#OK to post homework
#George Spahn, 2-20-2022, Assignment 9


# 4

with(combinat):
read "C9.txt":

question4 := proc ()

generalgame := [[[R,R],[S,T]],[[T,S],[P,P]]]:
tally := 0:
for p in permute({T,R,S,P}) do
	assume(p[1] > p[2], p[2] > p[3], p[3] > p[4]):
	ne := MNE2(generalgame):
	#print(ne):
	if nops(ne) = 3 then
		tally := tally + 1:
	fi:

od:

end:

# 12 out of the 24 possible orderings yield a mixed nash equilibrium.



# 5

# I used inclusion and exclusion over each strategy pair being a pure equilibrium
# Sum from 1 to 9 of strategy pair i being a nash = 5/3
# Sum from 1 to 9 choose 2 of strategy pairs i and j being a nash = 8/9
# Sum from 1 to 9 choose 3 of strategy pairs i and j and k being a nash = 2/9

# 5/3 - 8/9 + 2/9 = 1

# Therefore there are no permutations of the ordering such that there are no pure
# nash equilibrium, ie. every possible symmetric game with 3 strategies has a pure
# nash equilibrium.

# I think I can also show something stronger, that for all symmetric games, there must
# exist a pure nash equilibrium.

# Still working on a proof.