Consensus as a Nash Equilibrium of a stochastic differential game

Social networks influence a lot of behavioral activities including educational achievements (Calvó-Armengol, Patacchini and Zenou, 2009), employment (Calvo-Armengol and Jackson, 2004), technology adoption (Conley and Udry, 2010), consumption (Moretti, 2011) and smoking (Nakajima, 2007; Sheng, 2020). As social networks are the result of individual decisions, consensus takes an important role to understand the formation of networks. Although a lot of theoretical works on social networks have been done (Jackson, 2010; Goyal, 2012; Sheng, 2020), work on consensus as a Nash equilibrium under a stochastic network is very insignificant (Niazi, Özgüler and Yildiz, 2016). Sheng (2020) formalizes network as simultaneous-move game, where social links based on decisions are based on utility externalities from indirect friends. Sheng (2020) proposes a computationally feasible partial identification approach for large social networks. The statistical analysis of network formation dates back to the seminal work by Erdös and Rényi (1959) where a random graph is based on independent links with a fixed probability (Sheng, 2020). Beyond Erdös-Rényi model, many methods have been designed to simulate graphs with characteristics like degree distributions, small world, and Markov type properties (Polansky and Pramanik, 2021). A model based method is useful if this model can be fit successfully and if it is a relatively simple to simulate realizations (Pramanik, 2016; Polansky and Pramanik, 2021). The most frequently used general model for random graphs is the exponential random graph model (ERGM) (Snijders, 2002; Hua, Polansky and Pramanik, 2019; Polansky and Pramanik, 2021) because, this model fits well with the observed network statistics (Sheng, 2020). This ERGM model lacks microfoundations which are important for counterfactual analyses and furthermore, economists view network analysis as the optimal choices of agents who maximizes their utilities (Sheng, 2020). Network evolves as the result of a stochastic process is another popular framework where network may be observed, but it is the parameters of the stochastic process that are of interest, and the observed network is a single realization of the stochastic process (Polansky and Pramanik, 2021).

At birth, humans already posses different types of skills like breathing, digest foods and motor actions which make a human body to behave like an automaton (Kappen, 2007a). Furthermore, like other animals humans acquire skills through learning. Different person has different abilities to acquire a new information in order to get an idea about pleasure, danger or food (Kappen, 2007a). Humans are the most complicated species on earth because, their decisions are not linear and they can learn difficult skills through transitional signals of their complex constellations of sensory patterns. For example, if food is kept in front of a hamster, it would eat immediately. On the other hand, if a plate of food is kept in front of a human, they might not eat because, variety of factors such as the texture, smell, amount of it, their sociocultural background, religion and ethnicity take place before even they think about to tastes it. In order to make this decision, a lot of complex neural activities take place inside a person’s brain. Action of two main parts of a human brain, frontal and occipital lobes, makes them decide what they should do after seeing an object. In this case the occipital lobe sends information of an object through the synaptic systems to frontal lobe, which is known by their previous experiences and knowledge. As for humans one has to consider so many other possibilities compared to a hamster, such that they can choose any of the all available information with some probabilities and make decisions based on it.

This type of human behavior is a feedback circuit where the learning algorithm is determined by a synaptic motor command, more time with an object not only leads to get more information but also the knowledge to adapt with it in the long run and get more intelligence. For example, as humans grow older, more intelligent they become and reflects their genotype closely. On the other hand, environment influences a certain type of decision more with older ages which comes through a process called Hebbian learning (Hebb, 2005; Kappen, 2007a). Ancestors gather more information about an object or circumstance and transfer it to their off-springs in order to help them survive easily and make decisions rationally Kappen (2007a). For example, without having a prior knowledge one does not know how to get a certain type restaurant and which lead them explore their surroundings. If that person finds out a restaurant, they survive for that day. On the next day, they might not be completely sure about full availability of food in that restaurant because of sudden environmental degradation after his previous visit such as flash flood, tornado, an avalanche, landslide, earthquake, other activities like closure due to burglary, fire or some gun related activities so on. Even if that person is sure about the availability of food, they might not go because of other socioeconomic behaviors at the back of their mind. Hence, more information might not lead them react rationally. These types of activities occurs when an event is more uncertain. Consider person $A$ is selling their $1$ million-dollar car to another person $B$ by just $\$50,000$. The rationality assumption suggests person $B$ to go for this offer but, $B$ might think why $A$ is giving this offer and might be suspicious about the quality of that car and rejects it.

Therefore, subjective probabilities take an important role to make these types of decisions based on individual judgments such as success in a new job, outcome of an election, state of an economy, difference in learning a new complex topic among students, spreading gossips in small communities (Kahneman and Tversky, 1972; Niazi, Özgüler and Yildiz, 2016; Tversky and Kahneman, 1971). People follow representativeness in judging the likelihood of uncertain events where the probability is defined by the similarities in essential properties to its parent population and reflect the salient features of the process by which it is generated (Kahneman and Tversky, 1972), which makes opinion dynamics of a person to follow a stochastic differential equation. Furthermore, an individual minimizes its cost of foraging for food where finding food can be termed as a reward to them and they want to find their reward with minimal cost. Assume an agent discounts more to the recent future than farther future represented as feedback motor control reinforcement learning problem Kappen (2007a). In an environment of very complex opinion dynamics each agent minimizes their integral cost function subject to a stochastic differential opinion dynamics based on all above cases. This paper considers two environments first, all the agents have same opinion power and second, agents with a leader, where the leader has more power in opinion than others and determines their opinion first based on their own cost minimization mechanism. A feedback Nash equilibrium of opinion is determined by a Feynman-type path integral approach which so far from my knowledge is new (Feynman, 1949; Pramanik and Polansky, 2021). Furthermore, this approach can be used to obtain a solution for stability of an economy after pandemic crisis (Ahamed, 2021a), determine an optimal bank profitability (Hossain and Ahamed, 2015; Ahamed, 2021b).

As each agent’s opinion in a society is assumed to be a quantum particle, I introduce an alternative method based on Feynman-type path integral to solve this stochastic opinion dynamics problem based on Feynman-type path integrals instead of traditional Pontryagin Maximum Principle. If the objective function is quadratic and the differential equations are linear, then solution is given in terms of a number of Ricatti equations which can be solved efficiently (Kappen, 2007b). But the opinion dynamics is more complicated than just an ordinary linear differential equation and non-linear stochastic feature gives the optimal solution a weighted mixture of suboptimal solutions, unlikely in the cases of deterministic or linear optimal control where a unique global optimal solution exists (Kappen, 2007b). In the presence of Wiener noise, Pontryagin Maximum Principle, a variational principle, that leads to a coupled system of stochastic differential equations with initial and terminal conditions, gives a generalized solution (Kappen, 2007b; Øksendal and Sulem, 2019). Although incorporate randomness with its Hamiltonian-Jacobi-Bellman (HJB) equation is straight forward but difficulties come due to dimensionality when a numerical solution is calculated for both of deterministic or stochastic HJB (Kappen, 2007b). General stochastic control problem is intractable to solve computationally as it requires an exponential amount of memory and computational time because, the state space needs to be discretized and hence, becomes exponentially large in the number of dimensions (Theodorou, Buchli and Schaal, 2010; Theodorou, 2011; Yang et al., 2014). Therefore, in order to calculate the expected values it is necessary to visit all states which leads to the summations of exponentially large sums (Kappen, 2007b; Yang et al., 2014). Kappen (2005a) and Kappen (2005b) say that a class of continuous non-linear stochastic finite time horizon control problems can be solved more efficiently than Pontryagin’s Maximum Principle. These control problems reduce to computation of path integrals interpreted as free energy because, of their various statistical mechanics forms such as Laplace approximations, Monte Carlo sampling, mean field approximations or belief propagation (Kappen, 2005a, b, 2007b; Van Den Broek, Wiegerinck and Kappen, 2008). According to Kappen (2007b) these approximate computations are really fast.

Furthermore, one can transform a class of non-linear HJB equations into linear equations by doing a logarithmic transformation. This transformation stems back to the early days of quantum mechanics which was first used by Schrödinger to relate HJB equation to the Schrödinger equation (Kappen, 2007b). Because of this linear feature, backward integration of HJB equation over time can be replaced by computing expectation values under a forward diffusion process which requires a stochastic integration over trajectories that can be described by a path integral (Kappen, 2007b). Furthermore, in more generalized case like Merton-Garman-Hamiltonian system, getting a solution through Pontryagin Maximum principle is impossible and Feynman path integral method gives a solution (Baaquie, 1997; Pramanik, 2020, 2021a). Previous works using Feynman path integral method has been done in motor control theory by Kappen (2005b), Theodorou, Buchli and Schaal (2010) and Theodorou (2011). Applications of Feynman path integral in finance has been discussed rigorously in Baaquie (2007). In Pramanik (2020) a Feynman-type path integral has been introduced to determine a feedback stochastic control. This methods works in both linear and non-linear stochastic differential equations and a Fourier transformation has been used to find out solution of Wick-rotated Schrödinger type equation (Pramanik, 2020; Pramanik and Polansky, 2020a, b; Pramanik, 2021a; Pramanik and Polansky, 2021; Pramanik, 2021b). A more generalized Nash equilibrium on tensor field has been discussed in Pramanik and Polansky (2019).

Following Niazi, Özgüler and Yildiz (2016) consider a social network of $n$ agents by a weighted directed graph $G=(N,E,w_{ij})$, where $N=\{1,...,n\}$ is the set of all agents. Suppose, $E\subseteq N\times N$ is the set of all ordered pairs of all connected agents and, $w_{ij}$ is the influence of agent $j$ on agent $i$ for all $(i,j)\in E$. There are usually two types of connections, one sided or two sided. For the principle-agent problem the connection is one sided (i.e. Stackelberg model) and agent-agent problem it is two sided (i.e. Cournot model). Suppose $x^{i}(s)\in[0,1]$ be the opinion of agent $i^{th}$ at time $s\in[0,t]$ with their initial opinion $x^{i}(0)=x_{0}^{i}\in[0,1]$. Then $x^{i}(s)$ has been normalized into $[0,1]$ where $x^{i}(s)=0$ stands for a strong disagreement and $x^{i}(s)=1$ represents strong agreement and all other agreements stays in between. Consider $\mathbf{x}(s)=\left[x^{1}(s),x^{2}(s),...,x^{n}(s)\right]^{\prime}\in[0,1]^{n}$ be the opinion profile vector of $n$-agents at time $s$ where ‘prime’ represents the transpose. Following Niazi, Özgüler and Yildiz (2016) consider a cost function of agent $i$ as

where $w_{ij}\in[0,\infty)$ is a parameter which weighs the susceptibility of agent $j$ to influence agent $i$, $k_{i}\in[0,\infty)$ is agent $i$’s stubbornness, $u^{i}(s)$ is the control variable of agent $i$ and set of all agents with whom $i$ interacts is $\eta_{i}$ and defined as $\eta_{i}:=\{j\in N:(i,j)\in E\}$. The cost function $L^{i}(s,\mathbf{x},x_{0}^{i},u^{i})$ is twice differentiable with respect to time in order to satisfy Wick rotation, is continuously differentiable with respect to $i^{th}$ agent’s control $u^{i}(s)$, non-decreasing in opinion $x^{i}(s)$, non-increasing in $u^{i}(s)$, and convex and continuous in all opinions and controls (Mas-Colell et al., 1995; Pramanik and Polansky, 2020b). The opinion dynamics of agent $i$ follows a stochastic differential equation

with the initial condition $x_{0}^{i}$, where $\mu^{i}$ and $\sigma^{i}$ are the drift and diffusion component of agent $i$ with $B^{i}(s)$ is the Brownian motion. The reason behind incorporating Brownian motion in agent $i$’s opinion dynamics is because of Hebbian Learning which states that, neurons increase the synaptic connection strength between them when they are active together simultaneously and this behavior in probabilistic in the sense that, resource availability from a particular place is random (Hebb, 2005; Kappen, 2007a). For example, for a given stubbornness, and influence from agent $j$, agent $i$’s opinion dynamics has some randomness in opinion. Suppose, from other resources agent $i$ knows that, the information provided by agent $j$’s influence is misleading. Apart from that after considering humans as automatons, motor control and foraging for food becomes a big examples of minimization of costs (or the expected return) Kappen (2007a). As control problems like motor controls are stochastic in nature because there is a noise in the relation between the muscle contraction and the actual displacement with joints with the change of the information environment over time, we consider Feynman path integral approach to calculate the stochastic control after assuming the opinion dynamics Equation (2) (Feynman (1949),Fujiwara (2017)). The coefficient of the control term in Equation (1) is normalized to $1$, without loss of generality. The cost functional represented in the Equation (1) is viewed as a model of the motive of agent $i$ towards a prevailing social issue Niazi, Özgüler and Yildiz (2016). In this dynamic social network problem agent $i$’s objective is to $\min_{u^{i}}\{\mathbb{E}_{s}(L^{i})|\mathcal{F}_{0}^{x}\}$ subject to the Equation (2), where $\mathbb{E}_{0}(L^{i})|\mathcal{F}_{0}^{x}$ represents the expectation on $L^{i}$ at time $0$ subject to agent $i$’s opinion filtration $\mathcal{F}_{0}^{x}$ starting at the initial time $0$. A solution of this problem is a feedback Nash equilibrium as the control of agent $i$ is updated based on the opinion at the same time $s$.

The opinion dynamics of $i^{th}$ agent is continuous and it is mapped from an interval to a space of continuous functions with initial and terminal time points. Suppose, at time $s$, $g(s):[p,q]\rightarrow\mathcal{C}$ represents an opinion dynamics of $i^{th}$ agent with initial and terminal points $g(p)$ and $g(q)$ respectively, such that, the line path integral is $\int_{\mathcal{C}}f(\gamma)ds=\int_{p}^{q}f(g(s))|g^{\prime}(s)|ds$, where $g^{\prime}(s)$ is derivative with respect to $s$. In this paper I consider functional path integrals where the domain of the integral is the space of functions (Pramanik, 2020, 2021a; Pramanik and Polansky, 2021). Functional path integrals are very popular in probability theory and quantum mechanics. In Feynman (1948) theoretical physicist Richard Feynman introduced Feynman path integral and popularized it in quantum mechanics. Furthermore, mathematicians develop the measurability of this functional integral and in recent years it has become popular in probability theory (Fujiwara, 2017). In quantum mechanics, when a particle moves from one point to another, between those points it chooses the shortest path out of infinitely many paths such that some of them touch the edge of the universe. After introducing equal length small time interval$[s,s+\varepsilon]$ with $\varepsilon>0$ such that $\varepsilon\downarrow 0$ and using Riemann–Lebesgue lemma if at time $s$ one particle touches the end of the universe, then at a later time point it would come back and go to the opposite side of the previous direction to make the path integral a measurable function (Bochner et al., 1949). Similarly, agent $i$ has infinitely opinions, out of them they choose the opnion corresponding to least cost given by the constraint explained in Equation (2). Furthermore, the advantage of Feynman approach is that, it can be used in both in linear and non-linear stochastic differential equation systems where constructing an HJB equation is almost impossible (Baaquie, 2007).

Here $i^{th}$ agent’s objective is to minimize Equation (1) subject to Equations (2). Following Definition 1 the quantum Lagrangian at time $s$ of $[s,s+\varepsilon]$ is

where $\lambda^{i}$ is a time independent quantum Lagrangian multiplier (one can think of as a penalization constant of agent $i$). As at the beginning of the small time interval $[s,s+\varepsilon]$, agent $i$ does not have any future information, they make expectations based on their opinion $x^{i}$. For another normalizing constant $L_{\varepsilon}^{i}>0$ and for time interval $[s,s+\varepsilon]$ such that $\varepsilon\downarrow 0$ define a transition function from $s$ to $s+\varepsilon$ as

where $\Psi_{s}^{i}(x^{i})$ is the value of the transition function based on opinion $x^{i}$ at time $s$ with the initial condition $\Psi_{0}^{i}(x^{i})=\Psi_{0}^{i}$. Therefore, the action function of agent $i$ is,

where $h^{i}[\nu+\Delta\nu,x^{i}(\nu)+\Delta x^{i}(\nu)]\in C^{2}([0,t]\times\mathbb{R}^{n})$ such that,

Here the action function has the notation $\mathcal{A}_{s,s+\varepsilon}(x^{i})$ which means within $[s,s+\varepsilon]$ the action of agent $i$ depends on their opinion $x^{i}$ and furthermore, I assume this system has a feedback structure. Therefore, the opinion of agent $i$ also depends on the strategy $u^{i}$ as well as the rest of the school. Same argument goes to the transition function $\Psi_{s,s+\varepsilon}(x^{i})$.

Suppose, for the opinion space $S_{0}=\{\mathbf{x}(s):s\in[0,t]\}$ and agent $i$’s strategy space $\Gamma^{i}$ there exists a permissible strategy $\gamma^{i}:[0,t]\times S_{0}\rightarrow\Gamma^{i}$ and for all $i\in N$ define the integrand of the cost function as