# OK to post homework
# Vikrant, Feb 06 2022, Assignment 4

# ================================================================================
# 0. Code that has been given.
# ================================================================================

read("C4.txt"):

# ================================================================================
# 1. Samples from Garvan's Maple Primer.
# ================================================================================

(*

with(linalg):
A:= matrix(2,2,[1,2,3,4]);
A+A; evalm(%);
3*A; evalm(%);
A^2; evalm(%);
evalm(A&*inverse(A));
gausselim(A); gaussjord(A);
eigenvals(A);
jordan(A,`P`); print(P);

B:= matrix(2,4,[1,2,3,4,5,6,7,8]);
rank(B);
rowspace(B); colspace(B); nullspace(B);

*)

# ================================================================================
# 2. Read and understand C4.txt.
#    Confirm BRij with PureNashEqui for three games.
# ================================================================================

(*

randomize(614900547):

for i from 1 to 3 do
    G:= RandDisGame(30,30):
    print(PureNashEqui(G) subset {seq(seq([x,y], x in FP(BR12(G))), y in FP(BR21(G)))}):
od:

*)

# ================================================================================
# 3. Read and understand 2.1B of Gibbons.
# ================================================================================

# Done.

# ================================================================================
# 4. Read and understand 1.2A of Gibbons.
# ================================================================================

# Done.

# ================================================================================
# 5. Total profit maximization.
# ================================================================================

(*
    # I'd choose monopoly, where both firms agree to produce A/4 goods each (i.e. A/2 goods produced in total).
    # Since the price for each good is the same regardless of who sells it, we want to be maximizing
    # [Total Quantity of Goods]*[Price Per Good] = [q_1+q_2]*[A-(q_1+q_2)].
    # The total profit is thus maximized when q_1+q_2 = A/2.

    # No such solution results in a Nash Equilibrium, however.
    # Informal proof by picture (the area of region P_i is the profit of firm i):

        (*

             <----A/2---->
         ^    ___________   ^
         |   |           |  |
        q_1  |    P_1    |  |
         |   |___________| A/2
         x   |    P_2    |  |
        q_2  |___________|  |
         v                  v

        *)

    # From the picture we can see that firm i will want to square-ify their region with q_i + A/2 semiperimeter;
    # that is, firm i will increase their quantity of goods produced to (q_i + A/2)/2 in order to maximize the area of the region P_i.
*)

# ================================================================================
# 6. Quantity to produce when price is (A-q)^d.
# ================================================================================

(*
    # (i) Monopoly

    # Solves to q = A/(d+1), which is in [0,A] and so feasible.
    solve(diff(q*(A-q)^d,q)=0,q);

    # Testing the endpoints 0 and A, we conclude that q = A/(d+1) maximizes profit.
*)

(*
    # (ii) Stackelberg Duopoly

    # Repeating the result for monopoly in a backwards induction fashion, we get: 
    # q_1 = A/(d+1) and q_2 = dA/(d+1)^2.
*)

(*

    # (iii) Cournot Duopoly

    # Solves to q_1 = q_2 = A/(d+2), which is in [0,A] and so feasible.
    solve({diff(q__1*(A-q__1-q__2)^d,q__1)=0,diff(q__2*(A-q__1-q__2)^d,q__2)=0},{q__1,q__2});

*)

# ================================================================================
# oo. Helpers from past homeworks.
# ================================================================================

IsNash:=
    proc(G,a1,a2)
        a1 = BR1d(G,a2) and a2 = BR2d(G,a1):
    end:

PureNashEqui:=
    proc(G)
        local i,j:
        {seq(seq(`if`(IsNash(G,i,j),[i,j],NULL),j=1..nops(G[i])),i=1..nops(G))}:
    end: