used in physical sciences and the quantum models applicable to the human decision-making processes. With a minimum number of changes in the basic terms, postulates, and equations of quantum mechanics, here we have described a quantum decision model suitable for the decision-making processes. Essentially, the model described here is based on the quantum decision theory developed by Yukalov and Sornette (2009a). The application of postulates analogous to those of quantum mechanics and the selection of state of mind are similar to that given by Yukalov and Sornette (2009a). However, here, instead of prospect operators, we have considered more general and simpler kinds of operators (such as win operator, accept operator) to derive the same final results as derived by Yukalov and Sornette (2009a) so that the range of applicability may widen and it becomes easier to apply to the business-related problems. Operations Research 61(1), pp. 1–16, © 2013 INFORMS The description of basic postulates of quantum mechanics and corresponding postulates of the quantum decision model, in almost similar words, has been presented in Table 2. While introducing terms of quantum mechanics necessary for the quantum decision model, we have followed an approach of minimum necessary details in §2 with additional information in Appendices-B and C in the e-companion. Further, with an objective of ease in comprehension, we have introduced operators, eigenkets, eigenvalues, state of mind, etc. with practical examples, and avoided a formal rigor associated with a general case. It is expected that with the understanding of the application of these terms and equations as described here, it would be easier to follow the related formal rigorous terminology of quantum mechanics as described in the textbooks of physics for more advanced applications. To illustrate the success and to explain the various terms and equations of the quantum model, first it has been applied to two experiments performed by Tversky and Shafir (1992) related to the disjunction effect. As discussed earlier, the results of these experiments could not be explained by the conventional (classical) theories, but have been explained in recent years by various quantum models (Pothos and Busemeyer 2009, Khrennikov 2009, Yukalov and Sornette 2009a). The study of merger/acquisition of two business firms has been chosen to consider a business-related decision problem. Various concepts of the model such as price operator (P̂A 5 and its eigen kets [—KA ”1 —OA ”] with the corresponding eigen values [pa KA , and zero], the normalization and orthogonality conditions [Equations (EC-37) and (EC-38)], expansion coefficients [a1 , a2 , and their relationship given in Equation (59)], etc. related with this problem have been discussed. In this model, we also note the significance of the term a2 —OA ”, which leads to a decrease in the market value of the stock of the acquiring firm and an increase in the market value of the stock of the acquired firm immediately after the announcement of the merger deal. More investigations are required to incorporate the effect of the views of the board members, share holders, and public perception in the quantum exploration of the merger problem. In addition to quantum interference to explain the disjunction effect, and quantum superposition of states to explain the merger problem, quantum mechanics provides many special features such as Heisenberg’s uncertainty principle, quantum tunneling, quantum theory of measurement, etc. (e.g., Razavy 2003, Messiah 1970, Stenholm and Suominen 2005, Bohm 1959), which can accommodate various kinds of diversities associated with human decisions, in general, and business-related decisions, in particular. Further investigations on the quantum models of information exchange and market psychology (Choustova 2007, Haven 2008), quantum finance (Schaden 2003, Baaquie 2009a, Bagarello 2009), application of quantum mechanics to human dynamics related with Agrawal and Sharda: Quantum Mechanics and Human Decision Making Operations Research 61(1), pp. 1–16, © 2013 INFORMS healthcare management (Porter-O’Grady 2007), quantum administration (Overman 1996), quantum probabilistic behavior (Bordley 1997), etc., are needed to make a quantum difference in the world in this 21st century. Electronic Companion An electronic companion to this paper is available as part of the online version at http://dx.doi.org/10.1287/opre.1120.1068. Acknowledgments The authors are thankful to Girish Agarwal, Department of Physics, Oklahoma State University, for helpful discussions. They also recognize the anonymous reviewers for their comments, which helped improve the paper. References Aerts D (2009) Quantum structure in cognition. J. Math. Psych. 53(5): 314–348. Agrawal PM (1989) Quantum mechanics. Joshi AW, ed. Horizons of Physics, Vol. I (John Wiley, New Delhi, India), 25–54. Andrade G, Mitchell M, Stafford E (2001) New evidence and perspectives on mergers. J. Econom. Perspect. 15(2):103–120. Baaquie BE (2004) Quantum Finance (Cambridge University Press, Cambridge, UK). Baaquie BE (2009a) Interest Rates and Coupon Bonds in Quantum Finance (Cambridge University Press, Cambridge, UK). Baaquie BE (2009b) Interest rates in quantum finance: The Wilson expansion and Hamiltonian. Phys. Rev. E 80(4):046119-1–046119-23. Bagarello F (2009) A quantum statistical approach to simplified stock markets. Physica A 388(20):4397–4406. Bennett CH, Brassard G (1984) Proc. IEEE Internat. Conf. Comput., Systems, and Signal Processing (IEEE, Piscataway, NJ), 175–179. Bennett CH, Bessette F, Brassard G, Salvail L, Smolin J (1992) Experimental quantum cryptography. J. Cryptology 5(1):3–28. Bohm D (1959) Quantum Theory (Prentice Hall, Englewood Cliffs, NJ). Bohr N (1929) Wirkungsquantum und Naturbeschreibung. Naturwissenschaft 17(26):483–486. Bohr N (1933) Light and life. Nature 131(3308):421–423, 457–459. Bohr N (n.d.) http://www.brainyquote.com/quotes/quotes/n/nielsbohr164546 .html. Bordley RF (1997) Quantum probability and its potential application to decision analysis. Presentation, INFORMS, Hanover, MD. Bordley RF (1998) Quantum mechanical and human violations of compound probability principles: Toward a generalized Heisenberg uncertainty principle. Oper. Res. 46(6):923–926. Bordley RF (2005) Econophysics and individual choice. Physica A 354:479–495. Bordley RF, Kadane JB (1999) Experiment-dependent priors in psychology and physics. Theory and Decision 47(3):213–227. Busemeyer JR, Wang Z, Townsend JT (2006) Quantum dynamics of human decision-making. J. Math. Psych. 50(3):220–241. Chalmers D (1996) The Conscious Mind (Oxford University Press, Oxford, UK). Choustova O (2007) Quantum Bohmian model for financial markets. Physica A 374(1):304–314. Chung YF, Wu ZY, Chen TS (2008) Unconditionally secure cryptosystems based on quantum cryptography. Inform. Sci. 178(8):2044–2058. DePamphilis DM (2010) Merger, Acquisitions, and Other Restructuring Activities: An Integrated Approach to Process, Tools, Cases, and Solutions (Academic Press, San Diego). 15 DeSanctis G, Gallupe RB (1987) A foundation for the study of group decision support systems. Management Sci. 33(5):589–609. Dirac PAM (1958) The Principles of Quantum Mechanics (Clarendon Press, Oxford, UK). Feynman RP, Leighton RB, Sands M (1966a) The Feynman Lectures on Physics, Vol. III (Addison Wesley, Reading, MA). Feynman RP, Leighton RB, Sands M (1966b) The Feynman Lectures on Physics, Vol. I (Addison Wesley, Reading, MA). Guide VDR Jr., Van Wassenhove LN (2009) The evolution of closed-loop supply chain research. Oper. Res. 57(1):10–18. Hand E (2009) Quantum potential. Nature 462(7271):376–377. Haven E (2008) Private information and the “information function”: A survey of possible uses. Theory and Decision 64(2–3):193–228. Ishio H, Haven E (2009) Information in asset pricing: A wave function approach. Ann. Phys. 18(1):33–44. Khrennikov A (2009) Quantum-like model of cognitive decision making and information processing. BioSystems 95(3):179–187. Kondratenko A (2005) Physical Modeling of Economic Systems. Classical and Quantum Economies (Nauka, Novosibirsk, Russia). Lo HK, Spiller T, Popescu S (2000) Introduction to Quantum Computation and Information (World Scientific Publishing, Hackensack, NJ). Lockwood M (1989) Mind, Brain, and the Quantum (Basil Blackwell, Oxford, UK). Malmendier U, Tate G (2008) Who makes acquisitions? CEO overconfidence and the market’s reaction. J. Financial Econom. 89(1):20–43. Messiah A (1970) Quantum Mechanics, Vol. I (North Holland Publishing Company, Amsterdam). Morellec E, Zhdanov A (2005) The dynamics of mergers and acquisitions. J. Financial Econom. 77(3):649–672. Nielsen MA, Chuang LL (2000) Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, UK). Overman ES (1996) The new science of management: Chaos and quantum theory and method. J. Public Admin. Res. Theory 6(1):75–89. Penrose R (1989) The Emperor’s New Mind (Oxford University Press, Oxford, UK). Penrose R, Shimony A, Cartwright N, Hawking S (2000) The Large, the Small, and the Human Mind (Cambridge University Press, Cambridge, UK). Pessa E, Vitiello G (2003) Quantum noise, entanglement and chaos in the quantum field theory of mind-brain states. Mind and Matter 1(1):59–79. Porter-O’Grady T (2007) Quantum Leadership: A Resource for Health Care Innovation (Jones and Bartlet Publishers, London). Pothos EM, Busemeyer JR (2009) A quantum probability explanation for violations of ‘rational’ decision theory. Proc. Roy. Soc. B 276(1665):2171–2178. Razavy M (2003) Quantum Theory of Tunneling (World Scientific Publishing, Hackensack, NJ). Satinover J (2001) The Quantum Brain (Wiley, New York). Savage LJ (1954) The Foundations of Statistics (Wiley, New York). Schaden M (2003) A quantum approach to stock price fluctuations. arXiv:physics/0205053. Segal W, Segal IE (1998) The Black-Scholes pricing formula in the quantum context. Proc. Natl. Acad. Sci. USA 95(7):4072–4075. Shafir E, Tversky A (1992) Thinking through uncertainty: Nonconsequential reasoning and choice. Cognitive Psych. 24(4):449–474. Shleifer A, Vishny RW (2003) Stock market driven acquisitions. J. Financial Econom. 70(3):295–311. Shor P (1997) Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Sci. Statist. Comput. 26(5):1484–1494. Simon HA, Newell A (1958) Heuristic problem solving: The next advance in operations research. Oper. Res. 6(1):1–10. Stenholm S, Suominen K (2005) Quantum Approach to Informatics (John Wiley & Sons, Hoboken, NJ). Agrawal and Sharda: Quantum Mechanics and Human Decision Making 16 Operations Research 61(1), pp. 1–16, © 2013 INFORMS Tichy G (2001) What do we know about success and failure of mergers? J. Indust., Competition, Trade 1(4):347–394. Tversky A, Kahneman D (1974) Judgment under uncertainty: Heuristics and biases. Science 185(4157):1124–1131. Tversky A, Shafir E (1992) The disjunction effect in choice under uncertainty. Psych. Sci. 3(5):305–309. von Neumann J (1983) Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton, NJ). White RE (1966) Basic Quantum Mechanics (McGraw Hill, New York). Yukalov VI, Sornette D (2008) Quantum decision theory as quantum theory of measurement. Phys. Lett. A 372(46):6867–6871. Yukalov VI, Sornette D (2009a) Processing information in quantum decision theory. Entropy 11(4):1073–1120. Yukalov VI, Sornette D (2009b) Physics of risk and uncertainty in quantum decision making. Eur. Physics J. B 71(4):533–548. Yukalov VI, Sornette D (2009c) Scheme of thinking quantum systems. Laser Phys. Lett. 6(11):833–839. Paras M. Agrawal is an emeritus research faculty in the Department of Management Science and Information Systems in the Spears School of Business at Oklahoma State University. He taught quantum mechanics to graduate students as a lecturer/professor of physics in India for more than a decade. His publications include a book chapter on quantum mechanics, and research articles on quantum wave packets, quantum scattering, quantum chemistry, mechanical properties of carbon nanotubes, molecular dynamics, and Monte Carlo simulations. Ramesh Sharda is Director of the Institute for Research in Information Systems (IRIS), ConocoPhillips Chair of Management of Technology, and a Regents Professor of Management Science and Information Systems in the Spears School of Business at Oklahoma State University. His current research interests are in exploring novel analytics applications in the entertainment industry, manufacturing, and healthcare. CORRECTION The article “OR Forum—Quantum Mechanics and Human Decision Making” by Paras M. Agrawal and Ramesh Sharda (first published in Articles in Advance, December 13, 2012, Operations Research, http://dx.doi.org/10.1287/ opre.1120.1068) was corrected as follows: Page 4, Table 1, Eq. 3 was corrected to read as follows: p4BX1 5 = p4X1 5p4B — X1 5, Page 9, line 9 in the left column was corrected to read as follows: 0 0 0 determine p4A—X1 5 and p4A—X2 5, 0 0 0 Page 9, second line after Eq. 39 in the left column was corrected to read as follows: 0 0 0 same as ci occurring in Equation (30)0 0 0 Page 13, Eq. 60 in the right column was corrected to read as follows: P A = “ëA —P̂A —ëA ” = 41 − —a2 —2 5pa KA 0