Theoretical Results And Math Approacha General Form Of A Lin

Theoretical Results And Math Approacha General Form Of A Linear Courn The problem involves modeling a duopoly market with two firms choosing quantities simultaneously, with market price determined by an inverse demand function. The demand function is given as P = 13 - Q, where Q = q1 + q2, and each firm incurs a production cost of 1 per unit with no fixed costs. The profit functions for each firm are derived based on these demand and cost parameters, leading to the formulation of their best response functions and the subsequent calculation of the Nash equilibrium. To find the Nash equilibrium, we take the derivative of each firm's profit function with respect to its own quantity, set the derivative to zero, and solve for the best response functions. For Firm 1, the profit function is π1 = (P - c1) * q1 = (13 - q1 - q2 - 1) * q1 = (12 - q1 - q2) * q1. Differentiating with respect to q1 yields the first-order condition: 12 - 2q1 - q2 = 0, which simplifies to q1 = 6 - 0.5q2. Similarly, for Firm 2, π2 = (12 - q1 - q2) * q2, leading to its best response function q2 = 6 - 0.5q1. By solving this system of response functions simultaneously—substituting q2 in terms of q1 and vice versa—we determine the Nash equilibrium quantities. Setting q1 = q2, we find that q1 = q2 = 4, which is the point where both firms' best responses intersect. This equilibrium quantity indicates that both firms will produce four units in the market when they are rational and fully informed. Graphically, the best response functions are linear and intersect at the point (4,4), confirming the theoretical prediction. The initial potential strategies can be visualized by the response functions passing through points like (6,0) and (0,12), illustrating the strategic responses and the convergence to the equilibrium over iterative adjustments. The classical Cournot model predicts that, in the long run, both firms will settle at the equilibrium quantities, maximizing their profits given the other firm's output level. In an experimental setting, like the one described, players often deviate from the theoretical equilibrium due to bounded rationality, incomplete information, or adjustment processes. Initially, participants may produce quantities closer to monopoly levels (6 units), as they have not yet learned the strategic environment of duopoly. Over time, consistent play and adaptation lead players to approximate the Nash equilibrium (4 units). Empirical data from the lab reflects this learning process, showing initial quantities near monopoly levels and a gradual decline toward the equilibrium, as evidenced by the declining average quantities and converging distributions over rounds. The experimental results demonstrate that players can learn and adapt, tending toward the equilibrium solution, although deviations persist due to factors like risk preferences, misperceptions, and strategic