#OK to post homework
#George Spahn, 2-13-2022, Assignment 7

#3 

# Case 1: The two pure equilibria involve one player making the same move
#   let's say the player is A, and the other is B.
#   It must be the case that B is indifferent between its two choices
#    in the two equilibria, given A's choice.
#   Therefore any probility for player B between the two choices will also 
#    be a equilibrium. In this case there are infinitely many mixed
#    equilibria

# Case 2: To get from one equilibrium to the other, both players must switch
#         strategy
#   Suppose player 1 plays choice 1 with probability p1. Then player 2 gets some
#    expected utility u1 from choice 1 and some expected utility u2 from choice 2.
#   Either one of these is greater, or they are the same. Without loss of generality
#      u1 >= u2 when p1 = 0
#      u2 >= u1 when p1 = 1
#   Both of these functions, u1 and u2 vary continuously with p1.
#   Therefore, there exists a p1 such that they are equal, p1*.

#   We can do the same process to find a p2*.
#   For this p1*, p2*, both players are playing a best response to the other,
#   so it is a mixed Nash Equilibrium.




# 5

PayOffGG := proc(G,p1,p2)

	add(add(  p1[i]*p2[j]*G[i][j] ,j=1..nops(G[i])), i=1..nops(G)):
	
end: