#Maple Code for Lecture 13 of Dr. Z.'s Experimental Mathematics (Game Theory) class
Help13:=proc(): print(`RSG(n,K), SV(f,n), SVp(f,n), SVpF(f,n) `): end:

with(combinat):

#RSG(n,K): A random SUPER-ADDITIVE n-person cooperative game
RSG:=proc(n,K) :FM(RG(n,K),n):end:


##SV(f,n): Inputs a Coalition function a function defined on powerset(n) and outputs the list
#of length n, call it L, such that L[i] is the Shapley value
SV:=proc(f,n)  local V,i,g,PN,S,val:
#LET V[i]: The current Shapley value of Player i 

  for i from 1 to n do
    V[i]:=0:
 od:
 g:=BM(f,n):

#g[S]: means: corresponds to the game where the members of S are symmetric to each other and all the
#other players are NULL players, and they get together g[S]
  PN:=powerset(n):
 for S in PN do
   val:=g[S]:
 for i in S do
   V[i]:=V[i]+val/nops(S):
 od:
od:
[seq(V[i],i=1..n)]:
end:

#SVp(f,n): Shapley value with the permutation approach
SVp:=proc(f,n) local S,V,i,pi:

for i from 1 to n do
    V[i]:=0:
 od:

S:=permute(n):

for pi in S do
 for i from 1 to n do
  V[pi[i]]:=V[pi[i]]+ f[{op(1..i,pi)}]-f[{op(1..i-1,pi)}]:
 od:
od:
[seq(V[i]/n!,i=1..n)]:
end:



#Added after class
#SVi(f,n,i): The Shapley value of Player i in the n-person game given by f
#This uses Eq. (13) (and (12), note the typo: n should be n!, i.e. gamma_n(s)=(s-1)!*(n-s)!/n!, where s=|S|)
SVi:=proc(f,n,i) local PS,S:
PS:=powerset(n) minus {{}}:
add((nops(S)-1)!*(n-nops(S))!*(f[S]-f[S minus {i}]),S in PS)/n!:
end:

#SVpF(f,n): Yet another version of SV(f,n), using the above SVi
SVpF:=proc(f,n) local i: [seq(SVi(f,n,i),i=1..n)]:end:


#Stuff from Lecture 12: 
#C12.txt

Help12:=proc(): print(`Ex1(), RG(n,K), FM(f,n), BM(f,n) `): end:

Ex1:=proc():table([({3})=18,({2})=12,({})=0,({1, 3})=60,({2, 3})=90,({1, 2})=30,({1})=6,({1, 2, 3})=120]):end:


with(combinat):

#RG(n,K): inputs an and K outputs a random "n-person game" with entries in [0,K]
RG:=proc(n,K) local ra,PS,S,f:
ra:=rand(0..K):
f[{}]:=0:
PS:=powerset(n):

for S in PS minus {{}} do
f[S]:=ra():
od:
op(f):
end:

#FM(f,n): inputs a function from 2^{1,...n} to pos. real numbres
#we want g(T)= Sum of g(S) over all subsets S of T
FM:=proc(f,n) local S,PS,PN,g,S1:
PN:=powerset(n):
for S in PN do
PS:=powerset(S):
g[S]:=add(f[S1], S1 in PS):
od:
op(g):
end:

#BM(f,n): inputs a function from 2^{1,...n} to pos. real numbres
#we want g(T)= Sum of (-1)^(|T|-|S|)*g(S) over all subsets S of T
BM:=proc(f,n) local S,PS,PN,g,S1:
PN:=powerset(n):
for S in PN do
PS:=powerset(S):
g[S]:=add( (-1)^(nops(S)-nops(S1))*f[S1], S1 in PS):
od:
op(g):
end: