# OK to post homework
# Vikrant, Feb 13 2022, Assignment 6

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

read("C6.txt"):

# ================================================================================
# 1. Simulate five mixed 2-strategy 2-player games with three random mixed strategies each.
# ================================================================================

(*

randomize(1307804445):
r:= rand(0.0..1.0):
for i from 1 to 5 do
    G:= RG(2,2,1000):
    print(matrix(G)):
    print():
    for j from 1 to 3 do
        p1:= r():
        p2:= r():
        print(cat(p1,", ",p2)):
        p1:= convert(p1, rational):
        p2:= convert(p2, rational):
        print(cat("Simulated: ",SimulateMG(G,p1,p2,10^4))):
        print(cat("Exact: ",evalf(PayOffG(G,p1,p2)))):
        print():
    od:
od:

*)

# ================================================================================
# 2. Read and understand 1.2D of Gibbons.
# ================================================================================

# Done.

# ================================================================================
# 3. Tragedy of the Commons.
# ================================================================================

TragedyOfTheCommons:=
    proc(v,G,c,n)
        local i,V,Cs:
        V:= subs(G=add(g[i],i=1..n),v):
        Cs:= {0<=add(g[i],i=1..n),add(g[i],i=1..n)<=1}:
        for i from 1 to n do
            Cs:= Cs union {0<=g[i],V+g[i]*diff(V,g[i])-c=0}
        od:
        solve(Cs,{seq(g[i],i=1..n)}):
    end:

SociallyOptimal:=
    proc(v,G,c)
        solve({v+G*diff(v,G)-c=0,0<=G,G<=1},{G}):
    end:

(*

# The difference between the number of goats in the Nash Equilibrium and socially optimal solution indeed increases with the number of farmers.

c:= 0.5:
Vs:= [sqrt(1-G),(1-G)^(1/4),(1-G)^(3/4)]:
for v in Vs do
    print(v):
    for sol in seq(TragedyOfTheCommons(v,G,c,n),n=1..10) do
        print(rhs(add(sol)-SociallyOptimal(v,G,c)[1])):
    od:
    print():
od:

*)