CLIAMS:1. A computer implemented method for computing a correlated equilibrium between two interdependent variable parameters of elements of a system, said method comprising following steps:
accepting information related to said two interdependent variable parameters;
identifying interdependencies between said two variable parameters from the accepted information;
computing Choquet integrals to determine workable interdependent strategies for said variable parameters;
computing a workable strategy profile for each of said variable parameters of elements based on the determined workable strategies;
computing an infinite strategy profile of the system using the computed workable strategy profile of each of said variable parameters;
computing values for said variable parameters of each of said elements as a function of the infinite strategy profile;
plotting the computed function of the infinite strategy profile as curves; and
obtaining a correlated equilibrium of the values of said variable parameters based on a common solution of said function curves.

2. The method as claimed in claim 1, wherein said step of obtaining correlated equilibrium includes steps of defining distribution on the infinite strategy profile such that it is greater than or equal to zero and sum of all distributions is equal to one and, obtaining correlated equilibrium based on the defined distribution and said function curves.

3. The method as claimed in claim 1, wherein said method includes step of optimizing the computed functions based on Ellipsoid techniques.
4. A computer implemented system for computing a correlated equilibrium between two interdependent variable parameters of elements of the system, said system comprising:
a processor;
a memory coupled with the processor, the memory comprising:
an input module configured to accept information related to said two interdependent variable parameters;
an identifier configured cooperate with the input module to identify interdependencies between said two variable parameters from the accepted information;
a determiner configured to compute Choquet integrals to determine workable interdependent strategies for said variable parameters;
a first profiler configured to compute a workable strategy profile for each of said variable parameters of elements based on the determined workable strategies;
a second profiler configured to compute an infinite strategy profile of the system using the computed workable strategy profile of each of said variable parameters;
a function estimator configured to compute values for said variable parameters of each of said elements as a function of the computed infinite strategy profile;
a plotter configured to plot the computed function of the infinite strategy profile as curves; and
an equilibrium evaluator configured to obtain a correlated equilibrium of the values of said variable parameters based on a common solution of said function curves.

5. The system as claimed in claim 4, wherein said system comprises a distribution determiner configured to define distribution on the infinite strategy profile such that it is greater than or equal to zero and sum of all distributions is equal to one, said system is configured to obtain correlated equilibrium based on the defined distribution and said function curves.

6. The system as claimed in claim 4, wherein said system is configured to optimize the computed functions based on Ellipsoid techniques. ,TagSPECI:TECHNICAL FIELD
The present disclosure relates to the field of data analytics.
DEFINITIONS OF TERMS USED IN THE SPECIFICATION
The expression ‘strategy’ used hereinafter in this specification refers to available options of steps that can be carried out by a module/device to obtain a desired result while considering possibility of steps that may be carried out by other modules/devices in response. The module/device selects a suitable strategy from the available possible strategies.
The expression ‘payoff’ used hereinafter in this specification refers to a desired outcome/result that a module/device achieves on selection of strategies. Every module/device aims to maximize its own expected payoff.
The expression ‘collaboration’ used hereinafter in this specification refers to strategies that are followed by the modules/devices by working together to achieve better payoffs.
The expression ‘non collaboration’ used hereinafter in this specification refers to strategies that are followed by a module/device to obtain better payoffs without considering the effect of selected strategies on payoffs of other modules/devices.
The expression ‘Nash equilibrium’ used hereinafter in this specification refers to a technique wherein modules/devices within a system assume to discern equilibrium strategies of the other modules or devices in the system, and none of the module or device changes its strategy as it does not gain anything by deviating from the decided strategy.
The expression ‘Choquet integral’ used hereinafter in this specification refers to a sub-additive or super-additive integral that may be used in statistical mechanics, potential theory, decision theory etc. to measure expected utility of an uncertain event.
The expression ‘ellipsoid techniques’ used hereinafter in this specification refers to an iterative method for minimizing convex functions and for computing an optimal solution in a finite number of steps.
These definitions are in addition to those expressed in the art.
BACKGROUND
Usually, modules or devices within a system are controlled by a central module/device that provides instructions or assigns steps to be carried out by other modules/devices within the system. These modules/ devices perform their functions independently without direct communication with each other. However, there are systems where having coordination between the modules/devices helps them in choosing strategies/ steps which are favorable to each other in order to enable better functioning of the system.
Nash equilibrium is a technique wherein modules/devices within a system assume to discern equilibrium strategies of the other modules or devices in the system, and none of the module or device changes its strategy as it does not gain anything by deviating from the decided strategy. For example, if a device A and device B are interdependent and the steps carried out by device A are based on decisions that may be taken by device B and vice versa, device A and device B are in Nash equilibrium.
Correlated equilibrium on the other hand is a solution that is more general than the Nash equilibrium. In this technique each module or device within a system selects its step based on observation and historical records. In this technique each module/device chooses its action based on its observation of other modules working. A strategy assigns an action to every possible observation made by the module/device. When the modules/devices do not deviate from the recommended strategies by assuming no one deviates, the distribution is called a correlated equilibrium. Any Nash equilibrium is a correlated equilibrium. But, in case of correlated equilibrium, communication between interdependent modules/devices is possible which allows the modules/devices to choose an appropriate equilibrium based on strategies. For systems having strategy sets that are infinite in nature, it is difficult to compute correlated equilibria which allow modules/devices to provide better outcome.
Therefore, there is a need for a system that computes correlated equilibrium for systems having infinite strategy sets and which limits the aforementioned drawbacks.
OBJECTS
Some of the objects of the present disclosure aimed to ameliorate one or more problems of the prior art or to at least provide a useful alternative are described herein below:
An object of the present disclosure is to provide a system that computes correlate equilibrium.
Another object of the present disclosure is to provide a system that computes correlate equilibrium between interdependent parameters.
Further object of the present disclosure is to provide a system that computes correlate equilibrium such that better payoffs are available to the parameters.
Other objects and advantages of the present disclosure will be more apparent from the following description when read in conjunction with the accompanying figures, which are not intended to limit the scope of the present disclosure.

SUMMARY
The present disclosure relates to a computer implemented system for computing a correlated equilibrium between two interdependent variable parameters of elements of the system. In an embodiment, the system may comprise a processor and a memory coupled with the processor. The memory may comprise an input module, an identifier, a determiner, a first profiler, a second profiler, a function estimator, a plotter and an equilibrium evaluator. The input module may accept information related to the two interdependent variable parameters. The identifier may cooperate with the input module to identify interdependencies between the two variable parameters from the accepted information. Based on the identified interdependencies the determiner may compute Choquet integrals to determine workable interdependent strategies for the variable parameters. The first profiler may then compute a workable strategy profile for each of the variable parameters of elements based on the determined workable strategies. The second profiler may then compute an infinite strategy profile of the system using the computed workable strategy profile of each of the variable parameters. Based on the computed infinite strategy profile, the function estimator may compute values for the variable parameters of each of the elements as a function of the computed infinite strategy profile. The plotter may then plot the computed function of the infinite strategy profile as curves and the equilibrium evaluator may be configured to obtain a correlated equilibrium of the values of the variable parameters based on a common solution of the function curves.
This summary is provided to introduce concepts related to computing correlated equilibrium between two interdependent parameters, which is further described below in the detailed description. This summary is neither intended to identify essential features of the present disclosure nor is it intended for use in determining or limiting the scope of the present disclosure.

BRIEF DESCRIPTION OF ACCOMPANYING DRAWINGS
A computer implemented system of the present disclosure for computing correlated equilibrium between two interdependent parameters, will now be described with the help of accompanying drawings, in which:
Figure 1 illustrates a schematic of an embodiment of the system for computing correlated equilibrium between two interdependent parameters.
Figure 2 illustrates a flow diagram of an embodiment of the system for computing correlated equilibrium between two interdependent parameters.
DETAILED DESCRIPTION
A preferred embodiment of the present disclosure will now be described in detail with reference to the accompanying drawings. The preferred embodiment does not limit the scope and ambit of the disclosure. The description provided is purely by way of example and illustration.
The embodiments herein and the various features and advantageous details thereof are explained with reference to the non-limiting embodiments in the following description. Descriptions of well-known components and processing techniques are omitted so as to not unnecessarily obscure the embodiments herein. The examples used herein are intended merely to facilitate an understanding of ways in which the embodiments herein may be practiced and to further enable those of skill in the art to practice the embodiments herein. Accordingly, the examples should not be construed as limiting the scope of the embodiments herein.
The system of the present disclosure computes correlated equilibrium between two interdependent parameters. The system comprises elements consisting interdependent variable parameters. It considers two of these interdependent variable parameters of elements and computes correlated equilibrium between them. Referring to the accompanying drawings, Figure 1 illustrates a schematic of an embodiment of the system for computing correlated equilibrium between two interdependent parameters and Figure 2 illustrates a flow diagram of the embodiment of the system for computing the correlated equilibrium. The system 100 comprises a processor 102 that provides system processing commands to the modules present in the system 100 in order to compute a correlated equilibrium. Further, the system includes a memory 104 that is coupled with the processor 102. This memory 104 comprises an input module 106, an identifier 108, a determiner 110, a first profiler 112, a second profiler 114, a function estimator 116, a plotter 118 an equilibrium evaluator 120 and a distribution determiner 122. The input module 106 present in the memory 104 accepts information related to two interdependent variable parameters 200 of the elements of the system 100. The identifier 108 then accepts information related these two interdependent variable parameters and identifies interdependencies between them based on the accepted information 202. The determiner 110 uses the identified interdependencies of the parameters to compute Choquet integrals to obtain workable interdependent strategies for the variable parameters 204. Based on these workable interdependent strategies, a workable strategy profile for each of the variable parameters of elements is computed 206 by the first profiler 112. The second profiler 114 then computes an infinite strategy profile of the system 100 by using the computed workable strategy profile of each of the variable parameters 208 and the function estimator 116 computes values for the variable parameters of the elements as a function of the computed infinite strategy profile 210. This computed function of the infinite strategy profile is then plotted as curves 212 by the plotter 118 based on which a common solution is obtained. The equilibrium evaluator 120 then obtains a correlated equilibrium of the values of the variable parameters based on this obtained common solution of the function curves 214. The distribution determiner 122 present in the system 100 defines distribution on the infinite strategy profile such that it is greater than or equal to zero and sum of all distributions is equal to one. Based on this defined distribution and the function curves, the correlated equilibrium for the two interdependent variable parameters is computes.
Considering an example where the system of the present disclosure is implemented in a vehicle. Usually in a vehicle, direct communication or link is not present between the current speed of the vehicle and usage of the fuel. However, it is advantageous if these two modules communicate in order to choose an equilibrium which is favorable for both and thus in turn for the vehicle owner. The system of the present disclosure computes correlated equilibrium of speed and fuel consumption of the vehicle. It includes a speed maintenance module and a fuel consumption module which control the vehicle and provide values to maintain speed in order to allow optimum usage of fuel.
An equilibrium strategy is a joint probability distribution over actions where none of the module has an incentive if it deviates from the strategy. In case of vehicles, it is a known fact that the speed in range of 40-60 mph is a safe limit to provide better fuel efficiency. But, this value greatly depends on the load carried by the vehicle and other important factors like coefficient of drag, aerodynamics of the vehicle and weight of the vehicle. Generally vehicle having more aerodynamics gets best mileage at higher speeds and vice versa. But, a computation mechanism providing a suitable speed for a particular vehicle in a specific condition to provide optimum fuel efficiency is not available. This can be achieved by the system of the present disclosure by computing correlated equilibrium.
In order to compute correlated equilibrium, it is considered that payoff to the speed maintenance module is the probability of getting incentive on completion of a job task (achieving a specific speed) and the payoffs to the fuel consumption module is the probability of establishing a collaboration with the speed maintenance module. Considering the same, potential strategies (i.e. steps/actions to be undertaken) by the fuel consumption module includes the options as follows:
1. If the fuel consumption module collaborates with the speed maintenance module (such that both modules are dependent on each other to achieve an optimum fuel efficiency), then referring to the figure above, the speed maintenance module will always cooperate (C) and the payoffs will be (a, b) to the speed maintenance module and the fuel consumption module respectively.
2. If the fuel consumption module does not collaborate (NTC) with the speed maintenance module, then the speed maintenance module (anticipating NTC) will always choose NTC (i.e. it will not collaborate) and the payoffs will be (g, h).

TABLE I
Fuel consumption module
Speed maintenance module C NTC
C
a, b c, d
NTC
e, f g, h

In order to achieve fuel efficiency at a reasonable speed, the fuel consumption module must combine both the aforementioned options. Fuel efficiency can be increased by mixing C and NTC. For example considering TABLE I,
If the speed maintenance module aims 'C' with 50% chance and 'NTC' with 50% chance,
Then, the payoff for the speed maintenance module is
0.5 X a + 0.5 X c = (a + c) /2 if the speed maintenance module moves for 'C'
0.5 X e + 0.5 X g = (e + g) /2 if the speed maintenance module moves for 'NTC'
Since it is better to move 'C', the speed maintenance module collaborates and its payoff is
(a + c) /2.
Therefore, if the fuel consumption module mixes 50-50, then its payoff will be (e + g)/2.
In order to still improve the performance of the fuel consumption module, following steps are carried out:
When the fuel consumption module aims 'C' with q probability and 'NTC' with 1-q probability, the payoff for the speed maintenance module is:
q X a + (1- q) X c = c + (a-c) q (if the speed maintenance module moves 'C')
q X e + (1- q) X g = g + (e-g) q (if the speed maintenance module moves 'NTC')
The speed maintenance module will move towards maximizing its payoff (obtaining desired speed) and will get
C if c + (a-c) q > g + (e-g) q and NTC if c + (a-c) q < g + (e-g) q and, either C or NTC if c + (a-c) q = g + (e-g) q.
Therefore, payoff of the speed maintenance module is the larger of c + (a-c) q and g + (e-g) q.
The fuel consumption module will also want to maximize its payoff and minimize the speed maintenance modules payoff (i.e. consume less fuel at lesser speed). The fuel consumption module can do that by setting c + (a-c) q and g + (e-g) q equal:
c + (a-c) q = g + (e-g) q which implies q = (c-g)/ {(e-g)-(a-c)}
In order to maximize the fuel consumption module’s payoff, the fuel consumption module should collaborate 'C' q% of the time and not collaborate ‘NTC’ r% = (100-q) % of the time.
In this case, payoff of the speed maintenance module is c + (a-c) q = t.
In other words if the fuel consumption module mixes q% - r%, then the speed maintenance modules payoff is ‘t’ whether it collaborates ‘C’ or does not collaborate ‘NTC’ (or mixes between them) with the fuel consumption module. Therefore, payoff of the fuel consumption module is 100 – t.
Similarly in case of the speed maintenance module, if it collaborates with probability ‘p’, its payoff is
p X a + (1-p) X e = e + (a-e) p if the fuel consumption module aims 'C' and
p X c + (1-p) X g = g + (c-g) p if the fuel consumption module aims 'NTC'
To minimize payoff of the speed maintenance module, the fuel consumption module moves: C if e + (a-e) p < g + (c-g) p and
NTC if e + (a-e) p > g + (c-g) p and
either of C or NTC if e + (a-e) p = g +(c-g) p.
That is, the speed maintenance module’s payoff is the smaller of e + (a-e) p and g + (c-g) p.
The speed maintenance module should equate e + (a-e) p and g + (c-g) p so as to maximize its payoff:
e + (a-e) p = g + (c-g) p which implies p = (g-e)/ {(a-e)-(c-g)}
In order to maximize its payoff, the speed maintenance module should move 'C' p% of the time and NTC (100-p) % of the time. In this case, the speed maintenance module’s payoff will be e + (a-e) p = g + (c-g) p = u.
The payoff of the speed maintenance module is100-u
Therefore the mixed strategy where: the speed maintenance module follows p C + (1-p) NTC and the fuel consumption module follows q C + (1-q) NTC is the only one that cannot be exploited by either module. Hence it is a mixed strategy of Nash equilibrium. Every mixed strategy of Nash equilibrium could be correlated.
In extension to the aforementioned example, two car manufacturers A and B are considered. If these two manufacturers compete for the market share of the similar product (i.e. products having similar features), the payoff matrix in terms of the speed and fuel consumption of vehicles is illustrated in TABLE II.
TABLE II
Manufacturer B
Manufacturer A Speed (B1) Fuel consumption (B2)
Speed (A1) 12 15
Fuel consumption (A2) 14 10

To obtain optimal net outcomes for the two manufacturers A and B, probability on A1, A2, B1 and B2 is computed. The computation is shown below:
p(A1) = (14-10) / {(14-10) + (15-12)} = 4/(4+3) = 4/7
p(A2) = (15-12) / {(14-10) + (15-12)} = 3/7
p(B1) = (15-10) / {(15-10) + (14-12)} = 5/7
p(B2) = (14-12) / {(15-10) + (14-12)} = 2/7.
As per this example and computation, the manufacturer A should adopt strategy A1 (speed of A category cars) for 57% and A2 (fuel consumption of A category cars) for 43%. Similarly, the manufacturer B should adopt strategy B1 (speed of B category cars) for 71% and B2 (fuel consumption of B category cars) for 29%. In that case,
expected gain to the manufacturer A is
(i) 12 X (4/7) + 14 X (3/7) = 90/7 if the manufacturer B adopts B1; or
(ii) 15 X (4/7) + 10 X (3/7) = 90/7 if the manufacturer B adopts B2;
and expected loss to the manufacturer B is
(i) 12 X (5/7) + 15 X (2/7) = 90/7 if the manufacturer A adopts A1
(ii) 14 X (5/7) + 10 X (2/7) = 90/7 if the manufacturer A adopts A2.
These computations provide examples of optimal strategies for the two manufacturers and a net outcome thereof.
To compute correlated equilibria using linear programming techniques, the system considers two variable parameters P1 and P2 that are interdependent and each of them have strategy Sp, where p= 1, 2, 3 etc. based on the parameters in consideration. The strategy profile thus obtained is S = S1 * S2 with S-q denoting the profile for all players except q. An infinite strategy profile S8 = ? Spi (where i = 1, 2,...) is then obtained wherein payoffs of the two parameters during infinite system operation are functions upi on the strategy profile S8 into non-negative integers. A distribution determiner present in the system defines ‘x’ as a distribution on S8 (x = 0 and ? x(s) = 1 where s ? Spi). There is a probability that parameter P1 follows strategy i and P2 follows –s. This is denoted by x (i,-s). Similarly up(i,-s) is the payoff to a P1 for following strategy i ? Spi. The distribution x defined by the distribution determiner is a correlated equilibrium (CE) if the system imposes a condition ? up(i,-s) x(i,-s) = ?up(j,-s) x(i,-s) for all i, j ? Spi and s ? Spi (p =1,2), on the parameter that accepts the recommended strategy pi. The expected payoff from following this recommended strategy is no worse than following any other strategy. In such a case, the correlate equilibria conditions by following a linear program are:
? {up(i,-s) - up(j,-s)} x(i,-s) = 0 for all i, j ? Spi and s ? Spi (p =1,2)
x(s) = 0 for all s ? Spi
? x(s) = 1 where s ? Spi

Ellipsoid techniques can be used to optimize objective functions (cooperation and non-cooperation) with respect to these correlated equilibria conditions.
The system computes optimum payoffs to the parameters based on collaboration and non-collaboration strategies using Choquet integrals. TABLE III illustrates collaboration and non-collaboration probabilities for two interdependent parameters, Parameter A and Parameter B. Here, C stands for collaboration and NTC is for non-collaboration.
TABLE III

Parameter B
Parameter A
C NTC C NTC
C
2 2 3 -2
NTC
4 3 2 6

To compute collaboration (C) strategy:
g(x(1)) = 0.2
g(x(2)) = 0.2
g(x(3)) = 0.3
g(x(4)) = -0.2

For calculation for ? the fuzzy measure is as follows:
? + 1 = ? (? g (x (i)) + 1)
? + 1 = (0.2 ? + 1) (0.2 ? + 1) (0.3 ? + 1) ((-0.2) ? + 1)
? + 1 = (-0.0024) ?^4 – 0.02 ? ^3 + 0.02 ? ^2 + 0.5 ? + 1
(-0.0024) ? ^3 – 0.02 ? ^2 + 0.02 ? 0.5 =0
0.0024 ? ^3 + 0.02 ? ^2 - 0.02 ? - 0.5 =0

Solving the cubic, a nonzero real number ? = 4.3864 is obtained. Associated densities are then computed as follows:
g(x(1), x(2), x(3), x(4)) = 1
g(x(2), x(3), x(4)) = -0.10634
g(x(3), x(4)) = -0.163184
g(x(4)) = -0.2

By Choquet Integral the following value is obtained:
? f(x) dg = ? f(x(i) – f(x(i-1)) g(a(i))
= f(x(1)). g({x (1), x(2), x(3),x(4)} + (f(x(2)- f(x(1)). g({x(2), x(3),x(4)}) + (f(x(3)- f(x(2)). g({x(3),x(4)}) + (f(x(4)- f(x(3)). g(x(4))
= 2*1 + (0*0.2473632) + (1* -0.163184) + (-5)*(-0.2)
=2.836816
Similarly, to compute non-collaboration (NTC) strategy:
g(x(1)) = 0.4
g(x(2)) = 0.3
g(x(3)) = 0.4
g(x(4)) = 0.6
For calculation for ? the fuzzy measure is as follows:
? + 1 = ? (? g (x (i)) + 1)
? + 1 = (0.4 ? + 1) (0.3 ? + 1) (0.4 ? + 1) (0.6 ? + 1)
? + 1 = 0.0144 ? ^4 – 0.18 ? ^3 + 0.8 ? ^2 + 0.15 ? + 1
0.0288 ? ^3 + 0.288 ? ^2 + 1.06 ? + 0.7 =0
Solving the cubic equation, a nonzero real value ? = -0.83 is obtained. Associated densities are then computed as follows:
g(x(1), x(2), x(3), x(4)) = 1
g(x(2), x(3), x(4)) = 0.9018
g(x(3), x(4)) = 0.8
g (x(4)) = 0.6
By Choquet Integral we have,
? f(x) dg = ? f(x (i) – f(x (i-1)) g (a(i))
= f(x(1)). g({x(1), x(2), x(3),x(4)} + (f(x(2)- f(x(1)). g({x(2), x(3),x(4)}) + (f(x(3)- f(x(2)). g({x(3),x(4)}) + (f(x(4)- f(x(3)). g(x(4))
= 4*1 + ((-1)*0.9018) + ((-1)* 0.8) + (4)*(0.6)
= 4.6982
Based on the aforementioned example, it is seen that the selection preferences for NTC is more than that of C as per the Choquet Integral. Based on rules of dominance on the given payoff matrix, TABLE IV is obtained:

TABLE IV
Parameter B
Parameter A
C NTC
C
3 -2
NTC
2 6

The optimum payoff to parameter A based on TABLE III can be 3p1+2(1-p1) = -2p1+6(1-p1) this implies p1 = 4/9 and p2 = 1-p1 = 5/9, then, value of V = 3 x 4/9 + 2 x 5/9 = 22/9= 2.44.
The optimum payoff to parameter B based on TABLE III can be 3q1-2(1-q1) = 2q1+6(1-q1) implies q1 = 8/9 and q2 = 1-q1 = 1/9. Then value of the V = 3x8/9 – 2x1/9 = 22/9= 2.44.
A better payoff for parameter A and parameter B irrespective of strategy modification is possible by use of the concept of correlated equilibrium. If both parameters A and B want to collaborate, then, C strategy must dominate over NTC strategy. In the aforementioned example, it is observed that the NTC strategy (4.7) is far away from C strategy (2.87).
The use of the expression “at least” or “at least one” suggests the use of one or more elements or ingredients or quantities, as the use may be in the embodiment of the disclosure to achieve one or more of the desired objects or results.

The foregoing description of the specific embodiments will so fully reveal the general nature of the embodiments herein that others can, by applying current knowledge, readily modify and/or adapt for various applications such specific embodiments without departing from the generic concept, and, therefore, such adaptations and modifications should and are intended to be comprehended within the meaning and range of equivalents of the disclosed embodiments. It is to be understood that the phraseology or terminology employed herein is for the purpose of description and not of limitation. Therefore, while the embodiments herein have been described in terms of preferred embodiments, those skilled in the art will recognize that the embodiments herein can be practiced with modification within the spirit and scope of the embodiments as described herein.