#!/usr/local/bin/maple
# -*- maplev -*-

# Nathaniel Shar
# HW 19
# Experimental Mathematics

# It is okay to link to this assignment on the course webpage.

#############
# Problem 1 #
#############

# If A cooperates, she gets payoff Ry + S(1-y). If she defects, she
# gets payoff Ty + P(1-y). Cooperating is at least as good as
# defecting iff Ry + S(1-y) >= Ty + P(1-y) [1]. Solving for y, we see
# that if y(R-T-S+P) >= (P-S), then A can "rationally"
# cooperate. Similarly, if y(R-T-S+P) <= (P-S), then A can
# "rationally" defect.

# Of course the exact same formula (with x replacing y) is correct for
# Player B.

#############
# Problem 2 #
#############

# Are we supposed to assume that both players use the same payoff matrix?
# It's not completely clear from the problem description.

# If we do not assume that, we have to introduce new names
# Ra, Sa, Ta, Pa and Rb, Sb, Tb, Pb for the various payoffs for player
# A and B, respectively.

# Now, there are several cases. First, there is the case when
# cooperating is a (non-strictly) dominant strategy for
# A. This happens when Ra >= Ta and Sa >= Pa. In that case, a Nash
# equilibrium occurs at [1,1] if Rb >= Tb, and at [1,0] if Rb <= Tb.

# Similarly, if defecting is a dominant strategy for A (which happens
# when Ra <= Ta and Sa <= Pa), then a Nash equilibrium occurs at [0,1]
# if Rb >= Tb, and at [0,0] if Rb <= Tb.

# Similar results occur when B has a dominant strategy.

# If neither player has a dominant strategy then (R-T-S+P) can't be
# zero so the nash equilibrium occurs at the mixed strategy
# [(Pb-Sb)/(Rb-Tb-Sb+Pb), (Pa-Sa)/(Ra-Ta-Sa+Pa)].

#############
# Problem 3 #
#############

#From c19.txt:

#BRA(A,y): The Best Response for A to B's decision to
#cooperate with prob. y
BRA:=proc(A,y) local x, PayOff,v:

v:=[A[1][1]*y+A[1][2]*(1-y), A[2][1]*y+A[2][2]*(1-y)]:

PayOff:=v[1]*x+v[2]*(1-x):


if coeff(PayOff,x,1)>0 then
 RETURN(1, subs(x=1,PayOff)):
elif coeff(PayOff,x,1)<0 then
 RETURN(0, subs(x=0,PayOff)):

 else
   RETURN(x,PayOff):
fi:

end:


#BRB(B,x): The Best Response for B to A's decision to
#cooperate with prob. x
BRB:=proc(B,x) local y, PayOff,v:

v:=[B[1][1]*x+B[2][1]*(1-x), B[1][2]*x+B[2][2]*(1-x)]:

PayOff:=v[1]*y+v[2]*(1-y):


if coeff(PayOff,y,1)>0 then
 RETURN(1, subs(y=1,PayOff)):
elif coeff(PayOff,y,1)<0 then
 RETURN(0, subs(y=0,PayOff)):

 else
   RETURN(y,PayOff):
fi:

end:

# It seems to be right.