#C11.txt: Bearoff with Two players, Feb. 28, 2013

Help:=proc(): print(`LegalMoves(L,i), FW(L1,L2,r), fW(L1,L2,r,i), Exceptions3(N,j) `): end:

#LegalMoves(L,i): inputs a list of length nops(L)
#and a pos. integer i telling you that the option
#to move any counter i pieces towards the end
#without waste. And if you must waste, minimize it

LegalMoves:=proc(L,i) local S, j:

if i>nops(L) then
  RETURN(FAIL):
fi:

if L=[0$nops(L)] then
 RETURN({}):
fi:
  

S:={}:

if L[i]>0 then
  S:=S union {[op(1..i-1,L),L[i]-1,op(i+1..nops(L),L)]}:
fi:

for j from i+1 to nops(L) do
  
  if L[j]>0 then
#a counter distance j away goes to distance j-i away
   
 S:=S union {[op(1..j-i-1,L) , L[j-i]+1, op(j-i+1..j-1,L),
               L[j]-1, op(j+1..nops(L),L)]}:
  fi:


od:


if S<>{} then
  RETURN(S):
fi:


for j from i-1 by -1 to 1  while L[j]=0 do od:

{ [op(1..j-1,L),L[j]-1,op(j+1..nops(L),L)] }:

end:


#fW(L1,L2,r,i): The probability of L1 winning 
#if the roll was i, followed by the optimal move
fW:=proc(L1,L2,r,i) local tom, champ,rec,mr:
option remember:

if L1=[0$nops(L1)] then
  RETURN(FAIL):
fi:

tom:=LegalMoves(L1,i):

if tom={} then
    RETURN(FAIL):
fi:

champ:=tom[1]:

rec:=1-FW(L2,champ,r):

for  mr in tom minus {tom[1]} do
 if 1-FW(L2,mr,r)>rec then
    champ:=mr:
     rec:=1-FW(L2,mr,r):
 fi:
od:

rec,champ:



end:

#FW(L1,L2,r): Probability of winning if your are about
#to roll and your position is L1 and your opponent's
#is L2 in a SINGLE r-faced fair die
FW:=proc(L1,L2,r) local  i:
option remember:

if L1=[0$nops(L1)] and L2<>[0$nops(L2)] then
 RETURN(1):
fi:

if L2=[0$nops(L2)] and L1<>[0$nops(L1)] then
 RETURN(0):
fi:
add(fW(L1,L2,r,i)[1], i=1..r)/r:

end:


Exceptions3:=proc(N,j) local i1,i2,i3,j1,j2,j3,gu,P,Pw,lu1,lu2:

gu:={}:

for i1 from 0 to N do
 for i2 from 0 to N-i1 do
  for i3 from 1 to N-i1-i2 do
   
for j1 from 0 to N do
 for j2 from 0 to N-j1 do
  for j3 from 1 to N-j1-j2 do
   

 if f([i1,i2,i3],3,j)[1]<> fW([i1,i2,i3],[j1,j2,j3],3,j)[2] then

print(`In position `,[[i1,i2,i3],[j1,j2,j3]], `the strategies conflict when the die rolled`, j):

P:=f([i1,i2,i3],3,j)[1]:

Pw:=fW([i1,i2,i3],[j1,j2,j3], 3,j)[2]:

print(`according to the minimize-expected length, the optimal move   is`):
print(P):
print(`according to the maximize the probability of winning, the optimal move   is`):
print(Pw):

lu1:=evalf(F(Pw,3)-F(P,3)):

print(`The increase in expected duration if you follow the latter is`, lu1):

lu2:=evalf(FW([j1,j2,j3],P,3)-FW([j1,j2,j3],Pw,3)):

print(`The loss of proba. of winning if you follow the former is`,lu2):

    gu:=gu union {[[i1,i2,i3],[j1,j2,j3], P,Pw,lu1,lu2]}:
 fi:


  od:
 od:
od:

od:
od:
od:

gu:

end: