#OK to post
#Rebecca Embar, 2-4-2022, Homework 5

read `hw2RebeccaEmbar.txt`:
read `hw3RebeccaEmbar.txt`:
read `C5.txt`:

#1
#Payoff for player 1
payoff_1 := proc(p1,p2):
	return (a-p1-b*p2)*(p1-c):
end:

#Payoff for player 2
payoff_2 := proc(p1,p2):
	return (a-p2-b*p1)*(p2-c):
end:
#The best response functions are
#BR1(p2)=(1/2)*(a+b*p2+c)
#BR2(p1)=(1/2)*(a+b*p1+c)
#Nash equilibria is p1 = p2 = (a+c)/(2-b)

#2
G := RG(30,30,10000):
PureNashEqui(G);
#{[4,25]}
BestTot(G);
#{[4,25]},19661
BetterForBoth(G,4,25);
#Empty set

G := RG(30,30,10000):
PureNashEqui(G);
#{[30,23]}
BestTot(G);
#{[30,23]},19816
BetterForBoth(G,30,23);
#Empty set

G := RG(30,30,10000):
PureNashEqui(G);
#Empty set
BestTot(G);
#{[29,7]},19310

#3
BeatNE := proc(a,b) local games,l,i:
	games := AllGames(a,b):
	l := {}:
		for i from 1 to nops(games) do
			if nops(PureNashEqui(games[i])) = 1 and not nops(BetterForBoth(games[i],PureNashEqui(games[i])[1][1],PureNashEqui(games[i])[1][2])) = 0 then
				l := l union {games[i]}:
			fi:
		od:
	return l:
end:
#{[[[1, 1], [3, 2]], [[4, 3], [2, 4]]], [[[1, 1], [3, 4]], [[2, 3], [4, 2]]], [[[1, 2], [3, 4]], [[2, 3], [4, 1]]], [[[1, 3], [3, 4]], [[2, 2], [4, 1]]], [[[1, 4], [3, 3]], [[2, 2], [4, 1]]], [[[1, 4], [4, 3]], [[2, 2], [3, 1]]], [[[1, 4], [4, 3]], [[3, 2], [2, 1]]], [[[2, 1], [3, 2]], [[4, 3], [1, 4]]], [[[2, 2], [3, 1]], [[1, 4], [4, 3]]], [[[2, 2], [4, 1]], [[1, 3], [3, 4]]], [[[2, 2], [4, 1]], [[1, 4], [3, 3]]], [[[2, 3], [4, 1]], [[1, 2], [3, 4]]], [[[2, 3], [4, 2]], [[1, 1], [3, 4]]], [[[2, 4], [4, 3]], [[3, 2], [1, 1]]], [[[3, 1], [2, 2]], [[4, 3], [1, 4]]], [[[3, 2], [1, 1]], [[2, 4], [4, 3]]], [[[3, 2], [2, 1]], [[1, 4], [4, 3]]], [[[3, 3], [1, 4]], [[4, 1], [2, 2]]], [[[3, 4], [1, 1]], [[4, 2], [2, 3]]], [[[3, 4], [1, 2]], [[4, 1], [2, 3]]], [[[3, 4], [1, 3]], [[4, 1], [2, 2]]], [[[4, 1], [2, 2]], [[3, 3], [1, 4]]], [[[4, 1], [2, 2]], [[3, 4], [1, 3]]], [[[4, 1], [2, 3]], [[3, 4], [1, 2]]], [[[4, 2], [2, 3]], [[3, 4], [1, 1]]], [[[4, 3], [1, 4]], [[2, 1], [3, 2]]], [[[4, 3], [1, 4]], [[3, 1], [2, 2]]], [[[4, 3], [2, 4]], [[1, 1], [3, 2]]]}
#BeatNE(2,3) took too long

#4
EstBeatNE := proc(a,b,K,M) local onenash,bettertot,betterboth,i,G:
	onenash := 0:
	bettertot := 0:
	betterboth := 0:
	for i from 1 to M do
		G := RG(a,b,K):
		if nops(PureNashEqui(G)) = 1 then
			onenash := onenash + 1:
			if not member(PureNashEqui(G)[1],BestTot(G)[1]) then
				bettertot := bettertot + 1:
			fi:
			if nops(BetterForBoth(G,PureNashEqui(G)[1][1],PureNashEqui(G)[1][2])) > 0 then
				betterboth := betterboth + 1:
			fi:
		fi:
	od:
	return evalf([onenash/M,bettertot/M,betterboth/M]):
end:
#Resuts from EstBeatNE(10,10,1000,1000)
#[0.4320000000, 0.2230000000, 0.09800000000]
#[0.3930000000, 0.1910000000, 0.1000000000]
#[0.4150000000, 0.2100000000, 0.1030000000]
#[0.4170000000, 0.2250000000, 0.1090000000]
#[0.4240000000, 0.2160000000, 0.09700000000]

#5
u1 := q1*(1-q1-q2)^(d1);
u2 := q2*(1-q1-q2)^(d2);
diff(u1,q1);
q1opt := solve(%=0,q1);
diff(u2,q2);
q2opt := solve(%=0,q2);
solve({q1=q1opt,q2=q2opt},{q1,q2});
#NE is q1 = d2/(d1*d2+d1+d2), q2 = d1/(d1*d2+d1+d2)