^{*}

Edited by: Emmanuel E. Haven, University of Leicester, United Kingdom

Reviewed by: Ignazio Licata, ISEM- Institute for Scientific Methodology, Italy; Nicolas Francisco Lori, LANEN, INCYT, INECO Foundation, Argentina

*Correspondence: William F. Lawless

This article was submitted to Interdisciplinary Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Most of the social sciences, including psychology, economics, and subjective social network theory, are modeled on the individual, leaving the field not only a-theoretical, but also inapplicable to a physics of hybrid teams, where hybrid refers to arbitrarily combining humans, machines, and robots into a team to perform a dedicated mission (e.g., military, business, entertainment) or to solve a targeted problem (e.g., with scientists, engineers, entrepreneurs). As a common social science practice, the ingredient at the heart of the social interaction, interdependence, is statistically removed prior to the replication of social experiments; but, as an analogy, statistically removing social interdependence to better study the individual is like statistically removing quantum effects as a complication to the study of the atom. Further, in applications of Shannon's information theory to teams, the effects of interdependence are minimized, but even there, interdependence is how classical information is transmitted. Consequently, numerous mistakes are made when applying non-interdependent models to policies, the law and regulations, impeding social welfare by failing to exploit the power of social interdependence. For example, adding redundancy to human teams is thought by subjective social network theorists to improve the efficiency of a network, easily contradicted by our finding that redundancy is strongly associated with corruption in non-free markets. Thus, built atop the individual, most of the social sciences, economics, and social network theory have little if anything to contribute to the engineering of hybrid teams. In defense of the social sciences, the mathematical physics of interdependence is elusive, non-intuitive and non-rational. However, by replacing determinism with bistable states, interdependence at the social level mirrors entanglement at the quantum level, suggesting the applicability of quantum tools for social science. We report how our quantum-like models capture some of the essential aspects of interdependence, a tool for the metrics of hybrid teams; as an example, we find additional support for our model of the solution to the open problem of team size. We also report on progress with the theory of computational emotion for hybrid teams, linking it qualitatively to the second law of thermodynamics. We conclude that the science of interdependence advances the science of hybrid teams.

One of the major conclusions from modern game theorists, based on findings in the laboratory, is that the societies that cooperate have better social welfare [[

Much about the Hong'ao dump was not as it appeared on paper, a reconstruction of the disaster shows. The duplicity, involving doctored documents and false identities, illustrates systemic gaps in China's efforts to prevent industrial and transportation accidents, which claim tens of thousands of lives annually and have galvanized public anger over official corruption … like the deadly explosions last year at a toxic chemical storage site in Tianjin … the disaster in Shenzhen suggests that dark pools of mismanagement and corruption persist even in the most developed parts of the country.

Conceptually, interdependence has been known for some time. According to Smith's [

What we know so far from this our work-in-progress is that reducing interdependence increases errors and the misallocation of resources [

Even in American bureaucracies, consensus-seeking, corruption, and mismanagement appear to go hand in hand (e.g., for a cover-up by the Veterans Affairs, see [

In the literature, Khrennikov [

The phenomenon that links these examples is the interdependence between behavior and its interpretations; interdependence between multiple interpretations of social reality; and the interdependence among members of a team multitasking to solve a problem. In its review of teams, the National Academy of Sciences repeatedly cited the presence of interdependence but without addressing the phenomenon theoretically [

In the 1940s, Von Neumann and Morgenstern's ([

Bohr, the quantum physicist, criticized game theory on foundational grounds, leading [

Kelley [

The inability of scientists to determine the value of the social interaction at the heart of games is mirrored across the social sciences by practitioners who base their theories on observations of the processes of how the best teams should operate, often with self-reported (subjective) surveys that tell us nothing new (e.g., the surveys and interviews of teams at Google; in Duhigg [

In comparison to game theory and other traditional approaches to the study of interdependence in teams, we define interdependence as responsive or reactive to signals in nature between non-independent organisms (e.g., elk grazing in a forest with predators leads to healthier forest grasses; from Carroll [

When measuring states of interdependence, the measurement problem's “apparent impossibility of an objective measurement” ([

From the HIV example, if “quantum-like effects exist in the social world, expressed as interdependence” ([

As another example of how interdependence makes social reality non-deterministic, consider self-esteem, one of the major foci for the clinical practice of psychology over the decades. In the book published by the American Psychological Association (APA), [

Although, relatively little is known about self-esteem, it is generally considered to be a highly favorable personal attribute, the consequences of which are assumed to be as robust as they are desirable. Books and chapters on mental hygiene and personality development consistently portray self-esteem as one of the premier elements in the highest levels of human functioning … Its general importance to a full spectrum of effective human behaviors remains virtually uncontested. We are not aware of a single article in the psychological literature that has identified or discussed any undesirable consequences that are assumed to be a result of realistic and healthy levels of personal self-regard.

Despite this bold claim by Bednar and Peterson under the imprimatur of the APA, a 30-year meta-analysis of all of the known experimental studies where self-esteem could be measured against actual physical performance for both academics and the workplace by Baumeister [

As a result, we adopt the spirit imbued in game theory to model interdependence, but we reject game theory as fundamentally observational and a-theoretical. Instead, by using Von Neumann's model of quantum interference and Bohr, we review herein our advances: by taking limits, we derived a quantitative measure in the limit of what constitutes a perfect team, another for the worst team, and another we found as a relative metric of team performance modeled after Kullback–Leibler divergence where redundancy in teams is characterized by the divergence in team size from comparable free market teams [

In his theory of self-replicating automata, Von Neumann [

In summary, briefly, our goal is to apply our findings to determine mathematically the performance of hybrid teams. Traditional, but normative, models centered around cooperation, while of value in the creation of stories or religious homilies, are of little value for the engineering of hybrid teams. By extending our research to team emotion, we hope to generalize our research where our most recent goal was to use hybrid team performance as a guide to minimize human error (e.g., [

Martyushev [

Our theory is that excluded spaces are governed by the politics in play operating in a social reality, with bistable interpretations of social reality determined by neutral supporters [

A social system that controls, stops or blocks the bistable interdependence in Equation (2) should be modeled by Shannon entropy. Pure states are product states, where ^{1}

To reflect correlations caused by interference among the sources of information, unlike Shannon entropy, interdependence can be destructive or constructive, captured by Von Neumann's density matrix, ρ, with entropy depicted by Equation (4):

If a team is successful in producing a team with members who multitask together to form what appears to be a team with “single mind,” its degrees of freedom (

Interestingly, Einstein was the first to discover the reduction in

We have two hemispheres in our brain … [that form a] unified single mind. … But when you do a split-brain operation, a complete transection of the corpus callosum, you get clear evidence of two separate consciousnesses.

Interference may be constructive, as when the members of a team work well together. In contrast to Equation (3), to represent Hoffman's “unified single mind” and to further account for the invisibility of interdependence effects, we use subaddivity to get:

Working from Von Neumann's perspective, correlations in joint entropy can become greater or equal to their differences, reflected by Equation (7):

Equation (7) implies that social groups engage in tradeoffs to choose the more fit members of a team, where the best fit is signified by a reduction in joint entropy. Shannon states for subadditivity in a composite system can also be expressed as: ^{2} = ρ(idempotent). If^{2}) = 1, ρis pure and |ψ>_{AB} is separable; however, if^{2}) < 1, ρis mixed and |ψ>_{AB} is entangled ([

Returning to Equation (2), if the two factions in a group, represented by the operators

But interference from social interaction may be destructive; e.g., the rupture of a sports team; a married couple undergoing a divorce; the splitting apart of a business striving to survive a market turndown, like the Maersk Conglomerate [

As they form an audience, neutrals, we argue, are the only social element to enter into a superposition (Equation 2), driven into the superposition by the Nash equilibrium that acts like the two cylinders of an engine. As they are wooed to and fro, once neutrals are measured, the trail they leave behind forms limit cycles [

Wang and Busemeyer [

Cohen [_{A} is the standard deviation of variable _{B} for variable

As a notional example, the wide Gaussian is Fourier transformed to the narrow one; the Standard deviation for the latter one is 0.33, that for the wide one about 5.0; the two multiplied together is >1/2.

In quantum theory, the uncertainty relation Equation (10) follows directly from the non-commutativity of Hilbert space operators (Equation 9). Similar relations appear for Fourier pairs in classical field theory as well. By itself, the application of Equation (10) to what follows for the action of teams (Equations 10, 11) can be criticized as a mere analogy and not formally motivated. However, a new discovery of redundancy or overstaffing among oil producers as teams coupled with another discovery (e.g., flawed DOE nuclear waste management teams of scientists and their managers with Equation 14 below; in Lawless [

Based on our prior findings, when the goal of tradeoffs is to find the team members with the right skills for the best team fit, we begin to extend our findings with a revision of Equation (10) to:

Along with the claims of Smith [

To study the implications of Equation (10), we decompose a team into a (static) structure that directs its efforts, and its efforts at performing its mission (i.e., dynamic skill roles; actions based on those roles). Assume that the structure of a team is functioning perfectly, allowing the team to use an optimum amount of its available energy to solve the problems that the team was designed to address. Building on our prior success, but speculating, we convert Equation (11) into two components representing the least entropy production (LEP) for the structure of a team and maximum entropy (MEP) to perform a team's mission:

Taking limits with the variables in Equation (12) gives us an equation that captures a team's excellence; i.e., as a team's consumption of energy by its structure goes to zero, it's ability to maximize its ability to problem-solve itself becomes a maximum:

With Equation (13) in hand, by inverting it, an account is discovered for what happens when a team fails, splits apart, or implodes [

The teams represented by Equation (14) might be a couple undergoing divorce; a business team failing (e.g., Maersk Shipping; in Chopping [

At DOE's Savannah River Site from the 1950s until 1985, DOE's waste management permitted 90% of its military solid radioactive wastes to be buried in ordinary cardboard boxes, allowing these boxes to sit in open trenches exposed to the weather for months at a time.

The National Academy of Sciences [

We first define our four factors: redundancy, economic freedom, military power, and corruption. These factors are mixed objective and subjective, meaning the results will include varying levels of subjectivity.

Redundancy is a quantity measured for interacting human autonomous systems and interfering with other autonomous systems [

We used the ranking devised by Global Firepower (

An index established by the Heritage foundation based on four broad factors to measure liberty and free markets for 186 nations: rule of law; government size; regulatory efficiency; and open markets. Each factor has three sub-factors (

An index of nations established by Transparency International (

We measure redundancy with divergence from a Kullback–Leibler-type equation for relative entropy, where _{KL}

The sum of Equation (15) reflects the divergence of distribution _{i}, from distribution _{i}, with both distributions normalized. For example, log (P(i)/P(i)) = log 1 = 0. Thus, the more divergence, the larger the separation between two distributions. Based on Equation (15), assuming that a relatively perfect team is possible to solve the problem at hand, we also assume that some structures for desired teams may be closed-ended for a solved problem like those that exist for sports teams; e.g., for a baseball team, designated members take the role of pitcher, catcher, first baseman, etc. Unlike the relatively simple problem of designing a sports team, most business and scientific teams are open-ended whenever competition or innovation are factors. To solve this kind of a structural problem, in business, we look to an industry leader for the best team structure possible for the problem at hand.

To extend these findings to militaries, we hypothesize that redundancy is associated with less freedom in the marketplace and with more corruption. We test this hypothesis with correlations and Kullback–Leibler divergence. We expect that distributions in the real world range from minimum to maximum redundancy; from minimum to maximum freedom; and from minimum to maximum corruption. The nations used in this problem are footnoted below^{2}^{3}

Example:

As an example of the calculations with Equation (15), for China's Military Power Distribution (MPD, or P_{MPD}), we divided its military power rank (3) by its population in billions (1.374) = 2.183 and the result we divided by 8.1, China's GDP/capita in thousands = 0.370; we summed this result for our top 22 nations = 82.28, which we divided into 0.370 to get the fraction for China, P_{China} = .0045. We repeated to calculate the Free Market Distribution (FMD) for China (59.4) by dividing by the sum (1296.9) for our Q_{1}. Next we entered the calculations stepwise into Equation (15) to get for China the following calculation:

In addition, as one of our methods, we will look for a sign of the collective effects of intelligence.

For a pilot run, we used a convenience sample of 12 nations consisting of some of the largest militaries in the world^{4}

Heartened by these pilot results, we were ready to test our hypothesis with Equation (15). For Q_{1}, we summed the result of FMD versus MPD to get 1.78. We repeated the process for Q_{2} to get another distribution for corruption levels versus MPD for a sum of 1.95. Then we regressed the FMD results individually nation by nation versus MPD (Q_{1}) with the divergence of corruption from MPD (Q_{2}) and plotted the result in Figure ^{2}, 0.926,

In this figure, we regressed the divergence of freedom from a military distribution with the divergence of corruption from a military distribution. The result indicates a significant regression (^{2} = 0.926,

As a side issue, we also looked for signs of intelligence. We found it in our calculations. Consider an abstract from our data in column 4 of the Table

Data rounded off to three significant decimals.

China | 3 | 2.183 | 0.370 |

USA | 1 | 3.086 | 0.017 |

Russia | 2 | 14.084 | 0.225 |

Brazil | 17 | 82.927 | 1.954 |

UK | 6 | 93.75 | 0.150 |

Cuba | 78 | 7090.909 | 10.196 |

We had hypothesized and found in two separate distributions, one for the divergence of GDP for a country's Index of Economic Freedom with its military power ranking per capita, and the other for the divergence of a country's corruption index versus its military power ranking per capita, a significant regression. This indicates that, even with real-world data containing subjective estimations, redundancy increases the more authoritarian is a country's decision-making. As a corollary, the collective effects of intelligence in a society operate best under the freedom to allocate capital and labor for its best uses.

Our results for this study, also backed with correlations, support theory and justify our use of quantum-like models. We found less divergence with our hypothesis for military team size and economic freedom, but more divergence with military team size and corruption, indicating that National Defense improves under the collective effects of intelligent decisions at the level of the team in free markets. It means that a military is leaner and more effective under democracies that under autocracies.

We suspect that redundancy in the market of teams isolates excess teammates from interdependent effects, reducing responsiveness, and converting co-workers into featherbedders. Barriers, like authoritarian leadership and corruption, impede reaching MEP by intelligent teams. And, as we have found, redundancy increases under authoritarian governments, for the possible but corrupt political payoffs that may become necessary to keep civil peace. For example, corruption has stymied the reform of scientific practices in Russia [

Our model is different from the traditional model, specifically, the cognitive model. As a representative example of the influence of the cognitive model transported from social science to history in the hands of a popular historian^{5}

That is not what we have found. Our results establish the meaningful differences that interdependent information plays in the interactions and affairs of humans under any and every form of government. Information constraints (barriers) under authoritarian regimes are less able to direct the movement of labor and capital to best solve targeted problems, an added constraint for innovation, one reason the Chinese rely on the theft of intellectual property (see the interview of General M. Hayden, the former CIA and NSA chief, by the editor-in-chief of the

Unlike Google's survey of teams [

In the HRI community, a lot of research with reinforcement learning (RL) is designed to assist in social interaction where “emotions obviously are important for social interaction” ([

In his magisterial review of the literature on emotion, supported by our theory, Zajonc [

In addition, a rise in

What if judgments about reality are not rational but guided primarily by experience (where a culture has been ushered into being and molded by experiential learning; [

Applied to teams by integrating Zajonc and others, we can see that the structure of a team is in a relatively stable state (

Initially, we use a sigmoid function to model the effort required to hold a team together (see Figure

A Monte Carlo simulation of Equation (16) with the y-axis intercept at (0, ½) in the center, with

Results from a Monte Carlo simulation of Equation (16) shown in Figure

We have also found evidence that a well-fitted team having success at solving the problems it was designed to solve exhibits more intelligence than an under-fitted or over-fitted team with redundant members. The well-fitted team generates less entropy than its individual contributors, an indication that a state of maximum interdependence exists inside of the team, where each member is responsive to every other team member and to the team's mission as well. The state of maximum interdependence, however, can be reversed or blocked. Like quantum computations, the state of interdependence is a resource for a team but also for the society within which the well-fitted team is embedded and to which the well-fitted team contributes. Once a well-fitted team establishes a point of stability, an emotionless baseline, it is operating in a ground state (Figure

A notional aspect of the transactions modeled by the sigmoidal function in Equation (16).

We have not addressed the characteristics of the problem targeted, but we suspect that a team must be designed to match its designated problem (e.g., a well-fitted 5-member baseball team is of value in playing against an equally competent 5-member baseball team, but of little value when playing against an equally competent but nine-person baseball team).

Significant impediments exist in the formulation of a science of teams using traditional theories. Specifically, Shannon's information theory and the social sciences, including economics, assume that the human observation of human behavior records the actual behavior that has occurred, even for self-reports of self-observed behavior. In computational social science, this phenomenon has been labeled informally as the “god's eye view,” indicating that the “computer” within which computational action occurs knows immediately whatever action a computational agent takes. In the social sciences, this phenomenon manifests as an observational bias; it allows social scientists to assume that self-reported behavior is actual behavior (e.g., if this assumption was true, deception or denial, such as alcoholic denial or spying, would not exist). We claim that this assumption is unsupported by the evidence, as is the “knowledge” gathered in support, such as the conclusion consonant with widespread religious beliefs that cooperation provides for the best social good. At the heart of these rational, but false models, interdependence is seen as a constraint (information or communication theory) or experimental confound (cognitive science) that must be overcome by traditional social scientists to confirm a theoretical models based on methodological individualism (MI; [

By replacing MI with quantum-like models, we have found computational metrics for good and poor teamwork performance, and a third finding that redundancy is associated with corruption by using relative entropy to model divergence from an oil market leading team, now supported in this study by the size of a nation's military. We have also proposed a new model for a team's emotion as it shifts from a ground state to an excited state. We conclude that, like entanglement at the atomic level, interdependence at the social level is the primary social resource that ordinary humans exploit to innovate and promote social welfare.

Wendt [

We reject the traditional model of redundancy (e.g., [

Excessive team emotion is observable to external observers; e.g., a divorce; a business breakup; a team's collapse. More difficult to observe is the critical point, the transition from a team arguing appropriately [

For a mathematical physics of teams, a significant impediment has too long existed from accepting the traditional belief that social truth can be established by observing individuals. As exemplars, both built around the statistics of independent, identically distributed data (

In contrast, based on our model where interdependence reduces a team's degrees of freedom (

The best teams have the least redundancy so that they are maximally interdependent among teammates to be responsive to each other as they multitask to solve the problems that they face intelligently. In conclusion, we have found support for our quantum-like model with the solution to the open problem of team size.

The author confirms being the sole contributor of this work and approved it for publication.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The author thanks the reviewers of his manuscript for their very helpful comments, suggestions and corrections.

^{1}

^{2}Nations: the top 20 militaries in the world plus Cuba and North Korea were used: China, USA, Russia, Brazil, UK, India, France, Japan, Turkey, Germany, Italy, South Korea, Egypt, Pakistan, Indonesia, Israel, Vietnam, Poland, Taiwan, and Iran.

^{3}For the actual study, we used the top 20 militaries in the world per capita (from

^{4}For the pilot study, we used the following sample: USA, China, Cuba, North Korea, India, Israel, Iran, Japan, Mexico, Pakistan, Russia & Turkey; economic freedom index from 2016

^{5}His book is a New York Times Bestseller