# Theoretical Results and Math approach

Theoretical Results and Math approach:

A general form of a linear Cournot duopoly is as follows. We consider a duopoly model with two players in this experiment. Player 1 and player 2 simultaneously choose (non-negative) quantity q1 and q2 they produce and sell in the market, respectively. Q denotes the total quantity, where Q = q1+ q2; and price is determined from the inverse demand function quantity P=α−βQ to clear the market, where α>0 and β>0. According to the lab instruction of Monopoly and Cournot, demand intercept and slope are equal to 13.00 and 1.00 separately; each unit of product player choose to produce costs1.00, and fix cost is equal to 0. From that, we could generate a price function, which is P=13-Q. Note that since the demand function is always downward sloping, so the slope is negative. The players’ payoff functions are π1= (P-c1)*q1 and π2 = (P-c2)*q2. In order to get Nash-Equilibrium, we need to take the derivative of these two equations, which will provide the best response functions for both players.

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For the player 1: π1 = (P-c1)*q1⟹ π1= (12-q1-q2)*q1

F.O.C： 12-2q1-q2=0 ⟹ q1= 6-0.5q2

For the player 2: π2 = (P-c2)*q2⟹ π1= (12-q1-q2)*q2

F.O.C: 12-q1-2q2=0 ⟹ q2= 6-0.5q1

By combining those two functions, we get the equilibrium q1=q2=4. The optimal production quantity for both players is 4 units. It is easy to see the lines of best response for two players. We can discuss all players as a single monopolist separately, and the act they made which maximizes their joint profits (Siallagan et al., 2013). After knowing player 1’s F.O.C function which is q1= 6-0.5q2, assuming q2 is equal to 0, we get q1=6; assuming q1 is equal to 0, then q2=12. Therefore, we drive two points, (6,0) and (0,12) for player 1. For player 2, we also get (6,0) and (0,12) by the same procedure.

Theoretically, the results should show that all players are choosing the production level at the Nash equilibrium. However, because the first half of the lab is the monopoly setting where the Nash equilibrium is at 6 units. Therefore, the prediction will be that players may not realize it at the beginning, and it takes time for them to figure out the new Nash equilibrium. The prediction will be that all of the players should play at the equilibrium at the end of the lab.

5. Experiment Results:

In this section, we will discuss whether the Nash equilibrium is eventually chosen by the players or not, and the explanations of reasons that may lead them away from the Nash equilibrium. Section 4 illustrates that theoretically; all players should play at the Nash equilibrium, which is 4 in this case. In reality, there is no way that everyone is making the optimal choices. However, based on the lab results, we do conclude that subjects are aware of the Nash equilibrium and could adjust their choice accordingly. The Table 1 we created shows the total average production

quantity chosen by players in each round from 11 to 22.  Table 1.

At the first few rounds, the averages are near 6, which makes sense considering players were just switched from the Monopoly game and in that game the optimal choice is 6. This also matches the prediction we made in section 4. Figure 2 displays the movement of average quantity chosen by all players. The red line represents the Nash equilibrium. Similarly, it indicates that the average quantity is relatively far away from the NE at the beginning, but then moves much closer towards the red line. The lab results also show that, at the beginning of duopoly game, students choose from a wide range.

Figure 2.

Figure 3 displays that in the first 3 rounds the quantity choices are widely distributed. The result shows a large portion of players did not realize that this is the duopoly game instead of the monopoly. This means the Nash equilibrium is different from now. After a few rounds, more players started to change their strategies and began to slowly approach to the new Nash equilibrium, which is the 4 units.

Figure 3:

At the last two rounds, the production level on average is now more concentrated and around the Nash equilibrium. There is a huge difference between the distributions of quantities.

Figure 4:

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