Oligopoly is amonst the oldest discipline of mathematical economics. The first mathematical model of oligopoly was proposed by Cournot[1].Game theory is the eminent perspective to review oligopoly. Oligopoly is a system where few firms rule the market and make strategis to earn the profit[2].Many researchers have considered different aspects related to Oligopoly [3, 4, 5, 6]. There are many studies about the stability analysis in duopoly and triopoly [7, 8, 9, 10]with homogeneous expectations. Impact of heterogeneous expectations on equilibrium has also been topic of interest for many scholars[11, 12, 13, 14].There are eminent studies which draw own attention on complex dynamics of non-linear models in oligopoly [15, 16, 17, 18].Quadropoly is that part of oligopoly market system, where four firms are involved in profit maximization venture. An economic system becomes more realistic when the number of firms in the system is large. A dynamical system, in which more players are involved, is more complicated and more interesting. In this paper, a linear discrete-time dynamical Quadropoly model with heterogeneous strategies are formulated and conditions for the stability of fixed points are discussed.

In the underlying model of Quadropoly, the underlying assumption is that first and second firms are naive, third firm is adaptive and fourth one is bounded rational.

th

Let there be four firms and xi(t) , i = 1, 2, 3, 4 be output produced by i firm at any time t. Inverse demand function is Y = a − bX.

X = x1(t) + x2(t) + x3(t) + x4(t) represents total supply of the market.Let ci ,i = 1, 2, 3, 4 be the marginal cost

th th

of production of the i firm. Profit function of the i firm is πi = xi(a − bX) − cixi,i = 1, 2, 3, 4 Here xi is written in place of xi(t) for the sake of simplicity.

Marginal profits of all the four firms are:

= a − 2bx1 − bx2 − bx3 − bx4 − c1

= a − bx1 − 2bx2 − bx3 − bx4 − c2

= a − bx1 − bx2 − 2bx3 − bx4 − c3

= a − bx1 − bx2 − bx3 − 2bx4 − c4

th

Each firm wants to maximize respective profit. So value of output xi of the i firm is found, for which its ∂πi marginal profit = 0 , i = 1, 2, 3, 4 ∂xi For ∂π1 a − c1 − b(x2 + x3 + x4) = 0, x1 = (1) ∂x1 2b Similarly,

x2 = (2)

x3 = (3)

x4 = (4)

These are called reaction functions. Using the concept of maxima minima, it is found that for these values of output, profit is the maximum. So, these are profit maximization level of output. So, Dynamical equations of the first two firms with naive expectations are as under:

a − c1 − b(x2(t) + x3(t) + x4(t)) x1(t + 1) = (5)

a − c2 − b(x1(t) + x3(t) + x4(t)) x2(t + 1) = (6)

Third firm being adaptive, calculates level of output to be produced using weighted average of reaction function given in equation (2.3) and level of previous output at time t .i.e. x3(t) . Dynamical equation of the third firm is

a − c3 − b(x1(t) + x2(t) + x4(t)) x3(t + 1) = (1 − α)x3(t) + α (7) where 0 ≤ α ≤ 1 indicates rate with which the adaptive firm adjusts output with respect to market conditions. Fourth firm occurs to be bounded rational. Dynamical equation of bounded rational firm is

x4(t + 1) = x4(t) + βx4(t)[a − 2bx4(t) − b(x1(t) + x2(t) + x3(t)) − c4] (8)

For finding Boundary and Nash Equilibrium points of the linear Quadropoly game, it is required to determine the non-negative fixed point of the system of nonlinear equations (2.5), (2.6), (2.7) and (2.8). So, taking xi(t+ 1) = xi(t) , i = 1, 2, 3, 4 in each of (2.5), (2.6), (2.7) and (2.8) the system of equations is as under:

− x1 = 0

− x2 = 0

α − αx3 = 0

βx4[a − 2bx4 − b(x1 + x2 + x3) − c4] = 0

Solving the above equations, the values so obtained are:

a + c2 + c3 − 3c1 a + c1 + c3 − 3c2 a + c1 + c2 − 3c3 x1 = , x2 = , x3 = 4b 4b 4b Here a + c2 + c3 − 3c1 a + c1 + c3 − 3c2 a + c1 + c2 − 3c3 E1 = , , , 0 (9) 4b 4b 4b is the boundary equilibrium point and

a + c2 + c3 + c4 − 4c1 a + c1 + c3 + c4 − 4c2 a + c1 + c2 + c4 − 4c3 a + c1 + c2 + c3 − 4c4 E2 = , , , (10) 5b 5b 5b 5b

is the Nash equilibrium point.

E1 has non-negative coordinates if

a + c2 + c3 > 3c1 a + c1 + c3 > 3c2 a + c1 + c2 > 3c3

E2 has non-negative coordinates if

a + c2 + c3 + c4 > 4c1 a + c1 + c3 + c4 > 4c2 a + c1 + c2 + c4 > 4c3 a + c1 + c2 + c3 > 4c4

With the purpose of checking the stability of the equilibrium points obtained above, the Jacobian matrix at the equilibrium points are calculated below. Also, the Eigen values of the Jacobian matrix are calculated in the succeeding steps. Nature of equilibrium points will determine the stability of the equilibrium points. Following is the Jacobian matrix:

1 1 1

0 − − −

2 2 2

 1 1 1  − 0 − − J =  2 2 2   α α α  − − 1 − α −

2 2 2

−βbx4 −βbx4 −βbx4 1 + β(a − c4 − 4bx4 − bx1 − bx2 − bx3) At Boundary equilibrium point E1 , Jacobian matrix is

1 1 1

0 − − −

2 2 2

 1 1 1  − 0 − − J(E ) =  2 2 2  1  α α α  − − 1 − α −

2 2 2

0 0 0 1 + β(a − 4c4 + c1 + c2 + c3)

One of the Eigen values of J(E1) is 1 + (a + c1 + c2 + c3 − 4c4) > 0 as c1 + c2 + c3 > 4c4, So boundary equilibrium point is locally unstable.

At Nash equilibrium point E2 , Jacobian matrix is

1 1 1

0 − − −

2 2 2

 1 1 1  − 0 − − J(E ) =  2 2 2  2  α α α  − − 1 − α −

2 2 2 ∗ ∗ ∗ ∗ ∗ ∗ ∗

−βbx4 −βbx4 −βbx4 1 + β(a − c4 − 4bx4 − bx1 − bx2 − bx3)

∗ ∗ ∗ ∗

where x1, x2, x3 and x4 are the Nash equilibrium points.

Let γ be the Eigen values of , which are given by

1 1 1

−γ − − −

2 2 2 1 1 1

− −γ − −

2 2 2 α α α

− − 1 − α − γ −

2 2 2 ∗ ∗ ∗ ∗ ∗ ∗ ∗

−βbx4 −βbx4 −βbx4 1 + β(a − c4 − 4bx4 − bx1 − bx2 − bx3) − γ So, Eigen values of above Jacobian matrix are roots of equation

4 3 2

γ + A1γ + A2γ + A3γ + A4 = 0

, where A1 = 2 − α − αβ(a − c4 − 4bx4 − bx3 − bx1 − bx2) α 1 α A2 = (1 − α)(1 + β(a − c4 − 4bx4 − bx3 − bx1 − bx2)) − − + βbx4 2 4 4 βbx4 βbx4α α 1 1 A3 = − − βbx4α − + + (1 + β(a − c4 − 4bx4 − bx3 − bx1 − bx2)) + 2 8 2 4 4 3α 1 A4 = − − (1 + β(a − c4 − 4bx4 − bx3 − bx1 − bx2))

Schur Cohn Criterion shows that Conditions of local stability of Nash equilibrium points are:

1 + A1 + A2 + A3 + A4 > 0 1 − A1 + A2 − A3 + A4 > 0 1 − A4 > 0 (1 − A4)(1 − A4) − A2(1 − A4) + (A1 − A3)(A3 − A1A4) > 0

This paper examined the profit maximizing strategies of a game with four players using linear demand and cost function.It revealed that boundary equilinrium point seems locally unstable given certain conditions.Nash equilibrium becomes stable for defined speed of adjustment and apadptive player experience steady effect on the market.It also established conditions of stability of Nash equilibrium points using Schur Cohn Criterion.

The authors would like to express their sincere gratitude to the editors and anonymous reviewers for their invaluable comments and constructive feedback, which significantly contributed to the enhancement of this paper.

Bharti Kapoor: methodology, software, and editing. Shilpi Jain: conceptualization. Mehar Chand: writing and editing. All authors have read and agreed to the published version of the manuscript.

The authors declare that no external funding or support was received for the research presented in this paper, including administrative, technical, or in-kind contributions. Data Availability All data supporting the reported findings in this research paper are provided within the manuscript.

The authors declare that there is no conflict of interest concerning the reported research findings. Funders played no role in the study’s design, in the collection, analysis, or interpretation of the data, in the writing of the manuscript, or in the decision to publish the results.