@acer 

Thank you so much. The graph is meaningful as the profit increases with the increase in alpha. But perhaps for a small alpha, there is no feasible solution, right? Does it mean we need to find the minimum feasible alpha and adjust the range? Thank you"

Dear @Carl Love ​​​​​​, @acer, and @dharr

Thanks a million. Your ideas were wonderful. I also found another solution that is working:

alpha := [seq(i/10, i = 4 .. 9)];
L1 := map(FirmModelPartial1, alpha, 0.2, 0.1);
plt1 := listplot([alpha, L1], linestyle = dash, legend = "β=0.2,δ=0.1", labels = ["α", "Firm's profit"], labeldirections = ["horizontal", "vertical"], axes = boxed, color = red);

Thanks again for your time and sharing your valuable knowledge.

@acer

Thank you so much. I solved the issue with an error in plotting file.

Dear @acer

Sorry,

Download SiSTechno.mw

the file was not properly attached. I attached it. Thank you so much!

@acer 

Sorry for the issue. The question about a system of nonlinear functions was new. I thought I created a new question, not a follow-up. Sorry for the confusion:

About the answer:

Your solution is perfect if we have an optimization model. But in a 2-player Nash game, we have at least two optimization models that negotiate simultaneously. So, we write KKT conditions for each and combine them to make a system of equations. The answer to the system is the equilibrium points. So, I think optimization does not work here. Am I missing something?

Dear @acer

Please accept my apology once more.

I applied your suggestion to the plot. But there is an error that I could not handle (attached). Would you please advise? 

@acer ​​​​​​​It was not easy at all for me to create it. Thank you so much. It works perfectly.

I have another small question, please: When we use 'solve' in the code, if there are multiple solutions, which one Maple chooses? Can we push it to choose e.g., a solution that maximizes a function? Thanks again!

@acer

Wooow!I have made many terrible mistakes in coding. Thank you so much. The plot is perfectly in line with my analytical proof. I highly appreciate your valuable help. I am reading the code to remember them.

Now the legend is in the form of delta=0.1, delta=0.3, delta=0.5. Would you please also kindly let me know how I can add a text before them like TEXT, delta=0.1, TEXT, delta=0.3, TEXT, delta=0.5? Thanks again!

@sand15 Woooow!!! It is amazing! I didn't know it at all. Now, I completely realize the wonderful trick. You are absolutely phenomenal. Thanks again for teaching me.

@sand15 Thanks a million! I am indebted to you. It was very important. Thanks again.

@sand15 Thanks a million for your time! I am checking your code to understand how the last two lines, including Phi and tan h, work. It looks a bit high level for me to realize.

@Carl Love Thanks a million! It was very useful and important. I could not easily find it. Thanks again.

Thank you so much, dear @Joe Riel .

Your code runs perfectly. But I still cannot understand why the error happened. Could you please explain briefly?

Dear @acer

Thanks a million for your sharing your valuable time and expertise. You are kind and wonderful.

Dear @acer This is amazing. Thank you so much for your time and expertise. I slightly increased the beta from 0.2 to 0.35 (practically it cannot go higher) to obtain a larger green area which is important. I was wondering when Delta can move up to 1 (instead of 1/2), a large black rectangle appears above the current diagram. How can I change its color to something lighter (like light grey) to be able to write a note inside it? Thanks again for the important help!