#OK to post homework 
#Natalya Ter-Saakov, Feb 13, Assignment 7 
read "C7.txt":

#Problem 2
#Donation Game: (where row 1 and col 1 are 'cooperate' and row 2 and col 2 are 'defect') 
DG:= [[[R,R],[S,T]],[[T,S],[P,P]]];

#For the row player, defecting dominates over collaborating, as it does for the column player, so the only possible Nash equilibrium is (Defect, Defect).
#If T=R+0.01, R=10000P, S=P-0.01, then T > R > P > S still holds and so instead of getting 10000P each, they each get P, a significant amount slower.


#Problem 3 
#Snowdrift Game (T>R>S>P): 
SG:= [[[R,R],[S,T]],[[T,S],[P,P]]];

#Nash equilibria:
#Consider (Refuse, Pay): If Player Col pays, then Player Row prefers to refuse (T>R). If Player Row refuses, then Player Col prefers to pay (S>P). Thus, this is a pure Nash equilibrium. 
#Consider (Pay, Refuse): If Player Row pays, then Player Col prefers to refuse (T>R). If Player Col refuses, then Player Row prefers to pay (S>P). Thus, this is a pure Nash equilibrium.
#We know that (Pay, Pay) and (Refuse, Refuse) are not Nash equilibria because each player would switch if given the option. 
#In the last hw, problem 3, we proved that if there are two pure Nash equilibria, there is also a mixed Nash equilibrium. 
#The payoff for Player Row is 
P1:= ((R - S - T + P)*p2 + S - P)*p1 + (T - P)*p2 + P: 
#and the payoff for Player Col is 
P2:= ((R - S - T + P)*p1 + S - P)*p2 + (T - P)*p1 + P: 
#So p1 = p2 = (S-P)/(S+T-R-P) gives the mixed Nash equilibrium for this game. 
p1:= (S-P)/(S+T-R-P): p2:= (S-P)/(S+T-R-P):

#Testing some random values of T,R,S,P:
#(Note that we never run p2 because it will always be equal to p1)

#Test 1
T:=rand(100)();R:=rand(T)();S:=rand(R)();P:=rand(S)(); MNE22(SG);p1;

#Output:
#T:=41, R:=31, S:=5, P:=1
#MNE22(SG):={[0,1],[1,0],[4/17,4/17]}
#p1:=4/17

#Test 2
T:=rand(100)();R:=rand(T)();S:=rand(R)();P:=rand(S)(); MNE22(SG);p1;

#Output:
#T:=89, R:=33, S:=17, P:=10
#MNE22(SG):={[0,1],[1,0],[1/9,1/9]}
#p1:=1/9

#Test 3
T:=rand(100)();R:=rand(T)();S:=rand(R)();P:=rand(S)(); MNE22(SG);p1;

#Output:
#T:=59, R:=28, S:=13, P:=1
#MNE22(SG):={[0,1],[1,0],[12/43,12/43]}
#p1:=12/43

#Test 4
T:=rand(100)();R:=rand(T)();S:=rand(R)();P:=rand(S)(); MNE22(SG);p1;

#Output:
#T:=45, R:=43, S:=8, P:=4
#MNE22(SG):={[0,1],[1,0],[2/3,2/3]}
#p1:=2/3

#Test 5
T:=rand(100)();R:=rand(T)();S:=rand(R)();P:=rand(S)(); MNE22(SG);p1;

#Output:
#T:=76, R:=38, S:=27, P:=6
#MNE22(SG):={[0,1],[1,0],[21/59,21/59]}
#p1:=21/59