#OK to post
#Rebecca Embar, 1-28-2022, Homework 3

read `hw2RebeccaEmbar.txt`:
read `C3.txt`:

#3
#(a)
G := RandDisGame(4,4):
matrix(G);
#Output matrix
#[[13,8] [14,6]  [11,4]  [6,5]  ]
#[                              ]
#[[9,10] [1,11]  [16,3]  [7,9]  ]
#[                              ]
#[[3,2]  [2,7]   [12,15] [5,1]  ]
#[                              ]
#[[4,13] [10,16] [8,12]  [15,14]]
#(i)
#[[13,8*] [14,6]   [11,4]   [6,5]  ]
#[                                 ]
#[[9,10]  [1,11*]  [16,3]   [7,9]  ]
#[                                 ]
#[[3,2]   [2,7]    [12,15*] [5,1]  ]
#[                                 ]
#[[4,13]  [10,16*] [8,12]   [15,14]]
#If Player Row goes first, Player Row will pick strategy 1
#The game is then reduced to the one player game [8,6,4,5]
#Player Column will pick strategy 1, and the outcome will be [13,8]
#(ii)
#[[13*,8] [14*,6] [11,4]  [6,5]   ]
#[                                ]
#[[9,10]  [1,11]  [16*,3] [7,9]   ]
#[                                ]
#[[3,2]   [2,7]   [12,15] [5,1]   ]
#[                                ]
#[[4,13]  [10,16] [8,12]  [15*,14]]
#If Player Column goes first, Player Column will pick strategy 4
#The game is then reduced to the one player game [6,7,5,15]
#Player Row will pick strategy 4, and the outcome will be [15,14]
#(iii)
#[[13*,8*] [14*,6]  [11,4]   [6,5]   ]
#[                                   ]
#[[9,10]   [1,11*]  [16*,3]  [7,9]   ]
#[                                   ]
#[[3,2]    [2,7]    [12,15*] [5,1]   ]
#[                                   ]
#[[4,13]   [10,16*] [8,12]   [15*,14]]
#The set of pure Nash equilibria is {[1,1]}

#(b)
G:= RandDisGame(5,7):
matrix(G);
#Output matrix
#[[11,15] [6,35] [17,23] [34,26] [18,20] [21,17] [5,29] ]
#[[23,9]  [16,2] [28,25] [22,3]  [12,21] [10,12] [27,34]]
#[[24,5]  [1,10] [19,16] [20,7]  [26,8]  [15,24] [2,28] ]
#[[30,30] [7,6]  [13,33] [14,19] [8,1]   [35,27] [32,32]]
#[[3,18]  [9,13] [29,22] [31,11] [25,31] [4,14]  [33,4] ]
#(i)
#[[11,15] [6,35*] [17,23]  [34,26] [18,20]  [21,17] [5,29]  ]
#[[23,9]  [16,2]  [28,25]  [22,3]  [12,21]  [10,12] [27,34*]]
#[[24,5]  [1,10]  [19,16]  [20,7]  [26,8]   [15,24] [2,28*] ]
#[[30,30] [7,6]   [13,33*] [14,19] [8,1]    [35,27] [32,32] ]
#[[3,18]  [9,13]  [29,22]  [31,11] [25,31*] [4,14]  [33,4]  ]
#If Player Row goes first, Player Row will pick strategy 2
#The game is then reduced to the one player game [9,2,25,3,21,12,34]
#Player Column will pick strategy 7, and the outcome will be [27,34]
#(ii)
#[[11,15]  [6,35]  [17,23]  [34*,26] [18,20]  [21,17]  [5,29] ]
#[[23,9]   [16*,2] [28,25]  [22,3]   [12,21]  [10,12]  [27,34]]
#[[24,5]   [1,10]  [19,16]  [20,7]   [26*,8]  [15,24]  [2,28] ]
#[[30*,30] [7,6]   [13,33]  [14,19]  [8,1]    [35*,27] [32,32]]
#[[3,18]   [9,13]  [29*,22] [31,11]  [25,31]  [4,14]   [33*,4]]
#If Player Column goes first, Player Column will pick strategy 1
#The game is then reduced to the one player game [11,23,24,30,3]
#Player Row will pick strategy 4, and the outcome will be [30,30]
#(iii)
#[[11,15]  [6,35*] [17,23]  [34*,26] [18,20]  [21,17]  [5,29]  ]
#[[23,9]   [16*,2] [28,25]  [22,3]   [12,21]  [10,12]  [27,34*]]
#[[24,5]   [1,10]  [19,16]  [20,7]   [26*,8]  [15,24]  [2,28*] ]
#[[30*,30] [7,6]   [13,33*] [14,19]  [8,1]    [35*,27] [32,32] ]
#[[3,18]   [9,13]  [29*,22] [31,11]  [25,31*] [4,14]   [33*,4] ]
#There are no pure Nash equilibria

#4
with(combinat):
AllGames := proc(a,b) local payoff,payoffchoices,games,c,p1,p2,i,j:
	payoff := permute(a*b):
	payoffchoices := permute(nops(payoff),2):
	games := {}:
	for c from 1 to nops(payoffchoices) do
		p1 := payoff[payoffchoices[c,1]]:
		p2 := payoff[payoffchoices[c,2]]:
		games := games union {[seq([seq([p1[b*i+j],p2[b*i+j]],j=1..b)],i=0..a-1)]}:
	od:
	return(games);
end:

#5
FullStat := proc(a,b) local payoff,payoffchoices,l,c,p1,p2,i,j,game,nash,rc,cr:
	payoff := permute(a*b):
	payoffchoices := permute(nops(payoff),2):
	l := [0,0,0,0,0,0]:
	for c from 1 to nops(payoffchoices) do
		p1 := payoff[payoffchoices[c,1]]:
		p2 := payoff[payoffchoices[c,2]]:
		game := [seq([seq([p1[b*i+j],p2[b*i+j]],j=1..b)],i=0..a-1)]:

		nash := PureNashEqui(game):
		rc := DynRC(game):
		cr := DynRC(game):
		
		#Number of nash equilibria
		if nops(nash) = 0 then l[1] := l[1]+1
		elif nops(nash) = 1 then l[2] := l[2]+1
		elif nops(nash) = 2 then l[3] := l[3]+1
		fi:
		
		#Row player better off playing second
		if cr[2,1] > rc[2,1] then l[4] := l[4]+1 fi:
		
		#Column player better off playing second
		if rc[2,2] > cr[2,2] then l[5] := l[5]+1 fi:
		
		#DynRC(G) = DynCR(G)
		if rc = cr then l[6] := l[6]+1 fi:
	od:
	return(l);
end:
#(i) [72,416,64,0,0,552]
#(ii) [86400,345240,86040,0,0,517680]

#6
SampleStat := proc(a,b,K) local l,c,game,nash,rc,cr,i:
	l := [0,0,0,0,0,0]:
	for c from 1 to K do
		game := RandDisGame(a,b):
		
		nash := PureNashEqui(game):
		rc := DynRC(game):
		cr := DynRC(game):
				
		#Number of nash equilibria
		if nops(nash) = 0 then l[1] := l[1]+1
		elif nops(nash) = 1 then l[2] := l[2]+1
		elif nops(nash) = 2 then l[3] := l[3]+1
		fi:
		
		#Row player better off playing second
		if cr[2,1] > rc[2,1] then l[4] := l[4]+1 fi:
		
		#Column player better off playing second
		if rc[2,2] > cr[2,2] then l[5] := l[5]+1 fi:
		
		#DynRC(G) = DynCR(G)
		if rc = cr then l[6] := l[6]+1 fi:
	od:
	
	for i from 1 to 6 do
		l[i] := l[i]/K:
	od:
	
	return evalf(l):
end:
#[0.3327,0.4158,0.1918,0,0,1]
#[0.3364,0.4104,0.1928,0,0,1]
#[0.3283,0.4089,0.2021,0,0,1]
#[0.3365,0.4053,0.1986,0,0,1]
#[0.3225,0.4201,0.1992,0,0,1]